THE HOMOTOPY THEORY OF STRONG HOMOTOPY ALGEBRAS AND BIALGEBRAS

J.P.PRIDHAM

Abstract. Lada introduced strong homotopy algebras to describe the structures on a deformation retract of an algebra in topological spaces. However, there is no satisfactory general definition of a morphism of strong homotopy (s.h.) algebras. Given a monad ⊤ on a simplicial category C, we instead show how s.h. ⊤-algebras over C naturally form a Segal space. Given a distributive monad-comonad pair (⊤, ⊥), the same is true for s.h. (⊤, ⊥)-bialgebras over C; in particular this yields the homotopy theory of s.h. sheaves of s.h. rings. There are similar statements for quasi-monads and quasi-comonads. We also show how the structures arising are related to derived connections on bundles.

Introduction

Given a monad ⊤ acting on the category of topological spaces, Lada introduced (in [CLM]) the notion of a strong homotopy (s.h.) ⊤-algebra. This characterises the structures arising on deformation retracts of ⊤-algebras. Indeed, when ⊤ is an operad, there is a bar construction which realises every s.h. ⊤-algebra as such a deformation retract. The formulation of s.h. algebras does not use any special properties of topological spaces, so adapts to any simplicial category, and likewise adapts to describe s.h. coalgebras of a comonad ⊥.

Structures such as Hopf algebras or sheaves of rings cannot be described as algebras of a monad ⊤ or as coalgebras of a comonad ⊥. However, in both cases, there are both a natural monad ⊤ (governing the algebraic structure) and a comonad ⊥ (governing the coalgebraic structure), satisfying a distributivity condition. This seems to have first been described by Van Osdol in [Van] in order to develop bicohomology theory. Independent rediscoveries have appeared in [BJ] Appendix C, [PW] and [Pri2] §2. This permits the characterisation of a compatibility condition for the algebraic and coalgebraic structures.

In [Pri6], it was observed that the equations defining a s.h. ⊤-algebra can also be used to define s.h. (⊤, ⊥)-bialgebras associated to a distributive monad-comonad pair (⊤, ⊥). In particular, this gives rise to a notion of s.h. sheaves of s.h. algebras (on any site with enough points), yielding important applications in algebro-geometric deformation theory.

The first main result in this paper is Proposition 2.9, which provides a single unified framework for dealing with algebras, coalgebras and bialgebras. This then combines with Proposition 5.7 and Corollaries 5.19 and 5.25 to give three possible models for the ∞- category of s.h. algebras, coalgebras and bialgebras. These models are shown to be equivalent in Propositions 5.15 and 5.24. Finally, Theorem 6.23 shows how this ∞-category is related to the Maurer-Cartan functor featuring in [Man2] and [Hin].

These results have applications in derived deformation theory, which starts with a moduli functor from algebras to groupoids, and seeks a derived moduli functor from simplicial algebras to ∞-groupoids. By describing deformation problems in terms of bialgebraic structures, [Pri6] and [Pri4] apply the results of this paper to construct derived deformation functors, and these are being extended to (global) derived moduli functors in [Pri7]. Even where there are other possible approaches to defining derived moduli functors, strong homotopy bialgebras often provide a more concrete description, and have the crucial property that the functor is left exact, making it easier to verify Lurie’s representability theorem ([Lur]). For some examples, such as Hopf algebras, strong homotopy bialgebras provide the only known means of constructing derived deformations.

This work was supported by Trinity College, Cambridge; and by the Engineering and Physical Sciences Research Council [grant number EP/F043570/1].

A major failing of the theory of s.h. algebras is that there is no satisfactory general definition of morphisms. In [CLM], this difficulty was obviated by considering morphisms of the associated bar constructions, but this has several disadvantages. For applications in deformation theory, the main problem is that the bar construction does not respect finite limits, in general. For bialgebras, the difficulty is even more fundamental, since the bar and cobar constructions will in general be incompatible, so the s.h. structures cannot be rectified. This is essentially the phenomenon that a lax sheaf of lax simplicial rings cannot be replaced by a strict sheaf of strict simplicial rings.

Rather than seeking to define simplicial categories of s.h. algebras or bialgebras, we instead construct Segal spaces. These are a special type of simplicial space (i.e. bisimplicial set) introduced by Rezk in [Rez] as a model for homotopy theories, and Bergner showed in [Ber2] that the associated model category is Quillen-equivalent to the model category of simplicial categories, so any Segal space naturally gives rise to a simplicial category.

Our approach makes use of a generalisation of the theory of homotopy monoids expounded by Leinster in [Lei2]. We introduce a slight generalisation, the quasi-comonoid, of a homotopy comonoid, and associate a simplicial set \underline{MC}(E), the Maurer-Cartan space, to any simplicial quasi-comonoid E. When E^{0} is a group rather than a monoid, a more natural object is the Deligne space Del(E), which is the homotopy quotient of \underline{MC}(E) by E^{0}. We also develop the concept of a quasi-descent datum; put simply, a quasi-descent datum is to a quasi-comonoid as a category is to a monoid.

Proposition 2.12 then shows how a distributive monad-comonad pair on a simplicial category C naturally enriches it to a simplicial quasi-descent datum, with \underline{MC} evaluated at any object x of C giving the space of s.h. bialgebras over x. Moreover, there is a natural quasi-comonoid associated to any diagram in C, and we use this to define a simplicial space MC by mimicking the nerve construction (Definition 3.25). In Proposition 5.7 this is seen to be a Segal space.

Another source of quasi-comonoids is from cosimplicial groups, with the construction E given in Definition 6.6. In this case, the Maurer-Cartan space admits a simpler description (Proposition 6.8). If X is a simplicial set, G a simplicial group and C^{•}(X, G) the cosimplicial simplicial group of G-cochains on X, then Del(EC^{•}(X, G)) is equivalent to \underline{Hom}(X, W̄ G), where W̄ G is the classifying space of G (Proposition 6.11 and Remark 6.10). If the cosimplicial group comes from denormalising and exponentiating a differential graded Lie algebra, then the Maurer-Cartan space is equivalent to the classical Maurer-Cartan space of derived connections, given by the equation

dω + \frac{1}{2}[ω, ω] = 0,

for ω of total degree 1 (Corollary 6.27).

The structure of the paper is as follows.

In Section 1, we recall Leinster’s comonoids up to homotopy, and introduce quasicomonoids and then the category QDat of quasi-descent data. There is an adjunction (Lemma 1.15):

\xymatrix{Cat \ar[r]^{^{alg^{∗}}}_{⊥} & QDat \ar[l]^{alg}}.

In Section 2, we introduce monads and comonads, and show how a monad ⊤ on a category C naturally enriches it to a quasi-descent datum D(C, ⊤). This has the properties that alg D(C, ⊤) is the category C^{⊤} of ⊤-algebras, and that alg^{∗}C = D(C, id). There is a similar result for (⊤, ⊥)-bialgebras.

Section 3 begins the work of extending these constructions to simplicial categories. Distributive monad-comonad pairs on simplicial categories give rise to simplicial quasi-descent data, but the functor alg above destroys higher order information. We therefore begin (Definition 3.5) by defining the higher Maurer-Cartan functor \underline{MC} from simplicial quasicomonoids sQM^{∗} to simplicial sets 𝕊. Given an object x of a simplicial quasi-descent datum D, there is a simplicial quasi-comonoid D(x, x), and the vertices of \underline{MC}(D(x, x)) correspond closely to Lada’s set of strong homotopy ⊤-algebras over x, when D = D(C, ⊤) (Remark 3.13). We then extend this to a simplicial space (i.e. bisimplicial set) MC(D) (Definition 3.25), which mimics the nerve construction by developing constructions loosely corresponding to strong homotopy diagrams of strong homotopy algebras. When the underlying category is a groupoid, DEL(D) is the homotopy quotient of MC(D) by morphisms in D, and Del is similarly related to \underline{MC}.

Section 4 is primarily concerned with the study of quasi-comonoids in (Ab, ×) (abelian groups with the monoidal structure ×). These are equivalent to cosimplicial abelian groups (Lemma 4.1), and we exploit this to develop cohomology of quasi-comonoids, and relate it to homotopy groups of \underline{MC} and Del. We also study nerves BΓ of quasi-comonoids in groupoids, in order to understand fundamental groupoids of simplicial quasi-comonoids. We then introduce linear quasi-comonoids (i.e. quasi-comonoids in (Ab, ⊗)). These allow us to relate cohomology of a simplicial quasi-comonoid to homology of the underlying simplicial abelian group (Proposition 4.22). This gives a cohomological characterisation of when a cofibrant simply connected simplicial quasi-comonoid is contractible (Corollary 4.23). These results are then extended to quasi-bicomonoids, which play a crucial rôle in the construction of MC.

Although the functor alg above is poorly suited to simplicial categories, its left adjoint alg^{∗} extends naturally and preserves weak equivalences. There is a model structure on sQDat, and the purpose of Section 5 is to relate, for D fibrant, the simplicial spaces MC(D) and DEL(D) with the the simplicial space N (D) given by the derived function complexes

N (D)_{n} = Map_{sQDat}^{h}(alg^{∗}n, D),

where n is the category associated to the poset [0, n]. When D is constructed from a distributive monad-comonad pair on a simplicial category B, we think of MC(D), DEL(D) and N (D) as candidates for the simplicial space of s.h. bialgebras over B.

It turns out that MC(D) is a Segal space, while N (D) and DEL(D) are complete Segal spaces (Proposition 5.7, Corollaries 5.19 and 5.25). There are Dwyer-Kan equivalences between these Segal spaces (Propositions 5.15 and 5.24), so they are all weakly equivalent in the complete Segal space model structure from [Rez]. Moreover, Theorem 5.21 shows how MC and N can both be regarded as the derived right adjoint of alg^{∗}, even though alg^{∗} is not left Quillen.

Section 6 establishes a simpler Maurer-Cartan space construction for cosimplicial simplicial groups, yielding the equivalence Del(EC^{•}(X, G)) ≃ \underline{Hom}(X, W̄ G) described earlier, and simplifies this description further when G is a formal Lie group, via Theorem 6.23.

The appendices establish a framework for applying these results more widely still. In Appendix A, quasi-monads and quasi-comonads are introduced, motivated by the need for an analogue of the homotopy operads of [vdL]. We show what it would mean for a quasimonad to distribute over a quasi-comonad, while ensuring that these weaker structures still enhance a category to form a quasi-descent datum. In Appendix B, we relate A_{∞}-algebras to quasi-semigroups, and thus compare quasi-operads with homotopy operads.

I would like to thank Carlos Simpson for alerting me to the similarities between the SDCs of [Pri1] and Leinster’s homotopy monoids ([Lei2]). I would also like to thank the anonymous referee heartily for diligently identifying many errors and omissions in the original manuscript.

Contents

Introduction 1 1. Monoidal structures up to homotopy 4 1.1. Quasi-comonoids 5 1.2. Quasi-descent data 6 1.3. Adjoint functors 7 1.4. Quasi-bicomonoids and diagonals 8 1.5. Nerves 8 2. Algebras, coalgebras and bialgebras 11 2.1. Algebras and coalgebras 11 2.2. Quasi-descent data from monads 12 2.3. Bialgebras 13 3. Simplicial structures 14 3.1. Simplicial quasi-comonoids 14 3.2. Simplicial categories 17 3.3. Simplicial quasi-descent data 18 3.4. Simplicial bicomonoids 19 3.5. Nerves 20 4. Abelian groups and cohomology 23 4.1. Cosimplicial abelian groups 23 4.2. Cohomology 28 4.3. Groupoids 28 4.4. Linear quasi-comonoids 29 4.5. Diagonals 32 5. Mapping spaces 34 5.1. MC 34 5.2. N 37 5.3. DEL 40 6. Maurer-Cartan and classifying spaces 41 6.1. Groups 41 6.2. Lie algebras 45 Appendix A. Quasi-monads, quasi-comonads and quasi-distributivity 50 A.1. Quasi-monads and quasi-comonads 50 A.2. Distributivity for quasi-monads and quasi-comonads 51 Appendix B. A_{∞}-algebras and homotopy operads 52 B.1. Homotopy semigroups and semicogroups 52 B.2. Homotopy semigroups in abelian categories 53 B.3. DG co-operads 56 Bibliography 57 References 57

1. Monoidal structures up to homotopy

In this section, we will introduce various structures which will provide the framework for the rest of the paper. The main concepts are those of a quasi-comonoids and quasidescent data, which will provide the intermediate step between monads and comonads one one hand, and strong homotopy structures on the other.

1.1. Quasi-comonoids.

Definition 1.1. Define Δ_{∗∗} to be the subcategory of the ordinal number category Δ containing only those non-decreasing (i.e. f(i + 1) ≥ f(i)) morphisms f : m → n with f(0) = 0, f(m) = n. We define a monoidal structure on this category by setting m ⊗ n = m + n, with

(f ⊗ g)(i) = \begin{cases}f(i) & i ≤ m \\ g(i − m) + p & i ≥ m, \end{cases}

for f : m → p, g : n → q.

Remark 1.2. There is an isomorphism Δ_{∗∗}^{opp} ≅ Δ_{0}, the category of finite sets (i.e. Δ ⊔ ∅), given by n ↦ n − 1, where −1 := ∅, with the coboundary morphisms ∂^{i} mapping to σ_{i−1}, and the coface morphisms σ^{i} mapping to ∂_{i}.

Remark 1.3. Given a category C, a functor X : Δ_{∗∗} → C consists of objects X^{n} ∈ C, with all of the operations ∂^{i}, σ^{i} of a cosimplicial complex except ∂^{0}, ∂^{n+1} : X^{n} → X^{n+1}.

Definition 1.4. Given a monoidal category C, define a quasi-comonoid X in C to be a lax monoidal functor X : Δ_{∗∗} → C. This means that we have maps

ζ^{mn} : X^{m} ⊗ X^{n} → X^{m+n}, ζ^{0} : 1 → X^{0},

satisfying naturality and coherence, where 1 is the unit in the category. If C is a model category, say that X is a homotopy comonoid whenever the maps ζ^{mn}, ζ^{0} are all weak equivalences. This is equivalent to the definition in [Lei2], via the comparison of Remark 1.2.

Define a quasi-monoid in C to be a quasi-comonoid in C^{opp}. We let QM^{∗}(C) denote the category of quasi-comonoids in C.

Lemma 1.5. Giving a quasi-comonoid X in C is equivalent to giving objects X^{n} ∈ C for n ∈ ℕ_{0}, together with morphisms

\begin{matrix}∂^{i} : X^{n} → X^{n+1} & 1 ≤ i ≤ n \\ σ^{i} : X^{n} → X^{n−1} & 0 ≤ i < n, \end{matrix}

an associative product ζ^{mn} : X^{m} ⊗ X^{n} → X^{m+n}, with identity ζ^{0} : 1 → X^{0}, where 1 is the unit in the category, satisfying:

(1) ∂^{j}∂^{i} = ∂^{i}∂^{j−1} i < j.

(2) σ^{j}σ^{i} = σ^{i}σ^{j+1} i ≤ j.

(3) σ^{j}∂^{i} = \begin{cases}∂^{i}σ^{j−1} & i < j \\ id & i = j, i = j + 1 \\ ∂^{i−1}σ^{j} & i > j + 1 \end{cases} .

(4) ζ^{m+1,n}(∂^{i} ⊗ id) = ∂^{i}ζ^{mn}.

(5) ζ^{m,n+1}(id ⊗ ∂^{i}) = ∂^{i+m}ζ^{mn}.

(6) ζ^{m−1,n}(σ^{i} ⊗ id) = σ^{i}ζ^{mn}.

(7) ζ^{m,n−1}(id ⊗ σ^{i}) = σ^{i+m}ζ^{mn}.

Proof. This is a straightforward consequence of Remark 1.3, together with an analysis of the interaction in Δ_{∗∗} of the monoidal structure and the morphisms σ^{i}, ∂^{i}. □

Remark 1.6. When the maps ζ^{mn}, ζ^{0} are all isomorphisms, this becomes equivalent to the definition of a comonoid C. The correspondence is given by setting X^{n} := C^{⊗n}, and letting ∂^{1} : C → C ⊗ C be the coproduct.

In order to simplify the notation, we will write x ∗ y instead of ζ^{m,n}(x, y) from now on.

1.1.1. Maurer-Cartan. Observe the category of comonoids in (Set, ×) is equivalent to Set itself, since comultiplication Δ : X → X × X is necessarily the diagonal embedding. This gives a functor ι : Set → QM^{∗}(Set) from sets to quasi-comonoids.

Definition 1.7. Define the functor MC : QM^{∗}(Set) → Set by

MC(E) := Hom(ι•, E),

where • is the one-point set. Explicitly,

MC(E) = {ω ∈ E^{1} : σ^{0}ω = 1, ∂^{1}ω = ω ∗ ω}.

The reason for notation is that any cosimplicial unital ring R has a quasi-comonoid structure, with ∗ given by the Alexander-Whitney cup product ∪, and then

MC(R) = {1_{1} + α ∈ R^{1} : σ^{0}α = 0 1_{2} + ∂^{1}α = 1_{2} + 1_{1} ∪ α + α ∪ 1_{1} + α ∪ α}
= 1 + {α ∈ N^{1}R : ∂^{1}α = ∂^{2}α + ∂^{0}α + α ∪ α}
= 1 + {α ∈ N^{1}R : dα + α ∪ α = 0},

where NR is the cosimplicial normalisation of R, with differential d. Thus MC(R) is just the classical set of Maurer-Cartan forms of the differential graded algebra NR, which parametrises flat connections on a vector bundle.

1.2. Quasi-descent data. We will now introduce quasi-descent data, which will form the bridge between categories equipped with monads and/or comonads, and Segal spaces. §2 will show how monads and comonads give rise to quasi-descent data, while much of the remainder of the paper relates quasi-descent data to Segal spaces.

Definition 1.8. Given a monoidal category C and a set O, let a C-valued descent datum D = (C, G) on objects O consist of:

(1) objects G(a, b) ∈ C for each pair a, b ∈ O;

(2) compatible systems G(a, b) ⊗ G(b, c) ≅ G(a, c) and G(a, a) ≅ 1 of transition isomorphisms, for all a, b, c ∈ O (the cocycle condition);

(3) comonoids C(a) ∈ C for each a ∈ O;

(4) isomorphisms C(a) ⊗ G(a, b) ≅ G(a, b) ⊗ C(b) for all a, b ∈ O, compatible with the comonoidal structures and transition isomorphisms.

Note that these conditions imply that G(b, a)⊗C(a)⊗G(a, b) is isomorphic as a comonoid to C(b).

The reason for this terminology is that for an open cover {U_{a}}_{a∈O} of topological spaces X, a descent datum consists of sheaves C_{a} on U_{a}, with isomorphisms g_{ab} : C_{b}|_{U_{a}∩U_{b}} → C_{a}|_{U_{a}∩U_{b}}, with g_{ab} ∘ g_{bc} = g_{ac} : C_{c}|_{U_{a}∩U_{b}∩U_{c}} → C_{a}|_{U_{a}∩U_{b}∩U_{c}} (and g_{aa} necessarily the identity).

Definition 1.9. Let Dat_{O}(C) be the category of C-valued descent data on objects O, and let Dat(C) be the category of pairs (O, D), for O a set and D = (C, G) a descent datum on objects O. Write Dat := Dat(Set, ×), and similarly for Dat_{O}. Note that there is a functor Dat(C) → Cat(C) to C-enriched categories, sending (O, D) to the category with objects O and morphisms G(x, y).

Definition 1.10. Given a monoidal category C and a set O, let a C-valued quasi-descent datum on objects O consist of:

(1) objects X(a, b) ∈ C^{Δ_{∗∗}} for all a, b ∈ O; (2) morphisms X(a, b)^{m} ⊗ X(b, c)^{n} ⟶^{∗} X(a, c)^{m+n} making the following diagram com-

mute for all a, b, c ∈ O

Δ_{∗∗} × Δ_{∗∗} ⟶^{X(a,b)⊗X(b,c)} C
×↓  ↓^{∗}
Δ_{∗∗}  ⟶^{X(a,c)} C.

(3) morphisms 1 → X(a, a)^{0} for all a ∈ O, acting as the identity for the multiplication ∗.

Remarks 1.11. A C-valued (quasi-)descent datum on the set with one object is just a (quasi-)comonoid in C. Given a C-valued descent datum (C, G) on objects O, we may define a quasi-descent datum X by X(a, b)^{n} := C(a)^{⊗n} ⊗ G(a, b) ≅ G(a, b) ⊗ C(b)^{⊗n} via the transition maps, with the maps ∂^{i} given by the coproducts on C.

If the category C contains all finite coproducts, then we may define a monoidal structure on C^{Δ_{∗∗}} by setting

(X ⊗ Y)^{n} := ∐_{a+b=n} X^{a} ⊗ Y^{b},

with operations given by

∂^{i}(x ⊗ y) = \begin{cases}(∂^{i}x) ⊗ y & i ≤ a \\ x ⊗ (∂^{i−a}y) & i > a; \end{cases}
σ^{i}(x ⊗ y) = \begin{cases}(σ^{i}x) ⊗ y & i < a \\ x ⊗ (σ^{i−a}y) & i ≥ a. \end{cases}

Then a quasi-descent datum on objects O is just a C^{Δ_{∗∗}}-enriched category with objects O.

Definition 1.12. Let QDat(C) be the category of C-valued quasi-descent data, i.e. of pairs (O, X) for O a set and X a quasi-descent datum on objects O. This admits a functor to the category of C-enriched categories by taking the Hom-space underlying X(a, b) to be X(a, b)^{0}. Let QDat := QDat(Set, ×) and sQDat := QDat(𝕊, ×).

Let QDpd (resp. sQDpd) be the full subcategory of QDat (resp. sQDat) consisting of objects whose underlying categories (resp. simplicial categories) are groupoids (resp. simplicial groupoids). In other words, all elements of X(a, b)^{0} must be invertible under ∗.

1.3. Adjoint functors. From now on, we will systematically make use of the identification in Remarks 1.11 of quasi-descent data with C^{Δ_{∗∗}}-enriched categories.

We have seen that there is a functor (−)^{0} : QDat → Cat given by sending D to the category D^{0} with objects Ob D^{0} := Ob D and morphisms D^{0}(a, b) := D(a, b)^{0}.

Lemma 1.13. The functor (−)^{0} has a right adjoint alg^{∗}, given by Ob alg^{∗}C = Ob C, with

(alg^{∗}C)(a, b)^{n} = C(a, b)

for all n, with multiplication as in C, and all operations ∂^{i}, σ^{i} acting trivially.

Proof. Given a functor F : D^{0} → C, the associated morphism D → alg^{∗}C in QDat is given by F on objects, with morphisms F ∘ (σ^{0})^{n} : D(a, b)^{n} → C(a, b) = (alg^{∗}C)(a, b)^{n}. That all morphisms D → alg^{∗}C arise in this way follows because σ^{0} acts trivially on (alg^{∗}C)(a, b). □

Definition 1.14. Define alg : QDat → Cat as follows. The objects of alg (D) are pairs (D, ω), for D ∈ Ob D and ω ∈ MC(D(D, D)), for MC as in Definition 1.7. Morphisms from (D, ω) to (D′, ω′) are given by f ∈ D(D, D′)^{0} such that

f ∗ ω = ω′ ∗ f ∈ D(D, D′)^{1}.

Lemma 1.15. The functor alg^{∗} is left adjoint to alg .

Proof. Given a functor F : C → alg (D), construct the corresponding morphism G : alg^{∗}C → D as follows. For a ∈ Ob C, set G(a) ∈ Ob D to be the object underlying F (a), and then define G : alg^{∗}C(a, b)^{n} → D(Ga, Gb)^{n} by G(x) := ω_{F (b)}^{n} ∗ F (x) = F (x) ∗ ω_{F (a)}^{n}, where F (a) = (G(a), ω_{F (a)}). Conversely, any morphism G : alg^{∗}C → D is determined by alg^{∗}C^{0} → D^{0}, together with the elements G(id_{a}) ∈ D(a, a)^{1}. □

Remark 1.16. Note that the functor alg^{∗} is fully faithful, so (alg^{∗}C)^{0} ≃ C ≃ alg alg^{∗}C.

Lemma 1.17. The functor (−)^{0} : QDat → Cat has a left adjoint.

Proof. The left adjoint is the functor (id/∅) given by Ob (id/∅)(C) = Ob C, with morphisms

(id/∅)(C)(a, b)^{n} = \begin{cases}C(a, b) & n = 0 \\ ∅ & n > 0. \end{cases}

□

1.4. Quasi-bicomonoids and diagonals.

Definition 1.18. Given a monoidal category C, let the category QMM^{∗}(C) of quasibicomonoids consist of lax monoidal functors

X : Δ_{∗∗} × Δ_{∗∗} → C.

Definition 1.19. Define a functor diag : QMM^{∗}(C) → QM^{∗}(C) via the diagonal functor Δ_{∗∗} → Δ_{∗∗} × Δ_{∗∗}. Explicitly, (diag E)^{n} = E^{nn}, with the same product and identity as E, and operations ∂^{i} = ∂_{h}^{i}∂_{v}^{i} and σ^{i} = σ_{h}^{i}σ_{v}^{i}.

The functor diag preserves all limits, so by the Special Adjoint Functor Theorem ([Mac] Theorem V.8.2), it will have a left adjoint diag^{∗} for all the categories C which we will encounter, since they all satisfy the solution set condition.

Definition 1.20. The category of quasi-comonoids in Set×Set is isomorphic to Set×Set, with comultiplication necessarily given by the diagonal (X, Y) → (X×X, Y ×Y). Inclusion of comonoids in quasi-comonoids then gives a functor ι : Set × Set → QMM^{∗}(Set), with ι(X, Y)^{m,n} = X^{m} × Y^{n}

Lemma 1.21. For E^{•,•} ∈ QMM^{∗}(Set),

MC(diag E) ≅ {(α, β) ∈ MC(E^{•,0}) × MC(E^{0,•}) : α ∗ β = β ∗ α ∈ E^{11}}

Proof. Given ω ∈ MC(diag E), set α := σ_{v}^{0}ω, β := σ_{h}^{0}ω. Since σ^{0}ω = 1, we have σ_{h}^{0}α = σ_{v}^{0}β = 1. Applying the operations (σ_{v}^{0})^{2}, σ_{h}^{1}σ_{v}^{0}, σ_{h}^{1}σ_{v}^{0}, (σ_{h}^{0})^{2} (respectively) to the equation ω ∗ ω = ∂^{1}ω gives the equations

α ∗ α = ∂_{h}^{1}α, α ∗ β = ω, β ∗ α = ω, β ∗ β = ∂_{v}^{1}β.

This shows that the function ω ↦ (α, β) is well-defined on the sets above, and has inverse (α, β) ↦ α ∗ β. □

Corollary 1.22. diag^{∗}ι• = ι(•, •).

Proof. Given a morphism f : ι(•, •) → E in QMM^{∗}(Set), let α, β be the images of the unique elements of ι(•, •)^{1,0}, ι(•, •)^{0,1}. These generate ι(•, •), subject to the conditions of Lemma 1.21. □

1.5. Nerves. We will now see how to associate quasi-comonoids to small diagrams. In Remark 3.27, this will give rise to the notion of strong homotopy diagrams.

1.5.1. Categories. Given a set O, we will write Cat_{O} for the category of categories with object set O.

Definition 1.23. Given K ∈ 𝕊, define P_{K} : Cat_{K_{0}} → QM^{∗}(Set) by

P_{K}(C)^{a} = ∏_{x∈K_{a}} C((∂_{0})^{a}x, (∂_{1})^{a}x),

with operations

∂^{i}(e)(x) := e(∂_{i}x)
σ^{j}(e) := e(σ_{j}y),
(f ∗ e)(z) := f((∂_{a+1})^{b}z) ∘ e((∂_{0})^{a}z),

for f ∈ P_{K}(C)^{a}, e ∈ P_{K}(C)^{b}.

Let P_{n} := P_{Δ^{n}}.

Let Cat_{n} be the category of categories on (the n + 1) objects [0, n].

Lemma 1.24. The functor P_{n} : Cat_{n} → QM^{∗}(Set) has a left adjoint P_{n}^{∗}, given by

(P_{n}^{∗}E)(i, j) = \begin{cases}E^{j−i} & j ≥ i \\ ∅ & j < i, \end{cases}

with multiplication given by ∗, and identities 1 ∈ E^{0}.

Proof. Define ∇^{n} ∈ Set^{Δ_{∗∗}^{opp}} by (∇^{n})_{r} := Hom_{Δ_{∗∗}}(r, n). If U : 𝕊 → Set^{Δ_{∗∗}^{opp}} is the forgetful functor, then define b : UK → K_{0} × K_{0} by x ↦ ((∂_{1})^{n}x, (∂_{0})^{n}x), for x ∈ K_{n}. Then

UΔ^{n} = ∐_{(i,j)∈Δ_{0}^{n}} b^{−1}(i, j) ≅ ∐_{0≤i≤j≤n} ∇^{j−i}.

For C ∈ Cat_{n}, P_{n}C decomposes in Set^{Δ_{∗∗}} as

P_{n}E = ∏_{0≤i≤j≤n} C(i, j)^{∇^{j−i}},

where, for K ∈ Set^{Δ_{∗∗}^{opp}} and a set S, we define S^{K} ∈ Set^{Δ_{∗∗}} by (S^{K})_{r} = S^{K_{r}}. For E ∈ QM^{∗}(Set) and C ∈ Cat_{n}, this implies that

Hom_{Set}_{Δ∗∗} (E, P_{n}C) = ∏_{0≤i≤j≤n} Hom_{Set}_{Δ∗∗} (E, C(i, j)^{∇^{j−i}})
= ∏_{0≤i≤j≤n} Hom_{Set}(E^{j−i}, C(i, j)).

Analysis of the product now gives the required result. □

Definition 1.25. Given a category C, a set O and a morphism f : O → Ob C, define f^{−1}C to be the category with objects O and morphisms (f^{−1}C)(a, b) = C(fa, fb).

Lemma 1.26. For any category C, there is a natural isomorphism

(BC)_{n} ≅ ∐_{f∈(Ob C)^{[0,n]}} MC(P_{n}f^{−1}C).

Proof. Note that the right-hand side is just Hom_{Cat}(P_{n}^{∗}ι•, C). Since (ι•)^{n} = • for all n, we have P_{n}^{∗}ι• ≅ n by Lemma 1.24, where we regard n as a category (with objects [0, n], and a single morphism i → j whenever i ≤ j). This completes the proof, since the nerve is given by (BC)_{n} = Hom_{Cat}(n, C). □

We now generalise these results to more general categories.

Definition 1.27. Given E ∈ Set^{Δ_{∗∗}} and X ∈ Set^{Δ_{∗∗}^{opp}}, define the set X\underleftarrow{×}E to be the quotient of {(x, e) ∈ ∐_{n} X_{n} × E^{n}} by the equivalence relation generated by

(x, ∂^{i}e) ∼ (∂_{i}x, e), (x, σ^{i}e) ∼ (σ_{i}x, e).

Definition 1.28. Given K ∈ 𝕊 and a, b ∈ K_{0}, define K(a, b) ∈ Set^{Δ_{∗∗}^{opp}} by setting

K(a, b)_{n} := {x ∈ K_{n} : (∂_{1})^{n}x = a, (∂_{0})^{n}x = b}.

Lemma 1.29. Given a (small) category 𝕀, the left adjoint P_{B𝕀}^{∗} to the functor P_{B𝕀} : Cat_{Ob 𝕀} → QM^{∗}(Set) is given by

(P_{B𝕀}^{∗}E)(a, b) = (B𝕀)(a, b)\underleftarrow{×}E

for a, b ∈ Ob 𝕀, with product given by

(x, e) ∘ (y, f) := (x ⋆ y, e ∗ f),

where ⋆ denotes concatenation of strings of morphisms in 𝕀, and ∗ is the product on E. Proof. For K ∈ 𝕊, the category P_{K}^{∗}E has objects K_{0}. Morphisms are generated under composition by K(a, b)\underleftarrow{×}E in (P_{K}^{∗}E)(a, b) for a, b ∈ K_{0}, subject to the condition that for e ∈ E^{m}, f ∈ E^{n} and x ∈ K_{m+n},

(x, e ∗ f) ∼ ((∂_{m+1})^{n}x, e) ∘ (∂_{0})^{m}x, f).

When K = 𝕀, the map ((∂_{m+1})^{n}, (∂_{0})^{m}) : K_{m+n} → K_{m} ×_{(∂_{0})^{m},K_{0},(∂_{1})^{n}} K_{n} is an isomorphism, so any product of generators is a generator, giving the required result. □ Lemma 1.30. For any category C, there is a natural isomorphism

Hom_{Cat}(𝕀, C) ≅ ∐_{f∈(Ob C)^{Ob 𝕀}} MC(P_{B𝕀}f^{−1}C).

Proof. The proof of Lemma 1.26 carries over, noting that P_{B𝕀}^{∗}ι• ≅ 𝕀. □ 1.5.2. Quasi-descent data. Definition 1.31. Given K ∈ 𝕊, define P_{K} : QDat_{K_{0}} → QMM^{∗}(Set) by

P_{K}(D)^{a,b} = ∏_{x∈K_{b}} D((∂_{0})^{b}x, (∂_{1})^{b}x)^{a},

with operations

∂_{v}^{i}(e)(x) := e(∂_{i}x)
σ_{v}^{j}(e)(y) := e(σ_{j}y),
(f ∗ e)(z) := f((∂_{b+1})^{b′}z) ∗ e((∂_{0})^{b}z).

for f ∈ P_{K}(C)^{a,b}, e ∈ P_{K}(C)^{a′,b′}. The horizontal operations are ∂_{h}^{i} = ∂_{D}^{i}, σ_{h}^{i} = σ_{D}^{i}. Note that P_{n} = P_{Δ^{n}}. Let QDat_{n} be the category of quasi-descent data on the n + 1 objects [0, n].

Lemma 1.32. The functor P_{n} : QDat_{n} → QMM^{∗}(Set) has a left adjoint P_{n}^{∗}, given by

(P_{n}^{∗})(i, j)^{a} = \begin{cases}E^{a,j−i} & j ≥ i \\ ∅ & j < i, \end{cases}

with multiplication given by ∗, identities 1 ∈ E^{00}, and operations ∂_{h}^{i}, σ_{h}^{i}. Proof. The proof of Lemma 1.24 adapts to this generality. □ Definition 1.33. Given a quasi-descent datum D, a set O and a morphism f : O → Ob D, define f^{−1}D to be the quasi-descent datum with objects O and morphisms (f^{−1}D)(a, b)^{i} = D(fa, fb)^{i}.

Lemma 1.34. For any quasi-descent datum D, there is a natural isomorphism

(Balg D)_{n} ≅ ∐_{f∈(Ob D)^{[0,n]}} MC(diag P_{n}f^{−1}D).

Proof. By Lemma 1.15, the left-hand side is Hom_{Cat}(n, alg D) ≅ Hom_{QDat}(alg^{∗}n, D). Meanwhile, the right-hand side is Hom_{QDat}(P_{n}^{∗}diag^{∗}ι•, D), so it suffices to show that alg^{∗}n ≅ P_{n}^{∗}diag^{∗}ι•. By Corollary 1.22, P_{n}^{∗}diag^{∗}ι• = P_{n}^{∗}ι(•, •), and Lemma 1.32 shows that (P_{n}^{∗}ι(•, •))(i, j)^{a} is • for j ≥ i and ∅ for j < i, so P_{n}^{∗}ι(•, •) = alg^{∗}n. □

Lemma 1.35. Given a category 𝕀, the left adjoint P_{B𝕀}^{∗} to the functor P_{B𝕀} : QDat_{Ob 𝕀} → QMM^{∗}(Set) is given by

(P_{B𝕀}^{∗}E)(a, b)^{n} = (B𝕀)(a, b)\underleftarrow{×}E^{n}

for a, b ∈ Ob 𝕀, with product given by

(x, e) ∘ (y, f) := (x ⋆ y, e ∗ f),

where ⋆ denotes concatenation of strings of morphisms in 𝕀, and ∗ is the product on E.

Proof. The proof of Lemma 1.29 adapts to this generality. □

Lemma 1.36. For any quasi-descent datum D and a category 𝕀, there is a natural isomorphism

Hom_{Cat}(𝕀, alg D) ≅ ∐_{f∈(Ob D)^{Ob 𝕀}} MC(diag P_{B𝕀}f^{−1}D).

Proof. The proof of Lemma 1.34 carries over, noting that P_{B𝕀}^{∗}ι(•, •) ≅ alg^{∗}𝕀. □

2. Algebras, coalgebras and bialgebras

2.1. Algebras and coalgebras.

Definition 2.1. A monad (or triple) on a category B is a monoid in the category of endofunctors of B (with the monoidal structure given by composition of functors). A comonad (or cotriple) is a comonoid in the category of endofunctors of B.

Lemma 2.2. Take an adjunction

\xymatrix{A \ar[r]^{G}_{⊤} & E \ar[l]^{F}}

with unit η : id → GF and co-unit ε : F G → id. Then ⊤ := GF is a monad on E with unit η and multiplication μ := GεF, while ⊥ := F G is a comonad on A, with co-unit ε and comultiplication Δ := F ηG.

Proof. For the monad ⊤, this is [Mac] §VI.1, with the comonadic results following by duality. □

Definition 2.3. Given a monad (⊤, μ, η) on a category E, define the category E^{⊤} of ⊤-algebras to have objects

⊤E ⟶^{θ} E

(for E ∈ E), such that θ ∘ η_{E} = id and θ ∘ ⊤θ = θ ∘ μ_{E}.

A morphism

g : (⊤E_{1} ⟶^{θ} E_{1}) → (⊤E_{2} ⟶^{ϕ} E_{2})

of ⊤-algebras is a morphism g : E_{1} → E_{2} in E such that ϕ ∘ ⊤g = g ∘ θ.

We define the comparison functor K : A → E^{⊤} by

B ↦ (GF GB ⟶^{Gε_{B}} GB)

on objects, and K(g) = G(g) on morphisms.

Definition 2.4. The adjunction

\xymatrix{A \ar[r]^{G}_{⊤} & E \ar[l]^{F}},

is said to be monadic (or tripleable) if K : A → E^{⊤} is an equivalence.

Examples 2.5. Intuitively, monadic adjunctions correspond to algebraic theories, such as the adjunction

\xymatrix{Ring \ar[r]^{U}_{⊤} & Set, \ar[l]^{ℤ[−]}}

between rings and sets, U being the forgetful functor. Other examples are k-algebras over k-vector spaces, or groups over sets.

Definition 2.6. Dually, given a comonad (⊥, Δ, ε) on a category A, we define the category A_{⊥} of ⊥-coalgebras by

(A_{⊥})^{opp} := (A^{opp})^{⊥},

noting that ⊥ is a monad on A^{opp}. The adjunction of Lemma 2.2 is said to be comonadic (or cotripleable) if the adjunction on opposite categories is monadic.

Examples 2.7. If X is a topological space (or any site with enough points) and X′ is the set of points of X, let u : X′ → X be the associated morphism. Then the adjunction

\xymatrix{Shf(X′) \ar[r]^{u_{∗}}_{⊤} & Shf(X), \ar[l]^{u^{−1}}}

on the associated categories of sheaves is comonadic, so Shf(X) is equivalent u^{−1}u_{∗}- coalgebras in the category Shf(X′) of sheaves (or equivalently presheaves) on X′.

A more prosaic example is that for any ring A, the category of A-coalgebras is comonadic over the category of A-modules.

2.2. Quasi-descent data from monads. Given a monad (⊤, μ, η) on a category B, and an object B ∈ B, there is a quasi-comonoid E(B) given by

E^{n}(B) = Hom_{B}(⊤^{n}B, B)

in (Set, ×), with product g ∗ h = g ∘ ⊤^{n}h, and for g ∈ E^{n}(B),

∂^{i}(g) = g ∘ ⊤^{i−1}μ_{⊤^{n−i}B}
σ^{i}(g) = g ∘ ⊤^{i}η_{⊤^{n−i−1}B}.

If we replace B with a simplicial category, then E(B) becomes a quasi-comonoid in (𝕊, ×). Note that these constructions also all work for a comonad (⊥, Δ, ε), by contravariance.

Lemma 2.8. Given an object B ∈ B, the set of ⊤-algebra structures on B is MC(E(B)).

Proof. This follows immediately from the explicit description in Definition 1.7. □

Proposition 2.9. Given a monad (⊤, μ, η) (resp. a comonad (⊥, Δ, ε)) on a category B, there is a natural structure of a Set^{Δ_{∗∗}}-enriched category on B, i.e. a quasi-descent datum on objects Ob B.

Proof. Set

Hom(B, B′)^{n} := Hom_{B}(⊤^{n}B, B′) (resp. Hom_{B}(B, ⊥^{n}B′)),

with product and operations as above. □

Proposition 2.10. The category B^{⊤} (resp. B_{⊥}) of ⊤-algebras (resp. ⊥-coalgebras) on B is isomorphic to the image under the functor alg : QDat → Cat (Definition 1.14) of the quasi-descent datum on B given in Proposition 2.9.

Proof. This follows immediately from the definitions. □

2.3. Bialgebras. We now show how a bialgebraic structure on a category gives rise to a quasi-descent datum.

As in [Van] §IV, take a category B equipped with both a monad (⊤, μ, η) and a comonad (⊥, Δ, ε), together with a distributivity transformation λ : ⊤⊥ ⟹ ⊥⊤ for which the following diagrams commute:

\xymatrix{⊤⊥ \ar@{=>}[rr]^{λ} \ar@{=>}[d]_{⊤Δ} &  & ⊥⊤ \ar@{=>}[d]_{Δ⊤} & ⊤⊥ \ar@{=>}[rr]^{λ} &  & ⊥⊤ \\ ⊤⊥^{2} \ar@{=>}[r]^{λ⊥} & ⊥⊤⊥ \ar@{=>}[r]^{⊥λ} & ⊥^{2}⊤ & ⊤^{2}⊥ \ar@{=>}[u]_{μ⊥} \ar@{=>}[r]^{⊤λ} & ⊤⊥⊤ \ar@{=>}[r]^{λ⊤} & ⊥⊤^{2} \ar@{=>}[u]_{⊥μ}}

[figure: ⊤⊥; λ; ⊤ε; ⊥⊤; ε⊤; ⊤; ⊤⊥; λ; ⊥⊤; ⊥,; η⊥; ⊥η]

Definition 2.11. Given a distributive monad-comonad pair (⊤, ⊥) on a category B, define the category B_{⊥}^{⊤} of bialgebras as follows. The objects of B_{⊥}^{⊤} are triples (α, B, β) with

(⊤B ⟶^{α} B) an object of B^{⊤} and B ⟶^{β} ⊥B an object of B_{⊥}, such that the composition (β ∘ α) : ⊤B → ⊥B agrees with the composition

⊤B ⟶^{⊤β} ⊤⊥B ⟶^{λ} ⊥⊤B ⟶^{⊥α} ⊥B.

A morphism f : (α, B, β) → (α′, B′, β′) is a morphism f : B → B′ in B such that α′ ∘ ⊤f = f ∘ α and β′ ∘ f = ⊥f ∘ β.

Proposition 2.12. The data above give B the natural structure of an Set^{Δ_{∗∗}}-enriched category, with

Hom_{B}(B, B′)^{n} = Hom_{B}(⊤^{n}B, ⊥^{n}B′).

Proof. We follow [Pri2] in describing the operations. Since λ is natural,

(λ⊥⊤) ∘ (⊤⊥λ) = (⊥⊤λ) ∘ (λ⊤⊥).

Therefore any composition of λ’s gives us the same canonical map

λ_{m}^{n} : ⊤^{m}⊥^{n} → ⊥^{n}⊤^{m},

and we define the product on Hom(B′, B′′)^{m} × Hom(B, B′)^{n} → Hom(B, B′′)^{m+n} by

g ∗ h = ⊥^{n}(g) ∘ λ_{m}^{n} ∘ ⊤^{m}(h).

The other operations are given by

∂^{i}(g) = ⊥^{n−i}Δ_{⊥^{i−1}B} ∘ g ∘ ⊤^{i−1}μ_{⊤_{h}^{n−i}B}
σ^{i}(g) = ⊥^{n−i−1}ϵ_{⊥^{i}B} ∘ g ∘ ⊤^{i}η_{⊤^{n−i−1}B}.
□

To understand how the data (⊤, ⊥, η, μ, ε, Δ, λ) above occur naturally, note that by [Van] §IV or [Pri2] §2, these data are equivalent to a diagram

\xymatrix{D \ar[r]^{U}_{⊤} \ar[d]_{V} & E \ar[l]^{F} \ar[d]_{V} \\ A \ar[u]_{G}^{⊣} \ar[r]^{U}_{⊤} & B, \ar[l]^{F} \ar[u]_{G}^{⊣}}

with F ⊣ U monadic, G ⊢ V comonadic and U, V commuting with everything (although G and F need not commute). The associated monad on B is ⊤ = UF, and the comonad ⊥ = V G. Distributivity ensures that D ≃ E^{⊤} ≃ (B_{⊥})^{⊤} and D ≃ A_{⊥} ≃ (B^{⊤})_{⊥}. In other words, D ≃ B_{⊥}^{⊤}. The functors F are both free ⊤-algebra functors, while the functors G are both cofree ⊥-coalgebra functors.

Example 2.13. If X is a topological space (or any site with enough points) and X′ is the set of points of X, let D be the category of sheaves of rings on X. If B is the category of sheaves (or equivalently presheaves) of sets on X′, then the description above characterises D as a category of bialgebras over B, with the comonad being u^{−1}u_{∗} for u : X′ → X, and the monad being the free polynomial functor.

Proposition 2.14. The category B_{⊥}^{⊤} (⊤, ⊥)-algebras on B is isomorphic to the image under the functor alg : QDat → Cat (Definition 1.14) of the quasi-descent datum B̃ on B given in Proposition 2.12.

Proof. This is essentially [Pri2] Theorem 2.2. Note that alg (B̃) arises naturally as the diagonal of a Set^{Δ_{∗∗}×Δ_{∗∗}}-enriched category. By Lemma 1.21 and Proposition 2.10, the objects of alg (B̃) over B ∈ B correspond to pairs (α, β), for (⊤B ⟶^{α} B) ∈ B^{⊤} and

(B ⟶^{β} ⊥B) ∈ B_{⊥} satisfying the conditions of Definition 2.11. The description of the morphisms follows similarly. □

3. Simplicial structures

3.1. Simplicial quasi-comonoids. We now study the structure of the category of simplicial quasi-comonoids introduced in Definition 1.4.

Proposition 3.1. There is cofibrantly generated Reedy simplicial model structure on 𝕊^{Δ_{∗∗}} in which a map f : X → Y is

(1) a weak equivalence if the maps f^{n} : X^{n} → Y^{n} are all weak equivalences,

(2) a cofibration if the maps f^{n} : X^{n} → Y^{n} are all injective.

Proof. The category Δ_{∗∗} naturally has the structure of a Reedy category, with Δ_{∗∗,+} and Δ_{∗∗,−} the subcategories of injective and surjective maps, thus giving 𝕊^{Δ_{∗∗}} a Reedy model structure. Since Δ_{∗∗,−} =Δ_{−}, the matching objects in C^{Δ} and C^{Δ_{∗∗}} are isomorphic, so it follows that this model structure on 𝕊^{Δ_{∗∗}} is cofibrantly generated, from the corresponding result for 𝕊^{Δ} (as in [GJ] §VII.4).

It only remains to describe the cofibrations. A morphism f is a cofibration if the Δ_{∗∗}-latching maps are cofibrations in 𝕊. This is equivalent to saying that, for all i, the Δ_{∗∗}-latching maps

L^{n}(f_{i}) : X_{i}^{n+1} ∪_{L^{n}(X_{i})} L^{n}(Y_{i}) → Y_{i}^{n+1}

are injective.

Now, under the comparison Δ_{∗∗} ≅ Δ_{0}^{opp}, X_{i} ∈ Set^{Δ_{∗∗}} corresponds to an augmented simplicial set ˘ (X_{i}). Thus injectivity of the latching maps says that ˘ (X_{i})_{≥0} → ˘ (Y_{i})_{≥0} is a cofibration of simplicial sets, and that ˘ (X_{i})_{−1} → ˘ (Y_{i})_{−1} is injective. Since cofibrations in 𝕊 are precisely levelwise injective maps, this is equivalent to saying that the maps f_{i}^{n} : X_{i}^{n} → Y_{i}^{n} are all injective. □

Lemma 3.2. There is a cofibrantly generated simplicial model structure on QM^{∗}(𝕊) for which a morphism f is a fibration or weak equivalence whenever the underlying map in 𝕊^{Δ_{∗∗}} is so.

Proof. Since the forgetful functor QM^{∗}(𝕊) → 𝕊^{Δ_{∗∗}} preserves filtered direct limits and has a left adjoint F, for any finite object I ∈ 𝕊^{Δ_{∗∗}}, the object F I is finite in QM^{∗}(𝕊), so a fortiori permits the small object argument. Moreover, F preserves weak equivalences. The model structure on 𝕊^{Δ_{∗∗}} is cofibrantly generated by finite objects, so [Hir] Theorem 11.3.2 gives the required model structure on QM^{∗}(𝕊). □

Remark 3.3. Observe that the category of comonoids in (𝕊, ×) is just 𝕊 itself, since the comultiplication Δ : X → X × X is necessarily given by the diagonal. Thus there is a

functor ι : 𝕊 → QM^{∗}(𝕊), given by ι(X)^{m} = \overbrace{X × X × … × X}^{m}, sending the comonoid X to its associated quasi-comonoid.

The following follows immediately from Definition 1.7

Lemma 3.4. If E ∈ QM^{∗}(𝕊), and • denotes the constant simplicial set on one element, then

MC(E_{0}) ≅ Hom_{QM^{∗}(𝕊)}(ι•, E).

Definition 3.5. Define \underline{MC} : QM^{∗}(𝕊) → 𝕊 by

\underline{MC}(E) ⊂ ∏_{n≥0}(E^{n+1})^{I^{n}}

(where I = Δ^{1} ∈ 𝕊), consisting of those \underline{ω} satisfying:

ω_{m}(s_{1}, …, s_{m}) ∗ ω_{n}(t_{1}, …, t_{n}) = ω_{m+n+1}(s_{1}, …, s_{m}, 0, t_{1}, …, t_{n});
∂^{i}ω_{n}(t_{1}, …, t_{n}) = ω_{n+1}(t_{1}, …, t_{i−1}, 1, t_{i}, …, t_{n});
σ^{i}ω_{n}(t_{1}, …, t_{n}) = ω_{n−1}(t_{1}, …, t_{i−1}, min{t_{i}, t_{i+1}}, t_{i+2}, …, t_{n});
σ^{0}ω_{n}(t_{1}, …, t_{n}) = ω_{n−1}(t_{2}, …, t_{n});
σ^{n}ω_{n}(t_{1}, …, t_{n}) = ω_{n−1}(t_{1}, …, t_{n−1}),
σ^{0}ω_{0} = 1.

We will refer to these as the higher Maurer-Cartan relations.

Define MC : QM^{∗}(𝕊) → Set by MC(E) = \underline{MC}(E)_{0}, noting that this agrees with Definition 1.7 when E ∈ QM^{∗}(Set). Also note that we can recover \underline{MC} from MC, since \underline{MC}(E)_{n} = MC(E^{Δ^{n}}).

Remark 3.6. Given a distributive monad-comonad pair (⊤, ⊥) on a simplicial category B, and an object B of B, Proposition 2.12 gives \underline{Hom}_{B}(B, B) ∈ QM^{∗}(𝕊), and we then regard \underline{MC}(\underline{Hom}_{B}(B, B)) as being the space of strong homotopy (⊤, ⊥)-bialgebras over B. If ⊥ is trivial, this is essentially the same as Lada’s definition of the space of strong homotopy ⊤-bialgebras from [CLM] (see Remark 3.13 for differences).

Definition 3.7. We now define matching objects for E ∈ QM^{∗}(𝕊) by M^{0}E := •, M^{1}E := E^{0}, and for n ≥ 2

M^{n}E = {(e_{0}, e_{1}, …, e_{n−1}) ∈ (E^{n−1})^{n} | σ^{i}e_{j} = σ^{j−1}e_{i} ∀i < j}.

These correspond to the matching objects M^{n−1}E of [GJ] Lemma VII.4.9, but we have renumbered for consistency with the wider theory of Reedy categories.

The definition of the Reedy model structure on 𝕊^{Δ_{∗∗}} implies the following.

Lemma 3.8. A morphism f : E → F in QM^{∗}(𝕊) is a fibration (resp. a trivial fibration) if and only if the relative matching maps

E^{n} → F^{n} ×_{M^{n}F} M^{n}E

are fibrations (resp. trivial fibrations) in 𝕊.

Lemma 3.9. For any trivial fibration E → F in QM^{∗}(𝕊), the map

\underline{MC}(E) → \underline{MC}(F)

is a trivial fibration.

Proof. The idea is to write \underline{MC}(E) as \underleftarrow{lim} \underline{MC}(E)^{n}, where we define \underline{MC}(E)^{n} ⊂ ∏_{0≤r≤n}(E^{r+1})^{I^{r}} satisfying the relations of Definition 3.5 above (to level n). We can summarise the Maurer-Cartan relations involving ∂^{j} and ∗ as defining a function f : \underline{MC}(E)^{n−1} → (E^{n+1})^{∂I^{n}}, where ∂I^{n} is the boundary of the simplicial set I^{n}. The relations involving σ^{j} define a function g : \underline{MC}(E)^{n−1} → (M^{n+1}E)^{I^{n}}. If we set \underline{MC}(E)^{−1} = •, this allows us to write \underline{MC}(E)^{n} as the fibre product

\xymatrix{\underline{MC}(E)^{n} \ar[r] \ar[d] & \underline{MC}(E)^{n−1} \ar[d]^{(f,g)} \\ (E^{n+1})^{I^{n}} \ar[r] & (E^{n+1})^{∂I^{n}} ×_{(M^{n+1}E)^{∂I^{n}}} (M^{n+1}E)^{I^{n}}.}

Since the pullback of a trivial fibration is a trivial fibration, it suffices to show that

(E^{n+1})^{I^{n}} → [(E^{n+1})^{∂I^{n}}×_{(M^{n+1}E)^{∂I^{n}}} (M^{n+1}E)^{I^{n}}]×_{[(F^{n+1})^{∂I^{n}}×_{(M^{n+1}F)^{∂I^{n}}} (M^{n+1}F)^{I^{n}}]}(F^{n+1})^{I^{n}}

is a trivial fibration. By definition, the maps

E^{n+1} → M^{n+1}E ×_{M^{n+1}F} F^{n+1}

are trivial fibrations. If X → Y is a trivial fibration, then X^{I^{n}} → X^{∂I^{n}} ×_{Y^{∂I^{n}}} Y^{I^{n}} is a trivial fibration, since ∂I^{n} → I^{n} is a cofibration in 𝕊 (using the simplicial structure of 𝕊). This gives the required result. □

Definition 3.10. Define an object Ξ ∈ QM^{∗}(𝕊) by Ξ^{0} = • and Ξ^{n+1} = I^{n} for n ≥ 0, where I = Δ^{1}, with operations

(s_{1}, …, s_{m}) ∗ (t_{1}, …, t_{n}) = (s_{1}, …, s_{m}, 0, t_{1}, …, t_{n});
∂^{i}(t_{1}, …, t_{n}) = (t_{1}, …, t_{i−1}, 1, t_{i}, …, t_{n});
σ^{i}(t_{1}, …, t_{n}) = (t_{1}, …, t_{i−1}, min{t_{i}, t_{i+1}}, t_{i+2}, …, t_{n});
σ^{0}(t_{1}, …, t_{n}) = (t_{2}, …, t_{n});
σ^{n}(t_{1}, …, t_{n}) = (t_{1}, …, t_{n−1}).

Proposition 3.11. For E ∈ QM^{∗}(𝕊) fibrant, there is a natural weak equivalence

R\underline{Hom}_{QM^{∗}(𝕊)}(ι•, E) ≃ \underline{MC}(E)

in 𝕊.

Proof. It follows from Definition 3.5 that

\underline{MC}(E) = \underline{Hom}_{QM^{∗}(𝕊)}(Ξ, E).

Next, observe that the unique map Ξ → ι• is a weak equivalence, since the maps I^{n} → • are weak equivalences. For any trivial fibration E → F in QM^{∗}(𝕊), Lemma 3.9 shows that

Hom_{QM^{∗}(𝕊)}(Ξ, E) → Hom_{QM^{∗}(𝕊)}(Ξ, F)

is surjective, so Ξ has the left lifting property with respect to trivial fibrations, making it a cofibrant replacement for ι•. Thus, for E fibrant,

\underline{MC}(E) ≃ R\underline{Hom}_{QM^{∗}(𝕊)}(ι•, E).
□

Corollary 3.12. The functor \underline{MC} : QM^{∗}(𝕊) → 𝕊 is right Quillen.

Proof. Given a (trivial) fibration E → F, the morphism \underline{MC}(E) → \underline{MC}(F) is \underline{Hom}_{QM^{∗}(𝕊)}(Ξ, E) → \underline{Hom}_{QM^{∗}(𝕊)}(Ξ, F). This is a (trivial) fibration, since QM^{∗}(𝕊) is a simplicial model category. □

Remarks 3.13. Lada’s definition of a strong homotopy algebra in [CLM] differs from Definition 3.5 in that it omits all of the degeneracy conditions except σ^{0}ω_{0} = 0. Proposition 3.11 would not be true if we omitted those degeneracy conditions.

Now, consider a ⊤-algebra A in topological spaces, and a retraction r : A → X, with section s. Given a homotopy h from sr to id_{A}, Lada constructs a system {ω_{n} : ⊤^{n+1}X × |I|^{n} → X} by setting

ω_{n}(a_{1}, … a_{n}) := r ∘ ε_{A} ∗ h(a_{1}) ∗ ε_{A} ∗ h(a_{2}) ∗ … ∗ h(a_{n}) ∗ ε_{A} ∘ s.

If we impose the additional conditions that r ∘ h(a) = r, h(a) ∘ s = s and h(a) ∘ h(b) = h(min(a, b)), for all a, b ∈ [0, 1], then ω satisfies the higher Maurer-Cartan relations of Definition 3.5.

Moreover, a similar description holds for retracts X of (⊤, ⊥)-bialgebras A, replacing ε_{A} with η_{A} ∘ ε_{A} : ⊤A → ⊥A.

Another way to look at this is that |Ξ| is a cofibrant resolution of ι(|Δ^{0}|) in QM^{∗}(Top). An alternative (and possibly more natural) cofibrant replacement Φ of ι(|Δ^{0}|) is given by Φ^{0} = |Δ^{0}| and Φ^{n+1} = |I|^{n}, with the same operations as Ξ, except that we replace the map min : |I| × |I| → |I| with the map (a, b) ↦ ab. Thus, for E ∈ QM^{∗}(Top) and Sing : Top → 𝕊, the space \underline{MC}(SingE) is equivalent to the subset of ∏_{n≥0} \underline{Hom}_{Top}(|I|^{n}, E^{n+1}) satisfying the conditions of Definition 3.5, except that we replace min with multiplication. The procedure above then allows us to construct a point ω of this space from a deformation retract, provided we modify the conditions above by requiring that the homotopy satisfies h(a) ∘ h(b) = h(ab).

Definition 3.14. For E ∈ QM^{∗}(𝕊) with E^{0} a group (rather than just a monoid), there is an adjoint action of E^{0} on \underline{MC}(E), given by (g, ω) ↦ g^{−1} ∗ ω ∗ g. We then define Del(E) to be the homotopy quotient (or Borel construction) Del(E) = [\underline{MC}(E)/^{h}E^{0}] = \underline{MC}(E) ×_{E^{0}} W E^{0}, for W the universal cover of BE^{0} = W̄ E^{0}, as in [GJ] Ch.V.4.

3.2. Simplicial categories.

Definition 3.15. Given a simplicial category C, recall from [Ber1] that the category π_{0}C is defined to have the same objects as C, with morphisms

Hom_{π_{0}C}(x, y) = π_{0}\underline{Hom}_{C}(x, y).

A morphism in \underline{Hom}_{C}(x, y)_{0} is said to be a homotopy equivalence if its image in π_{0}C is an isomorphism.

Lemma 3.16. There is a model structure on the category sCat of simplicial categories, in which a morphism f : C → D is

(W) a weak equivalence whenever

(W1) for any objects a_{1} and a_{2} in C, the map \underline{Hom}_{C}(a_{1}, a_{2}) → \underline{Hom}_{D}(fa_{1}, fa_{2}) is

a weak equivalence of simplicial sets; (W2) the induced functor π_{0}f : π_{0}C → π_{0}D is an equivalence of categories.

(F) a fibration whenever (F1) for any objects a_{1} and a_{2} in C, the map \underline{Hom}_{C}(a_{1}, a_{2}) → \underline{Hom}_{D}(fa_{1}, fa_{2}) is

a fibration of simplicial sets; (F2) for any objects a_{1} ∈ C, b ∈ D, and homotopy equivalence e : fa_{1} → b in D, there is an object a_{2} ∈ C and a homotopy equivalence d : a_{1} → a_{2} in C such that fd = e.

Proof. [Ber1] Theorem 1.1. □

3.3. Simplicial quasi-descent data.

Lemma 3.17. For a fixed set O, there is a cofibrantly generated simplicial model category structure on sQDat_{O} for which a morphism f : D → D′ is a fibration or a weak equivalence if and only if for all a, b ∈ O, the map

f : \underline{Hom}_{D}(a, b) → \underline{Hom}_{D′}(a, b)

is a Reedy fibration or a levelwise weak equivalence in 𝕊^{Δ_{∗∗}}.

Proof. Applying [Hir] Theorem 11.3.2 (as in the proof of Lemma 3.2) to the right adjoint functor sQDat_{O} → 𝕊^{Δ_{∗∗}} given by D ↦ ∏_{a,b∈O} \underline{Hom}_{D}(a, b) shows that this is a model structure.

The simplicial structure comes from the simplicial structure on 𝕊^{Δ_{∗∗}}, so \underline{Hom}_{D^{K}}(a, b) := \underline{Hom}_{D}(a, b)^{K}. □

We now consider the whole category sQDat, not just the subcategories on fixed objects.

Lemma 3.18. The functor (−)^{0} : sQDat → sCat is both a left and a right adjoint. The functor D ↦ ∏_{a,b∈Ob D} \underline{Hom}_{D}(a, b) from sQDat → 𝕊^{Δ_{∗∗}} is a right adjoint.

Proof. (−)^{0} has left adjoint (id/∅) and right adjoint alg^{∗}, with the same formulae and reasoning as in Lemmas 1.17 and 1.13. The left adjoint to D ↦ ∏_{a,b∈Ob D} \underline{Hom}_{E}(a, b) is the functor U : 𝕊^{Δ_{∗∗}} → sQDat given by sending X to the category with two objects x, y, and morphisms

\underline{Hom}(x, x)^{n} = \underline{Hom}(y, y)^{n} := \begin{cases}1 & n = 0 \\ ∅ & n > 0, \end{cases}

\underline{Hom}(x, y) := X and \underline{Hom}(y, x) := ∅. □

Proposition 3.19. There is a cofibrantly generated model structure on sQDat for which a morphism f : D → E is

(W) a weak equivalence if and only if (W 1) for all a, b ∈ Ob D, the map

f : \underline{Hom}_{D}(a, b) → \underline{Hom}_{E}(fa, fb)

is a weak equivalence in 𝕊^{Δ_{∗∗}}, and (W 2) the morphism π_{0}(f^{0}) : π_{0}(D^{0}) → π_{0}(E^{0}) is an equivalence of categories (for

π_{0}C as in Definition 3.15);

(F) a fibration if and only if (F 1) for all a, b ∈ Ob D, the map

f : \underline{Hom}_{D}(a, b) → \underline{Hom}_{E}(a, b)

is a Reedy fibration in 𝕊^{Δ_{∗∗}}, and (F 2) for any objects a_{1} ∈ D, b ∈ E and a homotopy equivalence e : fa_{1} → b in E^{0},

there exist an object a_{2} ∈ E and a homotopy equivalence d : a_{1} → a_{2} in D^{0} such that f^{0}d = e.

Proof. Note that these conditions are equivalent to saying that f is a weak equivalence or fibration provided that both f^{0} : D^{0} → E^{0} and the maps \underline{Hom}_{D}(a, b) → \underline{Hom}_{E}(fa, fb) are weak equivalences or fibrations. This follows because the functor 𝕊^{Δ_{∗∗}} → 𝕊 given by X ↦ X^{0} preserves both weak equivalences and fibrations (by definition of the Reedy model structure).

For U as in the proof of Lemma 3.18, define (I1) to be the class consisting of the images under U of the generating cofibrations from Proposition 3.1, and let (I2) be the single morphism ∅ → 0 from the category with no objects to the category with one object and no non-identity morphisms. Define (I) := (I1) ∪ (I2).

Define (J1) to be the class consisting of the images under U of the generating trivial cofibrations from Proposition 3.1, and let (J2) be the image of the class (A2) from [Ber1] under the functor (id/∅) : sCat → sQDat of Lemma 3.18. Define (J) := (J1) ∪ (J2).

For a class C of morphisms, say that a morphism f is S-injective if it has the left lifting property (LLP) with respect to S. From the adjoint property of the functor U, it follows that a morphism f : D → E is (J1)-injective, (resp. (I1)-injective) if and only if f satisfies (F 1) (resp. (F 1) and (W 1)). The morphism f is (I2)-injective if and only if it is surjective on objects. By Lemma 3.18, f is (J2)-injective if and only if f^{0} : D^{0} → E^{0} is (A2)-injective.

Since cofibrations in 𝕊, concentrated in degree 0 in 𝕊^{Δ_{∗∗}}, become cofibrations in the model structure of Proposition 3.1, the images under (id/∅) of the classes (C1), (C2), (A1) and (A2) from [Ber1] lie in (I1), (I2), (J1) and (J2) respectively. It thus follows from [Ber1] Theorem 1.1 that for any J-injective (resp. I-injective) morphism f : D → E, the morphism f^{0} : D^{0} → E^{0} is a fibration (resp. a trivial fibration) in sCat. Looking at the LLP in 𝕊^{Δ_{∗∗}}, we then deduce that J-injectives (resp. I-injectives) are precisely (F) (resp. (F) ∩ (W) ) in sQDat.

We now verify the conditions of [Hov] Theorem 2.1.19. It is immediate that the class (W) has the two out of three property and is closed under retracts. The domains of (I2) and (J2) are small, and similarly to the proof of [Ber1] Theorem 1.1, the smallness of the generating (trivial) cofibrations in 𝕊^{Δ_{∗∗}} means that the domains of (I1) and (J1) are small relative to (I1)-cells and (J1)-cells, respectively. Thus the domains of (I) and (J) are small relative to (I)-cells and (J)-cells, respectively. We have shown that the class of (I)-injectives is the intersection of (W) with the class of (J)-injectives.

It remains only to show that all (J)-cells are in (W) and are (I)-cofibrations. Since cofibrations in 𝕊, concentrated in degree 0 in 𝕊^{Δ_{∗∗}}, become cofibrations in the model structure of Proposition 3.1, the functor (id/∅) maps cofibrations in sCat to (I)-cofibrations. Thus (J2)-cells are (I)-cofibrations, since (id/∅) preserves all colimits. Likewise, U preserves all colimits and maps cofibrations to (I)-cofibrations, so (J1)-cells are (I)-cofibrations, and therefore (J)-cells are (I)-cofibrations. Since (J) ⊂ (W), it follows immediately from the definitions and the corresponding properties in sCat and 𝕊^{Δ_{∗∗}} that all (J)-cells also lie in (W). □

Note that the functor (−)^{0} : sQDat → sCat of Lemma 3.18 is then both left and right Quillen, while the functor D ↦ ∏_{a,b∈Ob D} \underline{Hom}_{D}(a, b) from sQDat → 𝕊^{Δ_{∗∗}} is right Quillen.

Definition 3.20. We define model structures on sQDpd, sQDpd_{O} by requiring that a morphism f is a weak equivalence or a fibration whenever the underlying map in sQDat, sQDat_{O} is so. [Hir] Theorem 11.3.2 shows that these are indeed model structures, since the forgetful functors preserve filtered colimits, have left adjoints (denoted by D ↦ D^{Gpd}), given by formally inverting morphisms in D^{0}, and the left adjoints send trivial cofibrations to weak equivalences.

3.4. Simplicial bicomonoids.

Lemma 3.21. There is a cofibrantly generated simplicial model structure on QMM^{∗}(𝕊) for which a morphism f is a fibration or weak equivalence whenever the underlying map in the Reedy model category 𝕊^{Δ_{∗∗}×Δ_{∗∗}} is so.

Proof. The proof of Lemma 3.2 carries over. □

Lemma 3.22. The diagonal functor diag : QMM^{∗}(𝕊) → QM^{∗}(𝕊) is right Quillen.

Proof. The Special Adjoint Functor Theorem ([Mac] Theorem V.8.2) implies that diag : QMM^{∗}(𝕊) → QM^{∗}(𝕊) has a left adjoint diag^{∗}. It therefore suffices to show that for any (trivial) fibration f in 𝕊^{Δ_{∗∗}×Δ_{∗∗}}, the map diag f is a (trivial) fibration in 𝕊^{Δ_{∗∗}}.

Now, let Θ_{n}, ∂Θ_{n} ∈ Set^{Δ_{∗∗}} be given by Hom_{Set}_{Δ∗∗} (Θ_{n}, X) = X^{n} and Hom_{Set}_{Δ∗∗} (∂Θ_{n}, X) = M^{n}X, and similarly let Θ_{ij}, ∂Θ_{ij} ∈ Set^{Δ_{∗∗}×Δ_{∗∗}} be given by Hom_{Set}_{Δ∗∗×Δ∗∗} (Θ_{ij}, X) = X^{i,j} and Hom_{Set}_{Δ∗∗×Δ∗∗} (∂Θ_{ij}, X) = M^{i,j}X. Latching object arguments (adapting [GJ] Proposition VII.1.7) show that the maps ∂Θ_{n} ⊂ Θ_{n} generate all monomorphisms in Set^{Δ_{∗∗}}, and likewise the maps ∂Θ_{ij} ⊂ Θ_{ij} generate all monomorphisms in Set^{Δ_{∗∗}×Δ_{∗∗}}.

The diagonal functor diag : Set^{Δ_{∗∗}×Δ_{∗∗}} → Set^{Δ_{∗∗}} has a left adjoint diag^{∗}, and we just observe that this preserves monomorphisms (much like the case of bisimplicial sets considered in [GJ] Theorem IV.3.15). Therefore the functor diag^{∗} : 𝕊^{Δ_{∗∗}} → 𝕊^{Δ_{∗∗}×Δ_{∗∗}} preserves Reedy (trivial) cofibrations, so is left Quillen, making diag right Quillen. □

Note that we may regard QM^{∗}(𝕊) and QMM ∗ (𝕊) as being simplicial diagrams in QM^{∗}(Set) and QMM^{∗}(Set), respectively. This will allow us to extend many of the constructions of §1 to the simplicial case.

3.5. Nerves. In Remark 3.6, we saw how \underline{MC} enables us to define the space of s.h. bialgebras over a fixed object. However, as was first noted in [CLM], there is no satisfactory general way to define morphisms of s.h. algebras. The bar construction of [CLM] gives a definition when the monad is an operad, but does not generalise to s.h. bialgebras. Instead, we will now introduce a space of s.h. 𝕀-diagrams of s.h. bialgebras for any small category 𝕀, allowing us to mimic the nerve construction and thus to construct a simplicial space of s.h. bialgebras.

3.5.1. MC.

Definition 3.23. Given K ∈ 𝕊, define the functor P_{K} : sQDat_{K_{0}} → QMM^{∗}(𝕊) by extending Definition 1.31 to simplicial sets. Let P_{n} := P_{Δ^{n}}; as in Lemma 1.32, P_{n} has a left adjoint P_{n}^{∗}.

Proposition 3.24. The functor P_{K} : sQDat_{K_{0}} → QMM^{∗}(𝕊) is right Quillen.

Proof. Since P_{K} is defined as a limit, it preserves arbitrary limits, so we just need to show that it preserves (trivial) fibrations. We may regard an object X of 𝕊^{Δ_{∗∗}×Δ_{∗∗}} as a Δ_{∗∗} diagram in 𝕊^{Δ_{∗∗}}, by i ↦ X^{i,•}. Denote the associated Reedy matching objects by M_{hor}^{i}X ∈ 𝕊^{Δ_{∗∗}}. Similarly, there is a diagram j ↦ X^{•,j}, and we denote the associated matching objects by M_{ver}^{j}X ∈ 𝕊^{Δ_{∗∗}}. Note that the Reedy matching objects in 𝕊^{Δ_{∗∗}×Δ_{∗∗}} are then given by

M^{ij}X = (M_{hor}^{i}X)^{j} ×_{M_{hor}^{i}M_{ver}^{j}X} (M_{ver}^{j}X)^{i},

as an immediate consequence of the characterisation of matching objects in [Hov] Definition 5.2.2.

Now, for D ∈ sQDat_{K_{0}}, the horizontal matching object M_{hor}^{i}P_{K}(D) in 𝕊^{Δ_{∗∗}} is given by

(M_{hor}^{i}P_{K}(D))^{j} = M^{i}(P_{K}(D)^{j}) = ∏_{x∈K_{j}} M^{i}D((∂_{0})^{j}x, (∂_{1})^{j}x).

Next, observe that for X, K ∈ 𝕊 the object S ∈ 𝕊^{Δ_{∗∗}} given by S^{n} = X^{K_{n}}, the matching object M^{n}S is given by

M^{n}S = {(f_{0}, f_{1}, …, f_{n−1}) ∈ (X^{K_{n−1}})^{n} | σ^{i}f_{j} = σ^{j−1}f_{i} ∈ X^{K_{n−2}} ∀i < j} ≅ X^{L_{n}K},

where L_{n}K is the nth simplicial latching object of K (of [GJ] §VII.1). Thus the vertical matching object M_{ver}^{j}P_{K}(D) is given by

(M_{ver}^{j}P_{K}(D))^{i} = ∏_{x∈L_{j}K} D((∂_{0})^{j}x, (∂_{1})^{j}x)^{i}.

Therefore, since L_{j}K → K_{j} is always injective,

M^{ij}P_{K}(D) ≅ ( ∏_{x∈L_{j}K} D((∂_{0})^{j}x, (∂_{1})^{j}x)^{i}) × ( ∏_{x∈K_{j}−L_{j}K} M^{i}D((∂_{0})^{j}x, (∂_{1})^{j}x)),

which yields the required result.

In fact, we may adapt this further to say that for any cofibration i : J ↪ K in 𝕊 and any (trivial) fibration D → E in sQDat_{K_{0}}, the map

P_{K}(D) → P_{K}(E) ×_{P_{J} (i_{0}^{−1}E)} P_{J}(i_{0}^{−1}D)

is a (trivial) fibration. □

Definition 3.25. Define a functor MC : sQDat → s𝕊 to bisimplicial sets by

MC(D)_{(n)} := ∐_{f:[0,n]→Ob D} \underline{MC}(diag P_{n}(f^{−1}D)) ∈ 𝕊.

Definition 3.26. Given a simplicial set X, we define X ∈ s𝕊 to be the constant space X_{(n)} := X for all n. By contrast, we define X^{hor} by X_{(n)}^{hor} := X_{n}.

Remark 3.27. Assume D comes from a distributive monad-comonad pair (⊤, ⊥) on a simplicial category B, as in Proposition 2.12. Lemma 1.30 then allows us to think of MC(D)_{(n)} as being the space of s.h. n-diagrams of s.h. (⊤, ⊥)-bialgebras over B, so MC is a kind of nerve construction. More generally, for any small category 𝕀, we think of ∐_{f:Ob 𝕀→Ob D} \underline{MC}(diag P_{B𝕀}(f^{−1}D)) as the space of s.h. 𝕀-diagrams of s.h. (⊤, ⊥)-bialgebras over B, noting that this is just \underline{Hom}_{s𝕊}((B𝕀)^{hor}, MC(D)).

Lemma 3.28. Given a (trivial) fibration f : D → E in sQDat, the morphism

MC(D) → MC(E) ×_{cosk_{0}(Ob E)^{hor}} cosk_{0}(Ob D)^{hor}

is a (trivial) fibration in the Reedy category s𝕊 = 𝕊^{Δ^{opp}}, where cosk_{0} : Set → 𝕊 denotes the 0-coskeleton ([GJ] §IV.3).

Proof. For any simplicial set K, Hom_{𝕊}(K, cosk_{0}S) = S^{K_{0}}; since (∂Δ^{n})_{0} = (Δ^{n})_{0} = [0, n] (for ∂Δ^{n} ⊂ Δ^{n} the boundary), the nth Reedy matching map of the morphism above is given by taking the coproduct over all g : [0, n] → Ob D of

\underline{MC}(diag P_{n}g^{−1}D) → \underline{MC}(diag P_{n}g^{−1}(Ob f)^{−1}E)×_{\underline{MC}(diag P_{∂Δ^{n}}g^{−1}(Ob f)^{−1}E)}\underline{MC}(diag P_{∂Δ^{n}}g^{−1}D).

Assume that f is a (trivial) fibration. By the proof of Proposition 3.24,

P_{n}g^{−1}D → (P_{n}g^{−1}(Ob f)^{−1}E) ×_{P_{∂Δ^{n}}g^{−1}(Ob f)^{−1}E)} (P_{∂Δ^{n}}g^{−1}D)

is a (trivial) fibration in QMM^{∗}(𝕊), so Lemma 3.22 shows that

diag P_{n}g^{−1}D → (diag P_{n}g^{−1}(Ob f)^{−1}E) ×_{diag P_{∂Δ^{n}}g^{−1}(Ob f)^{−1}E)} (diag P_{∂Δ^{n}}g^{−1}D)

is a (trivial) fibration in QM^{∗}(𝕊), so Corollary 3.12 shows that the map above is a (trivial) fibration whenever f is so. □

Note that considering level 0 shows that MC : sQDat → s𝕊 is not right Quillen, since a fibration only maps to a fibration when it is surjective on objects, and a trivial fibration only maps to a trivial fibration when it is an isomorphism on objects.

3.5.2. Del.

Definition 3.29. Let sGpd be the category of simplicial groupoids, i.e. the full subcategory of (Gpd)^{Δ^{opp}} consisting of those Γ for which the simplicial set Ob Γ of objects is constant. As in [GJ] §V.7, this has a model structure in which a morphism is a weak equivalence or fibration whenever the corresponding morphism in sCat is so (for the model structure of Lemma 3.16), although the description simplifies considerably, since all morphisms in Γ (and in particular homotopy equivalences) are isomorphisms.

Definition 3.30. Given Γ ∈ sGpd, define 𝕊(Γ) to be the category of simplicial Γrepresentations. An object X ∈ 𝕊(Γ) consists of X(a) ∈ 𝕊 for all objects a of Γ, and distributive morphisms

Γ(a, b) × X(b) → X(a)

in 𝕊.

Lemma 3.31. For Γ ∈ sQDpd, there is a natural Γ^{0}-representation in 𝕊, given by mapping a ∈ Ob Γ to \underline{MC}(Γ(a, a)). Denote this representation by \underline{MC}(Γ) ∈ 𝕊(Γ^{0}).

Proof. We just need to define an associative action \underline{MC}(a) × Γ^{0}(a, b) → \underline{MC}(b). As in Definition 3.14, the adjoint action (ω, g) ↦ g^{−1} ∗ ω ∗ g suffices. □

Definition 3.32. Given a simplicial groupoid Γ, define the Γ-representation W Γ by

(W Γ)_{n}(a) := ∐_{x_{0},x_{1},...,x_{n}} Γ(a, x_{n})_{n} × Γ(x_{n}, x_{n−1})_{n−1} × … × Γ(x_{1}, x_{0})_{0},

for a ∈ Ob Γ. As in [GJ] §V.4 (which considered only simplicial groups), this has operations:

∂_{i}(v_{n}, v_{n−1}, …, v_{0}) = \begin{cases}(∂_{i}v_{n}, ∂_{i−1}v_{n−1}, …, (∂_{0}v_{n−i})v_{n−i−1}, v_{n−i−2}, …, v_{0}) & i < n, \\ (∂_{n}v_{n}, ∂_{n−1}v_{n−1}, …, ∂_{1}v_{1}) & i = n, \end{cases}
σ_{i}(v_{n}, v_{n−1}, …, v_{0}) = (σ_{i}v_{n}, σ_{i−1}v_{n−1}, …, σ_{0}v_{n−i}, 1, v_{n−i−1}, …, v_{0}),

For h ∈ Γ(a, b), the action is given by

h(v_{n}, v_{n−1}, …, v_{0}) = (hv_{n}, v_{n−1}, …, v_{0}).

Definition 3.33. Given Γ ∈ sGpd, define ho\underrightarrow{lim}_{Γ} : 𝕊(Γ) → (𝕊 ↓ W̄ Γ) by X ↦ X ×_{Γ} W Γ, where W̄ Γ := • ×_{Γ} W Γ is a model for the classifying space of Γ ([GJ] §V.7).

Lemma 3.34. The functor ho\underrightarrow{lim}_{Γ} : 𝕊(Γ) → (𝕊 ↓ W̄ Γ) is right Quillen, where fibrations and weak equivalences in 𝕊(Γ) are defined objectwise.

Proof. The proof of [GJ] Lemma VI.4.6, which takes the case when Γ is a discrete groupoid, carries over to this generality. The left adjoint is given by X ↦ X ×_{W̄ Γ} W Γ. □

Definition 3.35. Define a functor Del : sQDpd → 𝕊 by

Del(Γ) := ho\underrightarrow{lim}_{Γ^{0}} \underline{MC}(Γ),

making use of the forgetful functor 𝕊 ↓ W̄ Γ → 𝕊. Note that if Γ has one object, then it may be regarded as an object of QM^{∗}(𝕊), and this definition is consistent with Definition 3.14 in this case.

Proposition 3.36. The functor Del : sQDpd → 𝕊 is right Quillen.

Proof. Since Del clearly preserves limits, we need only show that it preserves (trivial) fibrations. Given a (trivial) fibration f : D → E in sQDpd, set F := E ×_{alg^{∗}E^{0}} alg^{∗}D^{0}, and observe that f factors as the composition of the (trivial) fibrations g : D → F, h : F → E.

Now, \underline{MC}(D) → \underline{MC}(F) is a morphism in 𝕊(D^{0}) which is a (trivial) fibration by Corollary 3.12. Lemma 3.34 then implies that Del(D) → Del(F) is a (trivial) fibration in 𝕊, since Ob (D) = Ob (cF). The morphism Del(F) → Del(E) is a pullback of Del(alg^{∗}D^{0}) → Del(alg^{∗}E^{0}), so it remains only to show that the latter is a (trivial) fibration.

Given E ∈ QM^{∗}(𝕊) with E^{n} = E^{0} for all n, studying the degeneracy operations shows that \underline{MC}(E) = \{1\}. Therefore \underline{MC}(alg^{∗}D^{0}) is the constant D^{0}-representation on the one-point set •, so

Del(alg^{∗}D^{0}) = • ×_{D^{0}} W D^{0} = W̄ D^{0},

and similarly for Del(alg^{∗}E^{0}) = W̄ E^{0}. The morphism D^{0} → E^{0} is a (trivial) fibration in sGpd, so W̄ D^{0} → W̄ E^{0} is a (trivial) fibration in 𝕊, by [GJ] Theorem V.7.8. □

Definition 3.37. Given K ∈ 𝕊 and D ∈ sQDat, define h(K, D) ∈ sQDat by Ob (h(K, D)) = Hom_{Set}(K_{0}, Ob D) and

h(K, D)(a, b)^{n} := ∏_{x∈K_{n}} C(a((∂_{0})^{n}x), b((∂_{1})^{n}x)),

and note that MC(D)_{n} = \underline{MC}(h(Δ^{n}, D)).

Definition 3.38. Define DEL : sQDpd → s𝕊 by DEL(Γ)_{(n)} := Del(h(Δ^{n}, Γ)).

Corollary 3.39. The functor DEL : sQDpd → s𝕊 is right Quillen.

Proof. This just combines Proposition 3.36 with the observation that for any (trivial) fibration f : D → E in sQDpd, the morphism

h(Δ^{n}, D) → h(Δ^{n}, E) ×_{h(∂Δ^{n},E)} h(∂Δ^{n}, D)

is a (trivial) fibration in sQDat for all n ≥ 0, which follows by combining Lemma 3.22 with the proof of Proposition 3.24. □

4. Abelian groups and cohomology

In this section, we will investigate quasi-comonoids in abelian groups and in groupoids. The main motivation for this is that we can detect whether a simplicial set X is contractible just by looking at π_{f}X and H^{∗}(X, ℤ), and we will now develop the corresponding notions for QM^{∗}(𝕊).

4.1. Cosimplicial abelian groups.

Lemma 4.1. There is an equivalence between the category QM^{∗}(Ab, ×) of quasicomonoids in (Ab, ×), and the category cAb of cosimplicial complexes of abelian groups.

Proof. Take A ∈ QM^{∗}(Ab, ×). The operations ∂^{i}, σ^{i} on A are necessarily ℤ-linear, and we enhance this to a cosimplicial structure by setting ∂^{0}a = 0_{1} ∗ a, ∂^{m+1}a = a ∗ 0_{1}, for a ∈ A^{m} and 0_{1} the group identity in A^{1}. To see that this satisfies the cosimplicial axioms, note that the properties of ∗ give that

∂^{i+1}∂^{0}a = ∂^{0}∂^{i}a ∀1 ≤ i ≤ m
∂^{i}∂^{m+1}a = ∂^{m+2}∂^{i}a ∀1 ≤ i ≤ m
σ^{i+1}∂^{0}a = ∂^{0}σ^{i}a ∀0 ≤ i < m
σ^{i}∂^{m+1}a = ∂^{m}σ^{i}a ∀0 ≤ i < m.

We also have ∂^{0}∂^{m+1}a = 0_{1} ∗ a ∗ 0_{1} = ∂^{m+2}∂^{0}a, so it only remains to show that σ^{0}∂^{0} = id and σ^{m}∂^{m+1} = id, and that ∂^{0}∂^{0} = ∂^{1}∂^{0} and ∂^{m+1}∂^{m+1} = ∂^{m+2}∂^{m+1}. The first two conditions follow because σ^{0}0_{1} = 0_{0}, which is the identity for ∗, and the second two follow because 0_{1} ∗ 0_{1} = 0_{2} = ∂^{1}0_{1}.

Since A^{m} × A^{n} ⟶^{∗} A^{m+n} is linear, for a, a′ ∈ A^{m}, b, b′ ∈ A^{n} we then have

a ∗ b′ + a′ ∗ b = (a + a′) ∗ (b′ + b),

so setting a′ = 0_{m}, b′ = 0_{n} gives a ∗ b = a ∗ 0_{n} + 0_{m} ∗ b, and 0_{n} = 0_{1}^{∗n}, so the product is necessarily the Alexander-Whitney cup product

a ∗ b = (∂^{m+1})^{n}a + (∂^{0})^{m}b,

which is uniquely determined by the cosimplicial structure. □

Definition 4.2. Let sAb be the category of simplicial abelian groups, and csAb be the category of cosimplicial simplicial abelian groups.

Definition 4.3. Denote the left adjoint to the inclusion functor csAb → QM^{∗}(𝕊) by cot. This is left Quillen, and we denote the associated left-derived functor on homotopy categories by L cot.

Lemma 4.4. There is a cofibrantly generated simplicial model structure on QM^{∗}(sAb, ×) in which a morphism is a fibration or a weak equivalence whenever the underlying map in QM^{∗}(𝕊) is so.

Proof. We may apply [Hir] Theorem 11.3.2 to the forgetful functor QM^{∗}(sAb, ×) → QM^{∗}(𝕊). This functor preserves filtered direct limits, so admits the small object argument where necessary. It has a left adjoint (analogous to the free module generated by a set), and the left adjoint preserves weak equivalences. □

Lemma 4.5. There is an equivalence

QM^{∗}(sAb, ×) ≃ csAb

of model categories, where csAb is given the Reedy model structure for cosimplicial objects in sAb (with its standard model structure).

Proof. Lemma 4.1 gives the equivalence of categories, by passing to simplicial diagrams. Now, f is a weak equivalence in QM^{∗}(sAb, ×) whenever each f^{n} is a weak equivalence, and a fibration whenever f is a Reedy fibration in 𝕊^{Δ_{∗∗}}. Since the matching objects for Δ and Δ_{∗∗} are the same, this means that the model structure of Lemma 4.4 is just the Reedy model structure on (sAb)^{Δ}, as they have the same weak equivalences and fibrations. □

Definition 4.6. Let N^{s} denote simplicial normalisation from simplicial abelian groups to non-negatively graded chain complexes, given by N^{s}(V)_{n} := ⋂_{i>0} ker(∂_{i} : V_{n} → V_{n−1}), with differential ∂_{0}. Let N_{c} denote cosimplicial conormalisation from cosimplicial abelian groups to non-negatively graded cochain complexes, given by N_{c}(V)^{n} := ⋂_{i≥0} ker(σ^{i} : V^{n} → V^{n−1}), with differential ∑_{i}(−1)^{i}∂^{i}. By the Dold-Kan correspondence ([Wei] Theorem 8.4.1, passing to opposite categories and using [Wei] Lemma 8.3.7 in the cosimplicial case), these functors are both equivalences; let D_{c} be the cosimplicial denormalisation functor, inverse to N_{c}.

Definition 4.7. Set I = Δ^{1}, and for n ≥ 0, let

ℷ^{n} := ({1} × I^{n−1}) ∪ ⋃_{j>0}(I^{j} × {0, 1} × I^{n−1−j}) ⊂ I^{n} ∈ 𝕊;

for n ≥ 2 this is given by removing the interior of 0 × I^{n−1} from the boundary ∂I^{n}, while ℷ^{1} = {1} and ℷ^{0} = ∅.

Let

ℤ(I^{n}/ℷ^{n}) := ℤ(I^{n})/ℤ(ℷ^{n}) ∈ sAb,

where ℤ(S) is the free ℤ-module generated by the set S, and let δ be the canonical map ℤ(I^{n−1}/ℷ^{n−1}) → ℤ(I^{n}/ℷ^{n}) arising from the map I^{n−1} → I^{n} given by x ↦ (0, x).

For any simplicial abelian group W, write

W^{I^{n}/ℷ^{n}} := ker(W^{I^{n}} → W^{ℷ^{n}}) = \underline{Hom}_{sAb}(ℤ(I^{n}/ℷ^{n}), W)

and let δ be the canonical map W^{I^{n}/ℷ^{n}} → W^{I^{n−1}/ℷ^{n−1}} dual to the map δ above.

Proposition 4.8. Given an abelian group object E in QM^{∗}(𝕊), corresponding under Lemma 4.1 to the cosimplicial simplicial abelian group C(E), there is an isomorphism

\underline{MC}(E) ≅ \{η ∈ ∏_{n=0}^{∞} N_{c}^{n+1}C(E)^{I^{n}/ℷ^{n}} : d_{c}η_{n−1} = δη_{n}\}.

Proof. As in Lemma 3.9, write \underline{MC}(E) = \underleftarrow{lim}_{n} \underline{MC}(E)^{n}, and assume that we are given an element

(ω_{0}, …, ω_{n−1}) ∈ \underline{MC}(E)^{n−1}.

The proof of Lemma 3.9 then gives rise to the data

β_{n−1} ∈ M^{n+1}C(E)^{I^{n}}, α_{n−1} ∈ C^{n+1}(E)^{∂I^{n}}

(in the notation of Definition 3.7). By Lemma 3.9, the fibre of \underline{MC}(E)^{n} → \underline{MC}(E)^{n−1} over (ω_{0}, …, ω_{n−1}) is given by ω_{n} ∈ C^{n+1}(E)^{I^{n}} compatibly lifting α_{n−1}, β_{n−1} in the following diagram:

\xymatrix{C^{n+1}(E)^{I^{n}} \ar[r] \ar[d] & C^{n+1}(E)^{∂I^{n}} \ar[d] \\ M^{n+1}C(E)^{I^{n}} \ar[r] & M^{n+1}C(E)^{∂I^{n}}.}

For any abelian cosimplicial abelian group C^{•}, dualising [Wei] Lemma 8.3.7 gives a decomposition of the associated cochain complex as C^{n} = N_{c}^{n}(C) ⊕ (M′)^{n}(C), where N_{c}^{n}(C) = ∩_{i=0}^{n−1} ker σ^{i}, and (M′)^{n}(C) = ∑_{i=1}^{n} ∂^{i}C^{n−1}, so the commutative diagram becomes

\xymatrix{(N_{c}^{n+1}C(E)^{I^{n}}) ⊕ ((M′)^{n+1}C(E)^{I^{n}}) \ar[r] \ar[d] & (N_{c}^{n+1}C(E)^{∂I^{n}}) ⊕ ((M′)^{n+1}C(E)^{∂I^{n}}) \ar[d] \\ M^{n+1}C(E)^{I^{n}} \ar[r] & M^{n+1}C(E)^{∂I^{n}}.}

Moreover, \underline{σ} : (M′)^{n}C → M^{n}C is an isomorphism, and we will denote the inverse by a ↦ ã. Thus the problem of constructing ω_{n} reduces to seeking an element η_{n} = ω_{n} − β̃_{n−1} ∈ N_{c}^{n+1}C(E)^{I^{n}} lifting pr_{N}(α_{n−1}) ∈ N_{c}^{n+1}C(E)^{∂I^{n}}, where pr_{N} : C → N_{c}(C) is the projection given by annihilating (M′)^{C} .

Now, α_{n−1} is defined by

α_{n−1}(t_{1}, …, t_{i−1}, 0, t_{i+1}, …, t_{n}) = (∂^{i+1})^{n−i+1}ω_{i−1}(t_{1}, …, t_{i−1}) + (∂^{0})^{i}ω_{n−i}(t_{i+1}, …, t_{n});
α_{n−1}(t_{1}, …, t_{i−1}, 1, t_{i+1}, …, t_{n}) = ∂^{i}ω_{n−1}(t_{1}, …, t_{i−1}, t_{i+1}, …, t_{n}).

Therefore

pr_{N}α_{n−1}(0, t_{2}, …, t_{n}) = pr_{N}∂^{0}η_{n−1}(t_{2}, …, t_{n});
pr_{N}α_{n−1}(t_{1}, …, t_{i−1}, 0, t_{i+1}, …, t_{n}) = 0 for i > 1;
pr_{N}α_{n−1}(t_{1}, …, t_{i−1}, 1, t_{i+1}, …, t_{n}) = 0,

since all other terms lie in (M′)^{n+1} (the span of {∂^{i} : i > 0}).

Since pr_{N}∂^{0} = pr_{N}d_{c}, it follows from [Wei] Lemma 8.3.7 that on N_{c}^{n},

pr_{N}∂^{0} = d_{c} = ∑_{i=0}^{n+1}(−1)^{i}∂^{i}.

This implies that

η_{n} ∈ N_{c}^{n+1}C(E)^{I^{n}/ℷ^{n}},

and the condition for η_{n} to lift pr_{N}(α_{n−1}) is precisely that δ(η_{n}) = d_{c}η_{n−1}. □

Corollary 4.9. A representative for L cot(ι•) is given by ι(ℤ) ≅ ED_{c}(ℤ^{[−1]}), for E : csAb → QM^{∗}(sAb, ×) as in Lemma 4.1.

Proof. First observe that, for Ξ from Definition 3.10, L cot(ι•) ≃ L cot(Ξ), since Ξ → ι• is a cofibrant replacement.

Since Ξ represents \underline{MC}, for any simplicial cochain complex V (in non-negative cochain degrees) we have

Hom(N_{c}C(cot Ξ), V) ≅ MC(E(D_{c}V)),

where Hom is taken in the category of simplicial cochain complexes. Since C is inverse to E, and N_{c} is inverse to D_{c}, Proposition 4.8 can then be rephrased to say that

N_{c}^{n}C(cot(Ξ)) ≅ \begin{cases}ℤ(I^{n−1}/ℷ^{n−1}) & n ≥ 1 \\ 0 & n = 0, \end{cases}

with differential d_{c} = δ.

Thus the bicomplex N^{s}N_{c}C(cot Ξ) is weakly equivalent (in the Reedy model category of cochain diagrams in chain complexes) to the bicomplex ℤ^{[−1]}, consisting of ℤ concentrated in cochain degree 1, chain degree 0. This means that cot Ξ is weakly equivalent to ED_{c}(ℤ^{[−1]}) (with constant simplicial structure), but this is isomorphic to ι(ℤ) (having n copies of ℤ in level n). □

Definition 4.10. Given a cochain complex V, denote the brutal truncation in degrees ≥ n by σ^{≥n}V, so

(σ^{≥n}V)^{i} = \begin{cases}V^{i} & i ≥ n \\ 0 & i < n. \end{cases}

Definition 4.11. Define the total complex functor Tot^{Π} from chain cochain complexes (i.e. bicomplexes) to chain complexes by

(Tot^{Π}V)_{n} := ∏_{a−b=n} V_{a}^{b},

with differential d := d^{s} + (−1)^{a}d_{c} on V_{a}^{b}.

Proposition 4.12. For A ∈ csAb,

π_{n}\underline{MC}(A) ≅ H_{n−1}(Tot^{Π}σ^{≥1}N^{s}N_{c}A).

Proof. First, note that

\underline{MC}(A) = \underline{Hom}_{QM^{∗}(𝕊)}(Ξ, A),

so the Dold-Kan correspondences give

\underline{MC}(A) = \underline{Hom}_{csAb}(cot Ξ, A) ≃ \underline{Hom}_{DG^{≥0}dg_{≥0}Ab}(N^{s}N_{c}C(cot Ξ), N^{s}N_{c}A),

where DG^{≥0}dg_{≥0}Ab is the Reedy category of non-negatively graded cochain complexes of non-negatively graded chain complexes.

Now, Corollary 4.9 implies that N^{s}N_{c}C(cot Ξ) is a cofibrant replacement for ℤ^{[−1]} in DG^{≥0}dg_{≥0}Ab, so

\underline{MC}(A) ≃ R\underline{Hom}_{DG^{≥0}dg_{≥0}Ab}(ℤ^{[−1]}, N^{s}N_{c}A)

A simpler cofibrant replacement is the object L given by

L_{i}^{n} = \begin{cases}ℤ & n = i + 1, i + 2, \\ 0 & otherwise, \end{cases}

with chain and cochain differentials the identity whenever possible. Thus

\underline{MC}(A) ≃ \underline{Hom}_{DG^{≥0}dg_{≥0}Ab}(L, N^{s}N_{c}A).

Now, a map L → B is determined by the images b_{n} of the elements 1 ∈ L_{n−1}^{n}, subject to the conditions that d_{c}b_{n} = d^{s}b_{n+1}. Thus

Hom_{DG^{≥0}dg_{≥0}Ab}(L, B) ≅ Z_{−1}(Tot^{Π}σ^{≥1}B),

where Z_{i}(V) = ker(d : V_{i} → V_{i−1}), and the description of π_{n}\underline{MC}(A) follows. □

Proposition 4.13. For A ∈ csAb,

π_{n}Del_{A} ≅ H_{n−1}(Tot^{Π}N^{s}N_{c}A).

Proof. We need to understand the adjoint action of A^{0} on \underline{MC}(A). For g ∈ A^{0} and ω ∈ \underline{MC}(A), we have g^{−1} ∗ ω ∗ g = g^{−1} ∗ 0 ∗ g + ω, so the problem reduces to understanding the morphism D : A^{0} → \underline{MC}(A) given by D(g) = g^{−1} ∗ 0 ∗ g.

If we now consider the simplicial normalisation of [X/^{h}G], for X and G abelian and [−/^{h}−] as in Definition 3.14, we see that

N_{n}^{s}[X/^{h}G] ≅ N_{n}^{s}X ⊕ N_{n−1}^{s}G
(x, g, d^{s}g, 0, 0, …, 0) ↤ (x, g).

The corresponding differential is given by d^{s}(x, g) = (d^{s}x + g · 0, dg), so N^{s}[X/^{h}G] is isomorphic to the mapping cone of the morphism N^{s}G ⟶^{·0} N^{s}X.

If X = \underline{MC}(A) and G = A^{0}, then the gauge action on 0 is given by g · 0 = Dg, so N^{s}Del_{A} is isomorphic to the cone complex of the morphism

N^{s}A^{0} ⟶^{N^{s}D} N^{s}\underline{MC}(A).

Now,

\underline{Hom}_{csAb}(ι(ℤ), A) ≅ Z^{1}N_{c}A := ker(d_{c} : N_{c}^{1}A → N_{c}^{2}A),

with the map cot(Ξ) → ι(ℤ) corresponding to the inclusion f : Z^{1}N_{c}A ↪ \underline{MC}(A) given by f(a)_{n} = (∂^{1})^{n}(a) ∈ A^{n+1} ⊂ (A^{n+1})^{I^{n}}. The key observation is that this subset is closed under the adjoint action, so in particular D : A^{0} → \underline{MC}(A) factors through Z^{1}N_{c}A, via the map d_{c} : A^{0} → Z^{1}N_{c}A.

The map L → ℤ^{[−1]} corresponds to the natural inclusion (Z^{1}N_{c}A)_{n} ↪ Z_{−1}(Tot^{Π}σ^{≥1}N_{c}A^{Δ^{n}}). The proof of Proposition 4.12 gives an equivalence

N^{s}\underline{MC}(A) ≃ τ_{≥0}(Tot^{Π}(σ^{≥1}N_{c}A)[−1])

of chain complexes, where τ_{≥0} is good truncation in non-negative degrees (so

τ_{≥0}(V [−1])_{n} = \begin{cases}V_{n−1} & n > 0 \\ Z_{−1}V & n = 0 \\ 0 & n < 0). \end{cases}

Since this equivalence preserves the image of N^{s}\underline{Hom}_{csAb}(ι(ℤ), A), it also preserves the image of A^{0}, giving an equivalence between the cone complexes of

N^{s}A^{0} ⟶^{N^{s}D} N^{s}\underline{MC}(A) and N^{s}A^{0} ⟶^{d_{c}} τ_{≥0}(Tot^{Π}(σ^{≥1}N_{c}A)[−1]).

The latter cone complex is just

τ_{≥0}(Tot^{Π}(N^{s}N_{c}A)[−1]),

which has the required cohomology. □

4.2. Cohomology.

Definition 4.14. Given E ∈ QM^{∗}(𝕊) and A ∈ cAb, define cohomology groups of E with coefficients in A by

H^{i}(E, A) := Hom_{Ho(QM^{∗}(𝕊))}(E, N_{s}^{−1}A_{[−i]}),

where N_{s}^{−1}A_{[−i]} is the simplicial abelian group whose simplicial normalisation has A concentrated in chain degree i.

Proposition 4.15. For A ∈ cAb,

H^{i}(ι•, A) ≅ \begin{cases}H^{i+1}(N_{c}A) & i > 0 \\ Z^{1}N_{c}A & i = 0. \end{cases}

Proof. By Proposition 3.11, H^{i}(ι•, A) ≅ π_{0}\underline{MC}(N_{s}^{−1}A_{[−i]}), which by Proposition 4.12 is isomorphic to H_{−1}(Tot^{Π}σ^{≥1}N_{c}A_{[−i]}). Since Tot^{Π}σ^{≥1}N_{c}A_{[−i]})_{n} = (σ^{≥1}N_{c}A)^{i−n}, this is just H^{i+1}(σ^{≥1}N_{c}A), as required. □

4.3. Groupoids. Given Γ ∈ QM^{∗}(Gpd, ×) (or even in QM^{∗}(Cat, ×), we have BΓ ∈ QM^{∗}(𝕊), for B : Gpd → 𝕊 the nerve functor, and we now seek to describe the set MC(BΓ) (and hence the space \underline{MC}(BΓ)).

Proposition 4.16. For C ∈ QM^{∗}(Cat, ×), the set MC(BC) is isomorphic to the set of pairs (x, a), for x ∈ Ob C^{1}, a ∈ C^{2}(x ∗ x, ∂^{1}x) satisfying the following conditions:

σ^{0}x = 1
σ^{0}a = σ^{1}a = id_{x} ∈ C^{1}(x, x)
(∂^{2}a) ∘ (x ∗ a) = (∂^{1}a) ∘ (a ∗ x) ∈ C^{3}(x ∗ x ∗ x, ∂^{2}∂^{1}x),

where a ∗ x := a ∗ id_{x} and x ∗ a = id_{x} ∗ a, for id_{x} ∈ C^{1}(x, x) the identity morphism.

Proof. First, observe that

MC(BC) = Hom_{QM^{∗}(𝕊)}(Ξ, BC) ≅ Hom_{QM^{∗}(Cat)}(τ_{1}Ξ, C),

where τ_{1}Ξ is the fundamental category of Ξ (in the sense of [JT] §1).

It is therefore equivalent to show that τ_{1}Ξ is the quasi-comonoid in categories generated by an object ξ ∈ Ξ_{0}^{1} and a morphism α ∈ τ_{1}(Ξ^{2})(ξ ∗ ξ, ∂^{1}ξ) satisfying the conditions for (x, a) above.

Note that Ξ_{0} is the free quasi-comonoid (in sets) generated by the unique element ξ ∈ Ξ_{0}^{1}, subject to the condition that σ^{0}ξ = 1. Recall that Ξ_{0}^{n} = [0, 1]^{n−1} for n ≥ 1. We may then describe τ_{1}Ξ^{n} as the category associated to the poset [0, 1]^{n−1}. Thus τ_{1}Ξ^{2} has objects ξ ∗ ξ and ∂^{1}ξ (corresponding to 0 and 1 in [0, 1] respectively), with a unique isomorphism α : ξ ∗ ξ → ∂^{1}ξ. Since there is a unique morphism from (0, 0) to (1, 1) in τ_{1}Ξ^{3}, the morphisms

(∂^{2}α) ∘ (ξ ∗ α), (∂^{1}α) ∘ (α ∗ ξ) ∈ C^{3}(ξ ∗ ξ ∗ ξ, ∂^{2}∂^{1}ξ)

must be equal.

It therefore only remains to show that τ_{1}Ξ is isomorphic to the quasi-comonoid D defined to have objects Ob Ξ, with morphisms freely generated by α ∈ D(ξ ∗ ξ, ∂^{1}ξ), subject to the condition (∂^{2}α) ∘ (ξ ∗ α) = (∂^{1}α) ∘ (α ∗ ξ). Since the condition is satisfied by τ_{1}Ξ, there is a natural map D → τ_{1}Ξ in QM^{∗}(Cat).

Now, for u ∈ D^{m}(x, y) and v ∈ D^{n}(x′, y′), the fact that D^{m} × D^{n} ⟶^{∗} D^{m+n} is a functor implies that

(id_{y} ∗ v) ∘ (u ∗ id_{x′}) = u ∗ v = (u ∗ id_{y′}) ∘ (id_{x} ∗ v).

Therefore every morphism in D can be generated from α by the operations ∂^{i}, ξ∗, ∗ξ and composition. Thus every morphism is a composition of morphisms of the form

∂^{i_{r}} ⋯ ∂^{i_{1}}(ξ^{s} ∗ α ∗ ξ^{t}).

There are 2^{n−2}(n − 1) such morphisms in D^{n}, corresponding to edges in the (n − 1)-cube Ob D^{n} = [0, 1]^{n−1}, so we call these the edge morphisms. Since τ_{1}Ξ^{n} is generated by edge morphisms, this implies that the functor D^{n} → τ_{1}Ξ^{n} is full.

Finally, the condition (∂^{2}α) ∘ (ξ ∗ α) = (∂^{1}α) ∘ (α ∗ ξ) implies that any square of edge morphisms in D^{n} commutes. Thus D^{n} is the category associated to the poset [0, 1]^{n−1}, so D → τ_{1}Ξ is an isomorphism, as required. □

Corollary 4.17. For Γ ∈ QM^{∗}(Gpd),

Hom_{Ho(QM^{∗}(𝕊))}(ι•, BΓ) ≅ MC(BΓ)/ ∼,

where for pairs (x, a) as in Proposition 4.16, the equivalence relation ∼ is given by saying that (x, a) ∼ (x′, a′) if there exists λ ∈ Γ^{1}(x, x′) such that

(∂^{1}λ) ∘ a = a′ ∘ (λ ∗ λ), σ^{0}λ = id_{1},

where id_{1} is the identity morphism in Γ^{0}(1, 1).

Proof. Let Ar(Γ) ∈ QM^{∗}(Gpd) be the groupoid of arrows in Γ, defined levelwise, and observe that

BΓ ⟶^{id} BAr(Γ) → B(Γ × Γ)

is a path object for BΓ. Since Ξ is a cofibrant replacement for ι•, this gives

Hom_{Ho(QM^{∗}(𝕊))}(ι•, BΓ) ≅ Hom_{QM^{∗}(𝕊)}(Ξ, BΓ)/Hom_{QM^{∗}(𝕊)}(Ξ, BAr(Γ)),

which is just MC(BΓ)/MC(BAr(Γ)).

Applying Proposition 4.16 to Ar(Γ) shows that elements of MC(BAr(Γ)) correspond to pairs

(x ⟶^{λ} x′, \begin{matrix}x ∗ x & ⟶^{λ∗λ} & x′ ∗ x′ \\ ^{a}↓ & & ↓_{a′} \\ ∂^{1}x & ⟶^{∂^{1}λ} & ∂^{1}x′ \end{matrix}) ,

with (x, a), (x′, a′) ∈ MC(BΓ) and (∂^{1}λ) ∘ a = a′ ∘ (λ ∗ λ), σ^{0}λ = id_{1}. This gives the required description.

□

4.4. Linear quasi-comonoids. Given E ∈ QM^{∗}(𝕊), we now wish to describe the homology groups H_{∗}(E^{n}, ℤ), for all n — by definition, these are just homology groups of the simplicial abelian group ℤ ⊗ E freely generated by E. Applying ℤ⊗ levelwise gives a functor

ℤ⊗: QM^{∗}(𝕊) → QM^{∗}(sAb, ⊗),

where the latter category is not to be confused with the category QM^{∗}(sAb, ×) considered earlier. There is a forgetful functor, right adjoint to ℤ⊗, and [Hir] Theorem 11.3.2 allows us to put a model structure on QM^{∗}(sAb, ⊗) for which a morphism is a weak equivalence or a fibration if the underlying map in QM^{∗}(𝕊) is so.

This means that a morphism in QM^{∗}(sAb, ⊗) is a weak equivalence or a fibration whenever the underlying map in the Reedy category 𝕊^{Δ_{∗∗}} is so. Thus the forgetful functor QM^{∗}(sAb, ⊗) → (sAb)^{Δ_{∗∗}} is right Quillen, where the latter category has the Reedy model structure.

Alternatively, we can forget the operations ∂^{i}, σ^{i} and retain the multiplication. This gives us a forgetful functor U : QM^{∗}(Ab, ⊗) → GRing to simplicial (not necessarily commutative) ℕ_{0}-graded rings with unit, given by U(R)^{n} = R^{n}.

Lemma 4.18. The forgetful functor U : QM^{∗}(sAb, ⊗) → sGRing is left Quillen.

Proof. We first note that the model structure which we will use for sGRing is defined by saying that f : R → S is a weak equivalence (resp. a fibration) whenever all the maps f^{n} : R^{n} → S^{n} are weak equivalences (resp. fibrations) in 𝕊. That this defines a model structure follows from [Hir] Theorem 11.3.2.

We may explicitly describe the right adjoint U_{∗} by

(U_{∗}R)^{n} := ∏_{\begin{matrix}m∈ℕ_{0} \\ f:n→m in Δ \end{matrix}} R^{m},

where for any morphism g : n → n′ in Δ, the map g : (U_{∗}R)^{n} → (U_{∗}R)^{n′} is given by g(r)_{f} = r_{g∘f}, for f : n′ → m.

The matching maps of this all have canonical sections, so it follows immediately that U_{∗} preserves fibrations and trivial fibrations, hence is right Quillen. □

Definition 4.19. Given a (not necessarily commutative) ring R, and an R-bimodule M, define the set Der(R, M) of derivations to be the set of ring homomorphisms R → R ⊕ Mϵ over R, where ϵ^{2} = 0 and ϵ commutes with everything. Equivalently, a derivation is a morphism f : R → M such that f(ab) = af(b) + f(a)b.

Then (as in [DV]) define the R-bimodule Ω(R) to be the kernel of the multiplication map R ⊗ R → R; this has the universal property that Hom_{R−R}(Ω(R), M) ≅ Der(R, M), with the universal derivation R → Ω(R) given by r ↦ r ⊗ 1 − 1 ⊗ r.

We now also denote the forgetful functor QM^{∗}(Ab, ×) → GAb to ℕ_{0}-graded abelian groups by U.

Lemma 4.20. For E ∈ QM^{∗}(Set), there is a natural isomorphism

U cot(E) ≅ U(ℤ ⊗ ι•) ⊗_{U(ℤ⊗E)} Ω(U(ℤ ⊗ E)) ⊗_{U(ℤ⊗E)} U(ℤ ⊗ ι•).

Proof. Given a graded U(ℤ ⊗ ι•)-bimodule M, we have

Hom_{U(ℤ⊗E)−U(ℤ⊗E)}(Ω(U(ℤ ⊗ E)), M) = Der(U(ℤ ⊗ E), M),

making use of the augmentation U(ℤ ⊗ E) → U(ℤ ⊗ ι•) coming from the canonical map E → ι• to the final object.

Now,

Der(U(ℤ ⊗ E), M) = Hom_{GRing↓U(ℤ⊗ι•)}(U(ℤ ⊗ E), U(ℤ ⊗ ι•) ⊕ Mϵ),

for ϵ^{2} = 0, and this is isomorphic to Hom_{QM^{∗}(Set)↓U_{∗}U(ℤ⊗ι•)}(E, U_{∗}(U(ℤ ⊗ ι•) ⊕ Mϵ)).

Now, observe that cAb is equivalent to the category of ℤ ⊗ ι•-bimodules in Ab^{Δ_{∗∗}}, with operations ∂^{0} and ∂^{n+1} on level n corresponding to left and right multiplication by the unique element of (ι•)^{1}. Given a ℤ ⊗ ι•-bimodule N in Ab^{Δ_{∗∗}}, the equivalence cAb ≃ QM^{∗}(Ab, ×) then combines with the forgetful functor U_{Ab} : QM^{∗}(Ab, ×) → QM^{∗}(Set) to give rise to an object of QM^{∗}(Set). Explicitly, this is

((ℤ ⊗ ι•) ⊕ Nϵ) ×_{(ℤ⊗ι•)} ι•

(noting that the underlying object in Set^{Δ_{∗∗}} is just N).

Next, observe that the forgetful functor Ab^{Δ_{∗∗}} → GAb has right adjoint U_{∗}, defined by the same formulae as the functor U_{∗} of Lemma 4.18. Thus we get U_{∗}M ∈ Ab^{Δ_{∗∗}}, which has a natural ℤ⊗ι•-bimodule structure, allowing us to regard it as a cosimplicial complex. Moreover,

U_{Ab}U_{∗}M = ((ℤ ⊗ ι•) ⊕ U_{∗}Mϵ) ×_{(ℤ⊗ι•)} ι •
= ((U_{∗}U(ℤ ⊗ ι•)) ⊕ U_{∗}Mϵ) ×_{U_{∗}U(ℤ⊗ι•)} ι •
= U_{∗}(U(ℤ ⊗ ι•) ⊕ Mϵ) ×_{U_{∗}U(ℤ⊗ι•)} ι•

in QM^{∗}(Set). Since cot is left adjoint to U_{ab},

Hom_{cAb}(cot E, U_{∗}M) ≅ Hom_{QM^{∗}(Set)}(E, U_{∗}(U(ℤ ⊗ ι•) ⊕ Mϵ) ×_{U_{∗}U(ℤ⊗ι•)} ι•),

which we have already seen is isomorphic to Der(U(ℤ ⊗ E), M).

Thus

Hom_{U(ℤ⊗E)−U(ℤ⊗E)}(Ω(U(ℤ ⊗ E)), M) ≅ Hom_{cAb}(cot E, U_{∗}M)
≅ Hom_{U(ℤ⊗ι•)−U(ℤ⊗ι•)}(U cot E, M),

as required. □

Corollary 4.21. The morphism L cot ι• ≃ cot(ι•) is a weak equivalence.

Proof. We could use our explicit cofibrant replacement Ξ to calculate L cot, but instead we give an argument which will generalise more widely. Although ι• is not cofibrant, the underlying graded ring U(ℤ ⊗ ι•) is freely generated by the unique element of (ι•)^{1}, so is cofibrant in sGRing. If we took a cofibrant replacement E of ι•, we would then have a weak equivalence U(ℤ ⊗ E) → U(ℤ ⊗ ι•) of cofibrant objects. Therefore

U(ℤ ⊗ ι•) ⊗_{U(ℤ⊗E)} Ω(U(ℤ ⊗ E)) ⊗_{U(ℤ⊗E)} U(ℤ ⊗ ι•) → Ω(U(ℤ ⊗ ι•))

would be a weak equivalence of simplicial U(ℤ ⊗ ι•)-modules, so Lemma 4.20 gives a weak equivalence

cot(E) → cot(ι•),

as required. □

Proposition 4.22. Given a cofibrant object R ∈ sGRing, equipped with an augmentation R → ℤ, there is a spectral sequence

F (H_{∗}(ℤ ⊗_{R} Ω(R) ⊗_{R} ℤ)) ⟹ H_{∗}(R),

where F is the free graded (non-commutative) ℤ-algebra functor on a graded module. This converges whenever R^{0} = ℤ.

Proof. By [Qui2] §II.6, cofibrant simplicial rings are retracts of free simplicial rings, where a simplicial ring R_{•} is said to be free if there are free generators C_{q} ⊂ R_{q} for all q, closed under the degeneracy operations of R_{•}.

Since the proposition is unchanged by taking retractions, we may assume that R is free. If I = ker(R → ℤ) is the augmentation ideal, first observe that R = ℤ ⊕ I, and that freeness gives ℤ ⊗_{R} Ω(R) ⊗_{R} ℤ = I/(I · I). There is a filtration on R by powers of I, with associated graded algebra gr_{I}R = ⨁_{n≥0} I^{·n}/I^{·n+1}. Since R is free, the canonical map F (I/(I · I)) → gr_{I}R is an isomorphism. The spectral sequence of the proposition is then just the spectral sequence associated to this filtration.

Finally, if R^{0} = ℤ, then I^{0} = 0, so I^{·n+1} ∩R^{n} = 0. We may regard the spectral sequence as a direct sum of spectral sequences, using the graded decomposition. In degree n, this gives the spectral sequence associated to the filtration I^{·p} ∩ R^{n} = 0 of R^{n}. Since this filtration is bounded, the spectral sequence converges. □

Corollary 4.23. If f : E → F in QM^{∗}(Set) is a morphism with E^{0} = F^{0} = 1, π_{f}E ≃ π_{f}F ≃ • and L cot(E) ≃ L cot(F), then f is a weak equivalence. Moreover, if L cot(E) ≃ cot(ι•), then E → ι• is a weak equivalence.

Proof. We may choose cofibrant replacements Ẽ, F̃ of E, F, with Ẽ^{0} = F̃^{0} = 1. Since the objects Ẽ^{n}, F̃^{n} are simply connected for all n, we only need to prove that the map H_{∗}(Ẽ^{n}, ℤ) → H_{∗}(F̃^{n}, ℤ) of homology groups is an isomorphism. Now, H_{∗}(Ẽ^{n}, ℤ) = H_{∗}(ℤ⊗ Ẽ^{n}), so we may apply Lemma 4.20 and Proposition 4.22 to give the required isomorphism.

For the final part, note that Corollary 4.21 gives L cot(ι•) ≃ cot(ι•). □

4.5. Diagonals. In this section, we will study properties of the diagonal functor diag : QMM^{∗}(𝕊) → QM^{∗}(𝕊), with a view to characterising MC(diag E), thereby extending Lemma 1.21 to the simplicial case.

4.5.1. Groupoids. Our first step is to establish a diagonal version of Proposition 4.16.

Definition 4.24. Given a Δ_{∗∗} × Δ_{∗∗}-diagram S^{•,•} of sets, and a distinguished point 1 ∈ S^{n−1,i}, define (N_{h}^{n}S)^{i} = S^{n,i} ∩ ⋂_{j} ker(σ_{h}^{j}), where ker denotes the inverse image of 1. Similarly, given 1 ∈ S^{i,n−1}, define (N_{v}^{n}S)^{i} = S^{i,n} ∩ ⋂_{j} ker(σ_{v}^{j}).

Lemma 4.25. Given Γ ∈ QMM^{∗}(Gpd), the set MC(diag BΓ) consists of data (x, s, t, a, b), where x ∈ Ob Γ^{11}, and for x_{h} := σ_{v}^{0}x, x_{v} := σ_{h}^{0}x,

s ∈ Γ^{20}(x_{h} ∗ x_{h}, ∂_{h}^{1}x_{h}), t ∈ Γ^{02}(x_{v} ∗ x_{v}, ∂_{v}^{1}x_{v})

and

a ∈ N_{h}N_{v}Γ^{11}(x_{h} ∗ x_{v}, x) b ∈ N_{h}N_{v}Γ^{11}(x_{v} ∗ x_{h}, x).

These data satisfy the additional conditions that

(x_{h}, s) ∈ MC(BΓ^{•0}), (x_{v}, t) ∈ MC(BΓ^{0•})

(as in Lemma 4.16), and that if we set γ = b^{−1} ∘ a : x_{h} ∗ x_{v} → x_{v} ∗ x_{h}, then

∂_{h}^{1}γ = (x_{v} ∗ s) ∘ (γ ∗ x_{h}) ∘ (x_{h} ∗ γ) ∘ (s ∗ x_{v})^{−1}
∂_{v}^{1}γ^{−1} = (x_{h} ∗ t) ∘ (γ^{−1} ∗ x_{v}) ∘ (x_{v} ∗ γ^{−1}) ∘ (t ∗ x_{h})^{−1}.

Proof (sketch). By Proposition 4.16, we know that MC(diag BΓ) consists of pairs (x, α), with x ∈ Ob Γ^{11}, α ∈ Γ^{22}(x ∗ x, ∂_{h}^{1}∂_{v}^{1}x), satisfying various conditions. If we look at a := σ_{v}^{0}σ_{h}^{1}α, b := σ_{h}^{0}σ_{v}^{1}α, then we have a : σ_{v}^{0}x ∗ σ_{h}^{0}x → x and b : σ_{h}^{0}x ∗ σ_{v}^{0}x → x. We also set s = (σ_{v}^{0})^{2}α ∈ Γ^{20}, t = (σ_{h}^{0})^{2}α ∈ Γ^{02}.

Note that

(σ_{v}^{0}x, s) ∈ MC(BΓ^{•0}), (σ_{h}^{0}x, t) ∈ MC(BΓ^{0•}), a, b ∈ N_{h}N_{v}Γ^{11},

by applying powers of σ_{h}^{0} or σ_{v}^{0} to the equations for α.

Applying the operations σ_{h}^{i}σ_{v}^{j}σ_{v}^{k} to the equation

(∂_{h}^{2}∂_{v}^{2}α) ∘ (x ∗ α) = (∂_{h}^{1}∂_{v}^{1}α) ∘ (α ∗ x)

for {i, j, k} = {1, 2, 3} and j < k gives us the following equations in σ_{v}^{0}α, σ_{v}^{1}α:

σ_{v}^{1}α = (∂_{h}^{1}b) ∘ (b ∗ x_{h})^{−1}
(σ_{v}^{0}α)^{−1} ∘ (σ_{v}^{1}α) = (x_{h} ∗ b) ∘ (a ∗ x_{h})^{−1}
σ_{v}^{0}α = (∂_{h}^{1}a) ∘ (x_{h} ∗ a)^{−1},

which reduce to the first condition for γ. Interchanging horizontal and vertical structures does the same for σ_{h}^{0}α, σ_{h}^{1}α, giving the second condition (for γ^{−1}).

It remains to show that we can recover α from s, t, a, b. It is the composition

(∂_{h}^{1}∂_{v}^{1}a) ∘ (s ∗ t) ∘ (x_{h} ∗ γ ∗ x_{v}) ∘ (a^{−1} ∗ a^{−1}).
□

Proposition 4.26. For Γ ∈ QMM^{∗}(Gpd), elements of Hom_{Ho(QM^{∗}(𝕊))}(ι•, Bdiag Γ) are represented by data of the form (x_{h}, x_{v}, s, t, γ), where

(x_{h}, s) ∈ MC(BΓ^{•,0}), (x_{v}, t) ∈ MC(BΓ^{0,•}),

and γ : x_{h} ∗ x_{v} → x_{v} ∗ x_{h} satisfies the conditions of Lemma 4.25. Two systems (x_{h}, x_{v}, s, t, γ), (x′_{h}, x′_{v}, s′, t′, γ′) represent the same element if and only if there exist

λ_{h} ∈ Γ^{1,0}(x_{h}, x′_{h}), λ_{v} ∈ Γ^{0,1}(x_{v}, x′_{v})

such that

(∂_{h}^{1}λ_{h}) ∘ s = s′ ∘ (λ_{h} ∗ λ_{h}) σ_{h}^{0}λ_{h} = 1,
(∂_{v}^{1}λ_{v}) ∘ t = t′ ∘ (λ_{v} ∗ λ_{v}) σ_{v}^{0}λ_{v} = 1,
γ′ ∘ (λ_{h} ∗ λ_{v}) = (λ_{v} ∗ λ_{h}) ∘ γ.

Proof (sketch). Take (x, α) as in the proof of Lemma 4.25. By Corollary 4.17, (x, α) ∼ (x′, α′) whenever there exists λ ∈ Γ^{11}(x, x′) such that

(∂_{h}^{1}∂_{v}^{1}λ) ∘ α = α′ ∘ (λ ∗ λ) σ_{h}^{0}σ_{v}^{0}λ = 1.

Given (x, α), we may therefore set

x′ = x_{h} ∗ x_{v}, α′ = (∂_{h}^{1}∂_{v}^{1}a^{−1}) ∘ α ∘ (a ∗ a),

and define a transformation λ : (x, α) → (x′, α′) by λ = a^{−1}.

Therefore every element of Hom_{Ho(QM^{∗}(𝕊))}(ι•, Bdiag Γ) has a representative with x = x_{h} ∗x_{v}, and a = 1, giving data (x_{h}, x_{v}, s, t, γ) as above. Two such systems are equivalent if there exists a transformation λ ∈ Γ^{11}(x_{h} ∗ x_{v}, x′_{h} ∗ x′_{v}) satisfying the conditions of Lemma 4.17. Since a = 1, we recover that λ = λ_{h}∗λ_{v} for λ_{h} := σ_{v}^{0}λ and λ_{v} := σ_{h}^{0}λ. The conditions for λ then reduce to the conditions for λ_{h}, λ_{v} above. □

Corollary 4.27. The object diag^{∗}(Ξ) ∈ QMM^{∗}(𝕊) is simply connected in every level.

Proof. By Lemma 4.25, the fundamental groupoid Γ := π_{f}(diag^{∗}Ξ) ∈ QMM^{∗}(Gpd) is generated by an object x ∈ Ob Γ^{1,1}, together with isomorphisms (s, t, a, b) satisfying the conditions of that Lemma. Consider Υ ∈ QMM^{∗}(Gpd) generated by objects x_{h} ∈ Ob Γ^{1,0}, x_{v} ∈ Ob Γ^{0,1} and isomorphisms (s, t, γ) satisfying the conditions of Lemma 4.26. That Lemma implies that the canonical inclusion Υ → π_{f}diag^{∗}Ξ is a levelwise equivalence.

Now, the objects of Υ are words in (∂_{h}^{1})^{i}x_{h}, (∂_{v}^{1})^{j}x_{v}, and the conditions on (s, t, a, b) ensure that there is a unique isomorphism between any two such words in the same level, so Υ^{mn} is simply connected. Thus the maps Υ → π_{f}diag^{∗}Ξ → ι(•, •) are equivalences in every level, as required. □

4.5.2. Abelian groups.

Lemma 4.28. The category QMM^{∗}(Ab, ×) is equivalent to the category ccAb of bicosimplicial abelian groups.

Proof. The proof of Lemma 4.1 carries over to this context. □

Definition 4.29. Define cot : QMM^{∗}(Set) → QMM^{∗}(Ab, ×) ≃ ccAb to be left adjoint to the forgetful functor QMM^{∗}(Ab, ×) → QMM^{∗}(Set).

The following results have the same proofs as Corollary 4.21, Proposition 4.22 and Corollary 4.23, replacing the category GRing of graded rings with the category GGRing of bigraded rings.

Lemma 4.30. The morphism L cot ι(•, •) → cot ι(•, •) is a weak equivalence.

Proposition 4.31. Given a cofibrant object R ∈ sGGRing, equipped with an augmentation R → ℤ there is a spectral sequence

F (H_{∗}(ℤ ⊗_{R} cot(R) ⊗_{R} ℤ)) ⟹ H_{∗}(R),

where F is the free bigraded (non-commutative) ℤ-algebra functor on a bigraded module. This converges whenever R^{00} = ℤ.

Corollary 4.32. If f : E → F in QMM^{∗}(Set) is a morphism with E^{00} = F^{00} = 1, π_{f}E ≃ π_{f}F ≃ • and L cot(E) ≃ L cot(F), then f is a weak equivalence. Moreover, if L cot(E) ≃ cot ι(•, •), then E → ι(•, •) is a weak equivalence.

Proposition 4.33. The map Ldiag^{∗}(ι•) → ι(•, •) is a weak equivalence.

Proof. Proposition 4.26 shows that this gives an equivalence on fundamental groupoids, so by Corollary 4.32 it suffices to show that

L cot(Ldiag^{∗}(ι•)) ≃ L cot(ι(•, •)).

Now the diagonal functor diag : QMM^{∗}(Ab, ×) → QM^{∗}(Ab, ×) also has a left adjoint diag _{Ab}^{∗} : QM^{∗}(Ab, ×) → QMM^{∗}(Ab, ×), which can be calculated in a similar way to the functor d^{∗} in [GJ] §IV.3.3, by studying the associated cosimplicial and bicosimplicial complexes. It follows from this description that diag _{Ab}^{∗} preserves weak equivalences. Moreover, the functors diag _{Ab}^{∗} cot and cot diag^{∗} from QM^{∗}(Set) to QMM^{∗}(Ab, ×) are naturally isomorphic, since their right adjoints are.

By Corollary 4.30, L cot ι(•, •) ≃ cot ι(•, •), so it suffices to show that

diag _{Ab}^{∗} cot Ξ → cot ι(•, •) = diag _{Ab}^{∗} cot ι•

is a weak equivalence. Since diag _{Ab}^{∗} preserves weak equivalences in QM^{∗}(Ab, ×), we need only observe that

cot Ξ → cot ι•

is a weak equivalence by Corollary 4.21. □

5. Mapping spaces

Lemma 5.1. For n as in Lemma 1.26, P_{n}^{∗}diag^{∗}Ξ is a cofibrant replacement for alg^{∗}n in sQDat_{n} (with the model structure of Lemma 3.17), and hence also in sQDat.

Proof. By Proposition 4.33, the morphism diag^{∗}Ξ → ι(•, •) in QMM^{∗}(𝕊) is a weak equivalence. From the description of P_{n}^{∗} in Lemma 1.32, it follows that P_{n}^{∗} from Definition 3.23 preserves weak equivalences, so P_{n}^{∗}diag^{∗}Ξ → P_{n}^{∗}ι(•, •) is also a weak equivalence. Since P_{n}^{∗} : QMM^{∗}(𝕊) → sQDat_{n} is left Quillen, P_{n}^{∗}diag^{∗}Ξ is cofibrant (and a realisation of P_{n}^{∗}Ldiag^{∗}ι(•)). Now we need only recall from the proof of Lemma 1.34 that P_{n}^{∗}ι(•, •) ≅ alg^{∗}n. □

5.1. MC.

Proposition 5.2. For D ∈ sQDat fibrant,

MC(D)_{n} ≃ ∐_{f:[0,n]→Ob D} Map_{sQDat_{n}}^{h}(alg^{∗}n, f^{−1}D),

where Map^{h} denotes the derived function complex RMap of [Hov] §5.

Proof. For E ∈ sQDat_{n},

\underline{MC}(diag P_{n}E) = \underline{Hom}_{sQDat_{n}}(P_{n}^{∗}diag^{∗}Ξ, E),

recalling that (unlike sQDat) the model category sQDat_{n} has a simplicial structure. Since P_{n}^{∗}diag^{∗}Ξ is cofibrant, this is equivalent to the derived function complex Map_{sQDat_{n}}^{h}(P_{n}^{∗}diag^{∗}Ξ, E) whenever E is fibrant ([Hov] Theorem 5.4.9). Since P_{n}^{∗}diag^{∗}Ξ is weakly equivalent to alg^{∗}n, this function complex is also weakly equivalent to Map_{sQDat_{n}}^{h}(alg^{∗}n, E). The description now follows immediately from definition 3.25. □

Corollary 5.3. For D ∈ sQDat fibrant,

MC(D)_{n} ≃ Map_{sQDat}^{h}(alg^{∗}n, D) ×_{Map_{sCat}^{h}(0,D^{0})^{[0,n]}}^{h} (Ob D)^{[0,n]}.

Proof. Comparing the model structures of Lemma 3.17 and Proposition 3.19, it follows immediately that the functor O ↓ sQDat → sQDat_{O}, given by mapping f : O → D to f^{−1}D, preserves (trivial) fibrations. Its left adjoint is the inclusion functor sQDat_{O} → O ↓ sQDat, so these form a Quillen pair. Hence

Map_{sQDat_{n}}^{h}(alg^{∗}n, f^{−1}D) ≃ Map_{[0,n]↓sQDat}^{h}(alg^{∗}n, [0, n] ⟶^{f} D)
= Map_{sQDat}^{h}(alg^{∗}n, D) ×_{Map_{sQDat}^{h}([0,n],D)}^{h} {f}.

Thus we have that MC(D)_{n} = Map^{h}(alg^{∗}n, D) ×_{Map^{h}([0,n],D)}^{h} Hom([0, n], D), and

Map_{sQDat}^{h}([0, n], D) = Map_{sCat}^{h}([0.n], D^{0}) = Map_{sCat}^{h}(0, D^{0})^{[0,n]}.
□

Definition 5.4. For n ∈ ℕ_{0}, define Ξ × alg^{∗}n ∈ sQDat to have objects [0, n] and morphisms (Ξ × alg^{∗}n)(i, j) = Ξ for i ≤ j and ∅ otherwise. This can be characterised as the coproduct (Ξ × alg^{∗}(Ob n)) ⊔_{(id/∅)[0,n]} (id/∅)(n) in the category sQDat, or equivalently in the category sQDat_{n}.

Lemma 5.5. The natural morphism f : Ξ × alg^{∗}n → P_{n}^{∗}diag^{∗}Ξ is a trivial cofibration in sQDat_{n}.

Proof. For D ∈ sQDat_{n},

Hom_{sQDat_{n}}(Ξ × alg^{∗}n, D) = MC((P_{n}D)^{•,0} ×_{(P_{n}D)^{0,0}} (P_{n}D)^{0,•}).

The morphism f then corresponds to the map

((σ_{v}^{0})^{•}, (σ_{h}^{0})^{•}) : diag (P_{n}D) → (P_{n}D)^{•,0} ×_{(P_{n}D)^{0,0}} (P_{n}D)^{0,•}

in QM^{∗}(𝕊).

Given a trivial fibration D → E in sQDat_{n}, observe that the conditions (W1) and (F1) from Proposition 3.19 ensure that

diag (P_{n}D) → diag (P_{n}E) ×_{(P_{n}E)^{•,0}×_{(PnE)^{0,0}}(P_{n}E)^{0,•}} (P_{n}D)^{•,0} ×_{(P_{n}D)^{0,0}} (P_{n}D)^{0,•}

is a trivial fibration in QM^{∗}(𝕊), so Lemma 3.9 implies that the functor MC applied to this map is surjective. Therefore f is a cofibration.

Now, Ξ × alg^{∗}n ≃ (ι•) × alg^{∗}n = alg^{∗}n and P_{n}^{∗}diag^{∗}Ξ ≃ alg^{∗}n by Lemma 5.1. Thus f is a trivial cofibration in sQDat_{n}. □

Definition 5.6. Recall from [Rez] 4.1 that a Segal space is defined to be a bisimplicial set W ∈ s𝕊 which is Reedy fibrant, and for which the natural maps

W_{k} → \overbrace{W_{1} ×_{∂_{0},W_{0} ∂_{1}} … ×_{∂_{0},W_{0},∂_{1}} W_{1}}^{k}

are weak equivalences in 𝕊 for all k.

Proposition 5.7. For D ∈ sQDat fibrant, MC(D) is a Segal space.

Proof. By applying Lemma 3.28 to the morphism D → alg^{∗}0, we know that MC(D) is Reedy fibrant, since cosk_{0}(Ob D)^{hor} is Reedy fibrant, and MC(alg^{∗}0) = •.

Letting W := MC(D), we have

(W_{n})_{i} = ∐_{f:[0,n]→Ob D} \underline{Hom}_{sQDat_{n}}(P_{n}^{∗}diag^{∗}Ξ, f^{−1}D)_{i}

= ∐_{f:[0,n]→Ob D} Hom_{sQDat_{n}}((P_{n}^{∗}diag^{∗}Ξ), (f^{−1}D)^{Δ^{i}})

= Hom_{sQDat}((P_{n}^{∗}diag^{∗}Ξ), D^{Δ^{i}}),

where D^{K} is defined by Ob D^{K} = Ob D and \underline{Hom}_{D^{K}}(a, b) := \underline{Hom}_{D}(a, b)^{K} (note that although this makes sQDat into a simplicial category, it does not satisfy axiom (SM7) of a simplicial model category).

Now,

\overbrace{(W_{1} ×_{∂_{0},W_{0},∂_{1}} … ×_{∂_{0},W_{0},∂_{1}} W_{1}}^{k})_{i} ≅ Hom_{sQDat}((P_{1}^{∗}diag^{∗}Ξ)∪_{P_{0}^{∗}Ξ} … ∪_{P_{0}^{∗}Ξ}(P_{1}^{∗}diag^{∗}Ξ), D^{Δ^{i}})
≅ ∐_{f:[0,n]→Ob D} \underline{Hom}_{sQDat_{n}}((P_{1}^{∗}diag^{∗}Ξ)∪_{Ξ} … ∪_{Ξ}(P_{1}^{∗}diag^{∗}Ξ), f^{−1}D)_{i},

since [0, 1] ∪_{{1}} [1, 2] ∪_{{2}} … ∪_{{n−1}} [n − 1, n] = [0, n].

Now, Lemma 5.5 implies that

(Ξ × alg^{∗}1)∪_{Ξ} … ∪_{Ξ}(Ξ × alg^{∗}1) → P_{1}^{∗}diag^{∗}Ξ∪_{Ξ} … ∪_{Ξ}P_{1}^{∗}diag^{∗}Ξ

is a trivial cofibration (being a pushout of trivial cofibrations), and the left-hand side is just Ξ × alg^{∗}(1∪_{0} … ∪_{0}1) = Ξ × alg^{∗}k. This is weakly equivalent to alg^{∗}k, so

\overbrace{W_{1} ×_{∂_{0},W_{0},∂_{1}} × … ×_{∂_{0},W_{0},∂_{1}} W_{1}}^{k} ≃ ∐_{f:[0,n]→Ob D} Map_{sQDat_{n}}^{h}(alg^{∗}k, f^{−1}D),

which is equivalent to W_{k} by Proposition 5.2. □

5.1.1. Morphism spaces.

Lemma 5.8. Given D ∈ QDat, and x, y ∈ Ob alg D, the object D(x̄, ȳ) ∈ Set^{Δ_{∗∗}} has the natural structure of a cosimplicial set, where x̄, ȳ ∈ Ob D are the objects underlying x, y.

Proof. The objects x, y correspond to elements ω_{x} ∈ MC(D(x, x)), ω_{y} ∈ MC(D(y, y)). In order to enhance the structure of D(x̄, ȳ), we define operations ∂^{0} := ω_{x}∗ : D(x, y)^{n} → D(x, y)^{n+1}, and ∂^{n+1} := ∗ω_{y} : D(x, y)^{n} → D(x, y)^{n+1}. The Maurer-Cartan equations ensure that these operations satisfy the necessary conditions for a cosimplicial set. □

Definition 5.9. Recall from [GJ] §VIII.1 that the functor Tot : c𝕊 → 𝕊 from cosimplicial simplicial sets to simplicial sets is given by

Tot X^{•} = {x ∈ ∏_{n}(X^{n})^{Δ^{n}} : ∂_{X}^{i}x_{n} = (∂_{Δ}^{i})^{∗}x_{n+1}, σ_{X}^{i}x_{n} = (σ_{Δ}^{i})^{∗}x_{n−1}}.

When X is Reedy fibrant, homotopy groups of the total space are related to homotopy groups of the spaces X^{n} via a spectral sequence given in [GJ] Proposition VIII.1.15.

Proposition 5.10. Given D ∈ sQDat fibrant, and x, y ∈ Ob alg (D_{0}), there is a natural weak equivalence

MC(D)_{1} ×_{MC(D)×MC(D)} {(x, y)} ≃ Tot D(x̄, ȳ).

Proof. Define f : [0, 1] → Ob D by f(0) = x̄, f(1) = ȳ, and set E := f^{−1}D ∈ sQDat_{1} (for sQDat_{n} as in Lemma 1.32). Then

MC(D)_{1} ×_{MC(D)×MC(D)} \{(x, y)\} = \underline{Hom}_{(Ξ⊔Ξ)↓sQDat_{1}}(P_{1}^{∗}diag^{∗}Ξ, E)
= \underline{Hom}_{alg^{∗}(0⊔0)↓sQDat_{1}}((P_{1}^{∗}diag^{∗}Ξ)∪_{(Ξ⊔Ξ)}alg^{∗}(0 ⊔ 0), E).

By Lemma 5.5,

(P_{1}^{∗}diag^{∗}Ξ)∪_{(Ξ⊔Ξ)}alg^{∗}(0 ⊔ 0) ≃ (Ξ × alg^{∗}1)∪_{(Ξ⊔Ξ)}alg^{∗}(0 ⊔ 0) = alg^{∗}1,

so our expression reduces to R\underline{Hom}_{alg^{∗}(0⊔0)↓sQDat_{1}}(alg^{∗}1, E).

Now, in the simplicial category alg^{∗}(0 ⊔ 0)↓sQDat_{1}, a cofibrant replacement for alg^{∗}1 is given by the object C with C(0, 0) = C(1, 1) = ι•, C(1, 0) = ∅ and C(0, 1)^{•} = Δ^{•}. The multiplicative structure is determined determined by setting ω_{0} ∗ a = ∂^{0}a and a ∗ ω_{1} = ∂^{n+1}a, for a ∈ Δ^{n} and ω_{0}, ω_{1} the unique elements of C(0, 0)^{1} and C(1, 1)^{1}.

Now,

\underline{Hom}_{alg^{∗}(0⊔0)↓sQDat_{1}}(C, E) = \{e ∈ ∏_{n}(E(0, 1)^{n})^{Δ^{n}} : ∂_{E}^{i}e_{n} = (∂_{Δ}^{i})^{∗}e_{n+1}, σ_{E}^{i}e_{n} = (σ_{Δ}^{i})^{∗}e_{n−1}\}

(which preserves (trivial) fibrations, proving that C is cofibrant).

This expression is just Tot E(0, 1), where E(0, 1) is given the cosimplicial structure of Lemma 5.8, but E(0, 1) = D(x̄, ȳ), giving the required description. □

5.2. N.

Definition 5.11. Given D ∈ sQDat, define N (D) ∈ s𝕊 by N (D)_{n} := Map_{sQDat}^{h}(alg^{∗}n, D).

Proposition 5.12. Any Reedy fibrant replacement for N (D) is a Segal space.

Proof. If W := N (D), we need to show that the maps

W_{k} → \overbrace{W_{1} ×_{∂_{0},W_{0} ∂_{1}}^{h} … ×_{∂_{0},W_{0},∂_{1}}^{h} W_{1}}^{k}

are weak equivalences for all k.

Now, the right hand side is given by

Map_{sQDat}^{h}(alg^{∗}1∪_{alg^{∗}0}^{L} … ∪_{alg^{∗}0}^{L}alg^{∗}1, D),

so we need only show that alg^{∗}1∪_{alg^{∗}0}^{L} … ∪_{alg^{∗}0}^{L}alg^{∗}1 ≃ alg^{∗}k.

A cofibrant replacement of the diagram for this homotopy colimit is given by taking Ξ instead of alg^{∗}0, and P_{1}^{∗}diag^{∗}Ξ instead of alg^{∗}1. Thus the calculation of Proposition 5.7 can be interpreted as saying that

\overbrace{alg^{∗}1∪_{alg^{∗}0}^{L} … ∪_{alg^{∗}0}^{L}alg^{∗}1}^{k} ≃ alg^{∗}k,

which gives the required result. □

Definition 5.13. Given a Segal space W, define Ob W := (W_{0})_{0}. For x, y ∈ Ob W, define map_{W} (x, y) := {x}×_{W_{0},∂_{1}} W_{1}×_{∂_{0}} {y} ∈ 𝕊. There is a natural category Ho(W) with objects Ob W and morphisms π_{0}map_{W} (x, y).

Definition 5.14. Recall from [Rez] 7.4 that a morphism f : U → V of Segal spaces is said to be a Dwyer-Kan equivalence if

(1) Ho(f) : Ho(U) → Ho(V) is an equivalence of categories, and

(2) for all x, y ∈ Ob U, the map map_{U}(x, y) → map_{V} (fx, fy) is a weak equivalence in 𝕊.

Proposition 5.15. Given a fibrant object D ∈ sQDat, the natural transformation MC(D) → N (D)^{f} is a Dwyer-Kan equivalence of Segal spaces, where (−)^{f} denotes Reedy fibrant replacement.

Proof. Corollary 5.3 amounts to saying that MC(D) is weakly equivalent to the homotopy fibre product of

N (D) → cosk_{0}Map_{sQDat}^{h}(0, D^{0})^{hor} ← cosk_{0}(Ob D)^{hor}.

Thus, for x, y ∈ Ob MC(D), the space map_{MC(D)}(x, y) is is weakly equivalent to the homotopy fibre product of

map_{N (D)^{f}} (x, y) → map_{cosk_{0}Map_{sQDat}^{h}(0,D^{0})^{hor}}(x, y) ← map_{cosk_{0}(Ob D)^{hor}}(x, y).

Now, for any S ∈ 𝕊 and x, y ∈ S_{0}, map_{cosk_{0}(S)^{hor}}(x, y) = {(x, y)}, so

map_{MC(D)}(x, y) ≃ map_{N (D)^{f}} (x, y).

It therefore remains only to show that the morphism Ho(MC(D)) → Ho(N (D)^{f}) of categories is essentially surjective. Since 0 is cofibrant, the map Ob D → Map_{sQDat}^{h}(0, D^{0}) is surjective on π_{0}. Any fibrant replacement (Ob D)^{f} → Map_{sQDat}^{h}(0, D^{0}) will also be surjective on π_{0}, and hence surjective on level 0 (by path-lifting). Therefore the map Ob MC(D) → Ob N (D)^{f} is surjective on objects, so a fortiori essentially surjective. □

Definition 5.16. Given a Segal space W, let W_{hoequiv} ⊂ W_{1} consist of components whose images in π_{0}map_{W} (x, y) are equivalences in Ho(W). W is said to be a complete Segal space if the map σ_{0} : W_{0} → W_{hoequiv} is a weak equivalence.

Lemma 5.17. Given a levelwise trivial cofibration E → F in 𝕊^{Δ_{∗∗}}, and X ∈ Set^{Δ_{∗∗}^{opp}}, the map

X\underleftarrow{×}E → X\underleftarrow{×}F

is a trivial cofibration in 𝕊, for \underleftarrow{×} as in Definition 1.27.

Proof. Since Set^{Δ_{∗∗}} is equivalent to the category of augmented simplicial objects by Remark 1.2, a morphism E → F is a levelwise trivial cofibration precisely when E^{0} → F^{0} is a trivial cofibration and E^{≥1} → F^{≥1} corresponds to a levelwise trivial cofibration of bisimplicial sets. By [GJ] Theorem IV.3.9, this second condition is the same as being a Reedy trivial cofibration of bisimplicial sets.

Let L^{n}E be the Reedy latching object of E in Set^{Δ_{∗∗}} (as in [Hov] Definition 5.2.2). Explicitly, this is the quotient of ∐_{i=1}^{n−1} E^{n−1} by the relations (∂^{j}e)_{i} ∼ (∂^{i−1}e)_{j} for e ∈ E^{n−2} and i ≤ j. Note that L^{0}E = L^{1}E = ∅. For E^{≥1} → F^{≥1} to be a Reedy trivial cofibration says that the latching maps E^{n} ∪_{L^{n}E} L^{n}F → F^{n} are trivial cofibrations for all n ≥ 1. Thus a morphism E → F is a levelwise trivial cofibration in 𝕊^{Δ_{∗∗}} precisely when the latching maps are trivial cofibrations for all n ≥ 0.

Let (X\underleftarrow{×}E)^{(n)} ⊂ X\underleftarrow{×}E be the subspace generated by X_{i} × E^{i} for i ≤ n, and let N_{n}X = X_{n} − (⋃_{0≤r≤n−1} σ_{r}X_{n−1}). Then

(X\underleftarrow{×}E)^{(n)} = (X\underleftarrow{×}E)^{(n−1)} ∪_{(N_{n}X×L^{n}E)} (N_{n}X × E^{n}),

so the map

(X\underleftarrow{×}E)^{(n)} ∪_{(X\underleftarrow{×}E)^{(n−1)}} (X\underleftarrow{×}F)^{(n−1)} → (X\underleftarrow{×}F)^{(n)}

is a trivial cofibration.

We then proceed inductively to show that (X\underleftarrow{×}E)^{(n)} → (X\underleftarrow{×}F)^{(n)} is a trivial cofibration. The case n = 0 is immediate, and assuming the n − 1 case gives a trivial cofibration

(X\underleftarrow{×}E)^{(n)} → (X\underleftarrow{×}E)^{(n)} ∪_{(X\underleftarrow{×}E)^{(n−1)}} (X\underleftarrow{×}F)^{(n−1)}.

The result above then implies that

(X\underleftarrow{×}E)^{(n)} → (X\underleftarrow{×}F)^{(n)}

is a trivial cofibration, which completes the induction.

Thus X\underleftarrow{×}E → X\underleftarrow{×}F is a transfinite composition of trivial cofibrations, so must be a trivial cofibration, as required. □

Proposition 5.18. If 𝕀 is a small category and D ∈ sQDat, then there are canonical equivalences

Map_{s𝕊}^{h}((B𝕀)^{hor}, N (D)) ≃ Map_{sQDat}^{h}(alg^{∗}𝕀, D).

Proof. Lemma 5.1 shows that P_{n}^{∗}diag^{∗}Ξ is a cofibrant replacement for alg^{∗}n, so for K ∈ 𝕊,

Map_{s𝕊}^{h}(K^{hor}, N (D)) ≃ holim_{\overleftarrow{n∈Δ↓K}} Map_{s𝕊}^{h}((Δ^{n})^{hor}, N (D))
≃ holim_{\overleftarrow{n∈Δ↓K}} Map_{sQDat}^{h}(P_{n}^{∗}diag^{∗}Ξ, D)
≃ Map_{sQDat}^{h}( holim_{\overrightarrow{n∈(Δ↓K)^{opp}}} P_{n}^{∗}diag^{∗}Ξ, D)
= Map_{sQDat}^{h}(P_{K}^{∗}diag^{∗}Ξ, D).

Now, Lemma 1.35 gives a method for calculating the functor P_{B𝕀}^{∗}. Since (diag^{∗}Ξ)^{n,•} ∈ 𝕊^{Δ_{∗∗}} is levelwise contractible for all n (by Proposition 4.33), a choice of point x ∈ (diag^{∗}Ξ)^{n,1} determines a levelwise trivial cofibration ι• → (diag^{∗}Ξ)^{n,•} in 𝕊^{Δ_{∗∗}}. Therefore Lemma 5.17 implies that

K(a, b)\underleftarrow{×}(ι•) → K(a, b)\underleftarrow{×}(diag^{∗}Ξ)^{n,•}

is a trivial cofibration for all a, b ∈ K_{0}.

By Lemma 1.29, (P_{B𝕀}^{∗}diag^{∗}Ξ)(a, b)^{n} = (B𝕀)(a, b)\underleftarrow{×}(diag^{∗}Ξ)^{n,•}, so the calculation above shows that the canonical map

(P_{B𝕀}^{∗}diag^{∗}Ξ)(a, b)^{n} → (B𝕀)(a, b)\underleftarrow{×}(ι•)

is a weak equivalence, so

P_{B𝕀}^{∗}diag^{∗}Ξ → P_{B𝕀}^{∗}ι(•, •)

is a weak equivalence, and P_{B𝕀}^{∗}ι(•, •) ≅ alg^{∗}𝕀 as in Lemma 1.36. □

Corollary 5.19. Any Reedy fibrant replacement of N (D) is a complete Segal space.

Proof. We know that N (D)^{f} is a Segal space by Proposition 5.12. If we let J be the simply connected groupoid on two objects, [Rez] Proposition 6.4 implies that it suffices to show that the map

N (D)_{0}^{f} → Map_{s𝕊}^{h}((BJ)^{hor}, N (D)^{f})

is a weak equivalence.

By Proposition 5.18,

Map_{s𝕊}^{h}((B𝕀)^{hor}, N (D)) ≃ Map_{sQDat}^{h}(alg^{∗}𝕀, D),

and alg^{∗}J ≃ alg^{∗}0, since J is simply connected. The result then follows by considering this equivalence in the cases 𝕀 = J and 𝕀 = 0. □

Definition 5.20. Given a simplicial category 𝕀, let B𝕀 ∈ s𝕊 be the nerve of 𝕀, as considered in [Ber2] §8 (where it was denoted R). Explicitly, (B𝕀)_{n} ∈ 𝕊 is given by

(B𝕀)_{n} = ∐_{x_{0},...,x_{n}∈Ob 𝕀} \underline{Hom}_{𝕀}(x_{0}, x_{1}) × … × \underline{Hom}_{𝕀}(x_{n−1}, x_{n}).

In [Rez] Theorem 7.2, Rezk introduced the complete Segal space model structure CSS on the category s𝕊, whose fibrant objects are complete Segal spaces. In [Ber2], Bergner showed that there is a chain of Quillen equivalences between the model categories CSS and sCat. Therefore the following theorem can be interpreted as saying that N and MC are derived right adjoints to alg^{∗} : sCat → sQDat.

Theorem 5.21. If 𝕀 is a simplicial category and D ∈ sQDat, then there are canonical equivalences

Map_{sQDat}^{h}(alg^{∗}𝕀, D) ≃ Map_{s𝕊}^{h}(B𝕀, N (D)) ≃ Map_{CSS}^{h}(B𝕀, N (D)) ≃ Map_{CSS}^{h}(B𝕀, MC(D^{f})),

where D^{f} is a Reedy fibrant replacement of D.

Proof. First note that we can regard K ∈ s𝕊 as a diagram Δ^{opp} → 𝕊, and evaluate K in the simplicial category s𝕊 as the coend K = K ⊗_{Δ^{opp}} (Δ)^{hor} = ∫^{n} K_{n}^{ver} × (Δ^{n})^{hor} (in the notation of [Hir] Definition 18.3.2). Since K and Δ : Δ^{opp} → 𝕊 are both Reedy cofibrant, this means that Map_{s𝕊}^{h}(K, N (D)) is (assuming N (D) Reedy fibrant) the end ∫_{n} Map_{s𝕊}^{h}(K_{n}^{ver} × (Δ^{n})^{hor}, N (D)) = ∫_{n} N (D)_{n}^{K_{n}}. The proof of Proposition 5.18 now adapts to prove the first equivalence, replacing homotopy limits with ends.

For any K ∈ s𝕊, we may define P_{K}^{∗} : QMM^{∗}(𝕊) → sQDat(𝕊), specialising to the functor P_{K}^{∗} of Lemma 1.35 when K ∈ sSet. If 𝕀 is a simplicial category, then for all m the simplicial set [n] ↦ ((B𝕀)_{n})_{m} is the nerve of the category 𝕀_{m}, given by Ob 𝕀_{m} = Ob 𝕀 and Hom_{𝕀_{m}}(x, y) := \underline{Hom}_{𝕀}(x, y)_{i}. Thus the description of P_{B𝕀}^{∗} in Lemma 1.35 generalises to simplicial categories 𝕀, and as in Proposition 5.18, the morphism P_{B𝕀}^{∗}diag^{∗}Ξ → P_{B𝕀}^{∗}ι(•, •) = alg^{∗}𝕀 is a weak equivalence.

By Corollary 5.19, N (D)^{f} is a complete Segal space, and since the identity functor CSS → s𝕊 is right Quillen for the Reedy model structure, the second equivalence follows. Proposition 5.15 gives the third equivalence, since [Rez] Theorem 7.7 shows that a morphism of Segal spaces becomes a weak equivalence in CSS if and only if it is a Dwyer-Kan equivalence. □

5.3. DEL.

Definition 5.22. Let D ↦ D^{LGpd} be the left-derived functor Ho(sQDat) → Ho(sQDpd) of the functor D ↦ D^{Gpd} from Definition 3.20.

Lemma 5.23. There is a natural equivalence (alg^{∗}n)^{LGpd} ≃ alg^{∗}(n^{Gpd}).

Proof. By Lemma 1.36, P_{n}^{∗}diag^{∗}Ξ is a cofibrant resolution of alg^{∗}n, so we need to show that

(P_{n}^{∗}diag^{∗}Ξ)^{Gpd} ≃ alg^{∗}(n^{Gpd}).

By Lemma 5.5, the natural morphism f : Ξ × alg^{∗}n → P_{n}^{∗}diag^{∗}Ξ is a trivial cofibration in sQDat_{n}, and hence also in sQDat. Since (−)^{Gpd} : sQDat → sQDpd is left Quillen, this implies that f^{Gpd} is a trivial cofibration, so it suffices to show that (Ξ × alg^{∗}n)^{Gpd} ≃ alg^{∗}(n^{Gpd}). Now, (Ξ × alg^{∗}n) has objects [0, n] and (Ξ × alg^{∗}n)^{Gpd}(i, j) = Ξ for all i, j. Since Ξ ≃ ι•, this gives a weak equivalence (Ξ×alg^{∗}n)^{Gpd} → alg^{∗}(n^{Gpd}), as required. □

Proposition 5.24. For D ∈ sQDpd fibrant,

Del(D) ≃ Map_{sQDat}^{h}(alg^{∗}n, D)

for all n ≥ 0.

Proof. Let Del^{∗} be the left adjoint to Del : sQDpd → 𝕊, making it a left Quillen functor by Proposition 3.36. Since Del(D)_{0} = ∐_{x∈Ob D} MC(D(x, x)), it follows that Del^{∗}(Δ^{0}) = Ξ, so Del^{∗}(Δ^{0}) ≃ alg^{∗}0. Since Δ^{•} is a cofibrant cosimplicial resolution of Δ^{0} in 𝕊, Del^{∗}(Δ^{•}) is thus a cofibrant cosimplicial resolution of alg^{∗}0, so

Map_{sQDpd}^{h}(alg^{∗}0, D) ≃ ([m] ↦ Hom_{sQDpd}(Del^{∗}(Δ^{m}), D)) = Del(D),

giving the case n = 0.

For the general case, note that the inclusion functor sQDpd → sQDat is right Quillen, with left adjoint given by groupoid completion, and so

Map_{sQDat}^{h}(alg^{∗}n, D) ≃ Map_{sQDpd}^{h}((alg^{∗}n)^{LGpd}, D) ≃ Map_{sQDpd}^{h}(alg^{∗}(n^{Gpd}), D),

by Lemma 5.23. Since n^{Gpd} ≃ 0, this is just Map_{sQDpd}^{h}(alg^{∗}0, D), which completes the proof. □

Proposition 5.25. The functor DEL : sQDpd → s𝕊 of Definition 3.38 is right Quillen for the complete Segal space model structure CSS of [Rez] Theorem 7.2. In particular, for any D ∈ sQDpd fibrant, the simplicial space DEL(D) is a complete Segal space. In fact, DEL(D) is then equivalent to the constant simplicial space Del(D).

Proof. By Corollary 3.39, we know that DEL is right Quillen for the Reedy model structure. In order to prove the remaining statements, it suffices that for any fibration f : D → E, the fibrations

(f, (∂_{i})^{n}) : DEL(D)_{n} → DEL(E)_{n} ×_{Del(E)} Del(D)

are trivial fibrations for i = 0, 1 and for all n. This is equivalent to showing that the cofibrations

Del^{∗}(Δ^{m})∪_{Del^{∗}(∂Δ^{m})}(DEL_{n})^{∗}(∂Δ^{m}) → (DEL_{n})^{∗}(Δ^{m})

are trivial cofibrations for all m ≥ 0 (and both choices of map (∂_{i})^{n∗} : Del^{∗} → DEL_{n}^{∗}), where (DEL_{n})^{∗} : 𝕊 → sQDpd is left adjoint to DEL_{n}.

Since DEL is right Quillen, the functor DEL_{n} is right Quillen, so the proof of Proposition 5.24 adapts to show that the cosimplicial object i ↦ (DEL_{n})^{∗}(Δ^{m}) is a cofibrant cosimplicial resolution of (P_{n}^{∗}diag^{∗}Ξ)^{Gpd} in sQDpd, which by Lemma 5.23 is equivalent to alg^{∗}(n^{Gpd}), and hence to alg^{∗}0. Therefore the cofibrations Del^{∗}(Δ^{m}) → (DEL_{n})^{∗}(Δ^{m}) are all weak equivalences. Moreover, Del^{∗}(∂Δ^{m}) ≃ ho\underrightarrow{lim}_{Δ_{m−1,+}} alg^{∗}0 (for Δ_{m−1,+} the subcategory of Δ on objects ≤ m − 1 with only injective morphisms) and similarly for (DEL_{n})^{∗}(∂Δ^{m}). Thus the cofibrations Del^{∗}(∂Δ^{m}) → (DEL_{n})^{∗}(∂Δ^{m}) are also weak equivalences, giving a weak equivalence

Del^{∗}(Δ^{m}) → Del^{∗}(Δ^{m})∪_{Del^{∗}(∂Δ^{m})}(DEL_{n})^{∗}(∂Δ^{m}).

Since Del^{∗}(Δ^{m}) → (DEL_{n})^{∗}(Δ^{m}) is a trivial cofibration (coming the trivial cofibration (∂^{i})^{n} : Δ^{0} → Δ^{n}), this gives the required result. □

Corollary 5.26. For D ∈ sQDpd fibrant, the morphism MC(D) → DEL(D) is a Dwyer-Kan equivalence, and DEL(D) ≃ N (D).

Proof. Proposition 5.24 implies that the constant simplicial space Del is a model for N, so Proposition 5.25 shows that DEL must also be a model for N. By Proposition 5.15, the morphism MC(D) → N (D)^{f} is a Dwyer-Kan equivalence. □

6. Maurer-Cartan and classifying spaces

In many cases, a simplicial quasi-descent datum E has additional structure, and in this section we show how this simplifies the Segal spaces MC(E) and DEL(E) (where appropriate). This will give descriptions which are not only simpler, but are related to more familiar formulae.

6.1. Groups.

6.1.1. Cosimplicial simplicial groups.

Definition 6.1. Let csGp be the category of cosimplicial simplicial groups, equipped with its Reedy simplicial model structure over simplicial groups.

Example 6.2. Given a simplicial set X and a simplicial group G, our main motivating example of a cosimplicial simplicial group is the complex C^{•}(X, G) given by

C^{n}(X, G)_{m} := G_{m}^{X_{n}},

with cosimplicial operations ∂^{i} := G_{m}^{∂_{i}}, σ^{i} := G_{m}^{σ_{i}}, and simplicial operations ∂_{i} = ∂_{i}^{G}, σ_{i} = σ_{i}^{G}.

Definition 6.3. Define MC : cGp → Set by

MC(G) := Z^{1}(G) = {ω ∈ G^{1} : σ^{0}ω = 1 ∂^{1}ω = ∂^{2}ω · ∂^{0}ω}.

Definition 6.4. Define \underline{MC} : csGp → 𝕊 by \underline{MC}(G) ⊂ ∏_{n≥0}(G^{n+1})^{Δ^{n}}, satisfying the conditions of [Pri3] Lemma 3.3, i.e. the elements ω_{n} ∈ (G^{n+1})^{Δ^{n}} satisfy

∂_{i}ω_{n} = \begin{cases}∂^{i+1}ω_{n−1} & i > 0 \\ (∂^{1}ω_{n−1}) · (∂^{0}ω_{n−1})^{−1} & i = 0, \end{cases}
σ_{i}ω_{n} = σ^{i+1}ω_{n+1},
σ^{0}ω_{n} = 1.

Define MC : csGp → Set by MC(G) = \underline{MC}(G)_{0}, noting that this agrees with Definition 6.3 when G ∈ cGp.

Remark 6.5. If G ∈ csGp is of the form G^{n} = C^{•}(X, H), for X ∈ 𝕊 and H ∈ sGp as in Example 6.2, then [Pri3] Lemma 3.3 gives a canonical isomorphism MC(G) ≅ Hom_{𝕊}(X, W̄ H), where W̄ is the classifying space functor of [GJ] Ch.V.4.

Definition 6.6. Define E : csGp → QM^{∗}(𝕊) by E(G)^{n} = G^{n}, with identity 1 ∈ G^{0}, operations ∂_{E(G)}^{i} = ∂_{G}^{i}, σ_{E(G)}^{i} = σ_{G}^{i}, and Alexander-Whitney product

g ∗ h = ((∂_{G}^{m+1})^{n}g) · ((∂_{G}^{0})^{m}h),

for g ∈ G^{m}, h ∈ G^{n}.

Observe that E is right Quillen and preserves weak equivalences. Denote its left adjoint by E^{∗}. Note that MC(G) = MC(E(G)).

Note that the equivalence of Lemma 4.1 is just given by E : cAb → QM^{∗}(Ab, ×).

Proposition 6.7. For G ∈ csGp fibrant, there is a natural isomorphism

R\underline{Hom}_{csGp}(E^{∗}(ι•), G) ≃ \underline{MC}(G)

in Ho(𝕊).

Proof. This is essentially the same as Proposition 3.11. First, note that E^{∗}(ι•)^{n} is the free group on n generators, with constant simplicial structure. If s is the unique element of (ι•)^{1}, then the generators in level n are given by ∂^{i_{n−1}} … ∂^{i_{2}}∂^{i_{1}}s, for 0 < i_{1} < i_{2} < … < i_{n−1} ≤ n. We then define Φ ∈ csGp by

Φ^{n} := Fr(∐_{j<n} Δ^{j}),

where Fr denotes the free group functor. We give this the operations dual to those on W̄ G in [GJ] Ch.V, i.e. for x ∈ Δ^{j} ⊂ Φ^{n}, we set

∂_{Φ}^{i}(x) = \begin{cases}∂_{Δ}^{i−n+j}(x) & i > n − j \\ (∂_{Δ}^{0}x) · x & i = n − j \\ x & i < n − j. \end{cases}

σ_{Φ}^{i}(x) = \begin{cases}σ_{Δ}^{i−n+j}(x) & i ≥ n − j \\ 1 & i = n − j \\ x & i < n − j. \end{cases}

Thus Hom_{sGp}(Φ^{n}, G^{m}) = (W̄ G^{m})_{n}, with cosimplicial operations on Φ corresponding to simplicial operations on W̄ G, so we have

Hom_{csGp}(Φ, G) = {f ∈ ∏_{n}(W̄ G^{n})_{n} : ∂^{i}f_{n} = ∂_{i}f_{n+1} σ^{i}f_{n} = σ_{i}f_{n−1}} = Hom_{c𝕊}(Δ, W̄ G),

where Δ is the cosimplicial space given by Δ^{n} in level n.

The proof of [GJ] Lemma V.5.3 adapts to show that this is isomorphic to MC(G). Explicitly, \underline{ω} ∈ MC(G) corresponds to the maps Δ^{n} → W̄ G^{n} given by the element (ω_{n−1}, ∂^{0}ω_{n−2}, …, (∂^{0})^{n−1}ω_{0}) ∈ (W̄ G^{n})_{n}. (This also implies that Φ ≅ G(Δ), for G the loop group functor of [GJ] §V.5.) Hence

Hom_{csGp}(Φ, G) ≅ MC(G).

The inclusion MC(G_{0}) ↪ MC(G) corresponds to a map Φ → ι•. Since ∐_{i<n} Δ^{i} is weakly equivalent to a disjoint union of n−1 points, this map must be a weak equivalence.

Now, we may show (similarly to Lemma 3.9, replacing I^{n} by Δ^{n}) that for all trivial fibrations G → H,

MC(G) → MC(H)

is a surjection, so Φ is cofibrant.

Thus, for G fibrant,

R\underline{Hom}(E^{∗}(ι•), G) ≃ \underline{Hom}(Φ, G) ≃ \underline{MC}(G).
□

6.1.2. Equivalence of Maurer-Cartan spaces.

Proposition 6.8. There are equivalences

\underline{MC}(E(G)) ≃ \underline{MC}(G)

in 𝕊, functorial in fibrant objects G ∈ csGp. Here, the functors \underline{MC} on the left and right are those from Definitions 3.5 and 6.4 respectively.

Proof. Since G is fibrant, E(G) = RE(G). By Proposition 3.11,

\underline{MC}(E(G)) ≅ RHom_{QM^{∗}(𝕊)}(ι•, RE(G)) ≅ \underline{Hom}_{csGp}(LE^{∗}(ι•), G).

Thus, by Proposition 6.7, we need only show that

LE^{∗}(ι•) → E^{∗}(ι•)

is a weak equivalence in csGp. A model for LE^{∗}(ι•) is given by E^{∗}Ξ, for Ξ from Definition 3.10.

Since E^{∗} is a left adjoint, it commutes with coequalisers, so π_{0}(E^{∗}Ξ)^{n} = E^{∗}(π_{0}Ξ^{n}) = E^{∗}((ι•)^{n}). Since E^{∗}(ι•)^{n} is a discrete group, we need only show that the components of (E^{∗}Ξ)^{n} are contractible for all n. This is equivalent to saying that the universal cover W of BE^{∗}(ι•)^{n} is contractible, which will follow from the Hurewicz theorem if H_{i}(W, ℤ) = 0 for all i > 1. This is the same as saying that the homology groups H_{i}((E^{∗}Ξ)^{n}, ℤE^{∗}(ι•)^{n}) are zero for all i > 1.

Let csRep(E^{∗}(ι•)) be the category of abelian cosimplicial simplicial E^{∗}(ι•)- representations, and consider the functor csRep(E^{∗}(ι•)) → csGp ↓ E^{∗}(ι•) given by V ↦ V ⋊ E^{∗}(ι•). This is a right adjoint, and is right Quillen for the Reedy model structure on csRep(E^{∗}(ι•)). Denote the derived left adjoint by L cot, and observe that for V ∈ sRep(E^{∗}(ι•)^{n}),

Hom_{Ho(sRep(E^{∗}(ι•)^{n}))}((L cot G)^{n}, V) ≅ Hom_{Ho(sGp↓E^{∗}(ι•)^{n})}(G^{n}, V [−i]⋊E^{∗}(ι•)^{n}) ≅ ℍ^{1}(B(G^{n}), •; V),

where the final group is hypercohomology (with coefficients in a complex of local systems). Thus

H_{i}(L cot G)^{n} ≅ H_{i+1}(B(G^{n}), •; ℤE^{∗}(ι•)^{n}).

It therefore suffices to show that H_{i}(L cot E^{∗}Ξ) = 0 for all i ≥ 1.

Now, for W ∈ csRep(E^{∗}(ι•)),

\underline{Hom}_{csGp↓E^{∗}(ι•)}(E^{∗}Ξ, W ⋊ E^{∗}(ι•))
≅ \underline{Hom}_{QM^{∗}(𝕊)↓EE^{∗}(ι•)}(Ξ, E(W ⋊ E^{∗}(ι•)))
≅ \underline{Hom}_{QM^{∗}(𝕊)}(Ξ, E(W ⋊ E^{∗}(ι•)) ×_{EE^{∗}(ι•)} ι•)
≅ \underline{Hom}_{QM^{∗}(Ab,×)}(cot Ξ, E(W ⋊ E^{∗}(ι•)) ×_{EE^{∗}(ι•)} ι•).

This leads us to consider the functor ȷ : csRep(E^{∗}(ι•)) → csAb, given by

W ↦ C(E(W ⋊ E^{∗}(ι•)) ×_{EE^{∗}(ι•)} ι•).

Explicitly, we see that this has objects w · ϖ_{m} in level m, for ϖ_{m} the image of (ι•)^{m} → EE^{∗}(ι•)^{m}. The operations are ∂^{i}(w · ϖ_{m}) = ∂^{i}(w) · ϖ_{m+1} for 0 < i ≤ m, σ^{i}(w · ϖ_{m}) = σ^{i}(w) · ϖ_{m−1}, and

∂_{ȷW}^{0} (w · ϖ_{m}) = ϖ_{1} ∗ (w · ϖ_{m}) = (∂^{2})^{m}(ϖ_{1}) · ∂_{W}^{0} (w) · ∂^{0}(ϖ_{m})
= ad_{(∂^{2})^{m}(ϖ_{1})}(∂_{W}^{0} w) · ϖ_{m+1},

and

∂_{ȷW}^{m+1}(w · ϖ_{m}) = (w · ϖ_{m}) ∗ ϖ_{1} = ∂_{W}^{m+1}w · ϖ_{m+1}.

Thus (ȷW)^{n} ≅ W^{n}, with the same operations as W, except for

∂_{ȷW}^{0} (w) = ((∂^{2})^{m}ϖ_{1}) ⋆ ∂_{W}^{0} (w),

where ⋆ denotes the group action of E^{∗}(ι•)^{m+1} on W^{m+1}.

It therefore follows that the left adjoint ȷ^{∗} is given by

(ȷ^{∗}U)^{n} = U^{n}[E^{∗}(ι•)^{n}],

the free representation on generators U^{n}. This will have the same operations on generators as U, except that ∂_{ȷ^{∗}U}^{0}(u) = ((∂^{2})^{m}ϖ_{1})^{−1} ⋆ ∂_{U}^{0}(u).

For the natural Reedy model structure on csRep(E^{∗}(ι•)), ȷ^{∗} is clearly left Quillen, so it follows that ȷ is a right Quillen functor. Moreover, the descriptions above show that ȷ and ȷ^{∗} both preserve weak equivalences. Thus

L cot E^{∗}Ξ ≃ ȷ^{∗}L cot(ι•),

so H_{i}(L cot E^{∗}Ξ) ≅ H_{i}D(ℤ[−1]) by Lemma 4.9. In particular, H_{i}(L cot E^{∗}Ξ) = 0 for all i > 0, as required. □

6.1.3. Del.

Definition 6.9. For G ∈ csGp, there is an adjoint action of G^{0} on \underline{MC}(G), given by

(g ∗ ω)_{n} = (∂_{0}(∂^{1})^{n+1}(σ_{0})^{n+1}g) · ω_{n} · (∂^{0}(∂^{1})^{n}(σ_{0})^{n}g^{−1}),

as in [Pri3] Definition 3.8.

We then define Del(G) to be the homotopy quotient Del(G) = [\underline{MC}(G)/^{h}G^{0}] := \underline{MC}(G) ×_{G^{0}} W G^{0} ∈ 𝕊.

Remark 6.10. If G = C^{•}(X, H) for X ∈ 𝕊, H ∈ sGp, then Del(G) ≃ Map_{𝕊}(X, W̄ H). This is because Del(G)_{n} = Hom(X, W̄ (H^{Δ^{n}}) × (cosk_{0}H_{0})^{n}), and [n] ↦ W̄ (H^{Δ^{n}}) × (cosk_{0}H_{0})^{n} gives a fibrant simplicial resolution of W̄ H in 𝕊.

Proposition 6.11. The equivalence of Proposition 6.8 is G^{0}-equivariant, giving isomorphisms

Del(E(G)) ≃ Del(G)

in Ho(𝕊), functorial in fibrant objects G ∈ csGp.

Proof. If we let ℤ ∈ QM^{∗}(𝕊) be the object given by ℤ in degree 0 and ∅ in higher degrees, then

E^{0} ≅ \underline{Hom}_{QM^{∗}(𝕊)}(ℤ, E),

so the E^{0}-action on \underline{MC}(E) corresponds to a ℤ-coaction on Ξ, i.e. a map

Ξ → Ξ ∐ ℤ

in QM^{∗}(𝕊), where ∐ denotes coproduct in the category QM^{∗}(𝕊), satisfying a coassociativity condition.

Since E^{∗} is a left adjoint, it preserves coproducts, so we get a coaction of E^{∗}(ℤ) on E^{∗}(Ξ), i.e. a coassociative map

E^{∗}(Ξ) → E^{∗}(Ξ) ⋆ E^{∗}(ℤ),

where ⋆ denotes free product. Of course, \underline{Hom}_{csGp}(E^{∗}(ℤ), G) ≅ E(G)^{0} = G^{0}. Now, the morphisms

MC(E(G)) ← MC(E(G)_{0}) = MC(G_{0}) → MC(G)

are equivariant with respect to the action of G_{0}^{0}. Since Hom_{csGp}(E^{∗}(ℤ), G) ≅ G_{0}^{0}, this amounts to saying that the weak equivalences

E^{∗}(Ξ) → E^{∗}(ι•) ← Φ

are E^{∗}(ℤ)-coequivariant, with the E^{∗}(ℤ)-coaction on Φ corresponding to the adjoint action of Definition 6.9.

We now make use of the fact that for H ∈ sGp acting on Y ∈ 𝕊, one model for the homotopy quotient [Y/^{h}H] is given by first forming the simplicial object [Y/H] in the category of groupoids, then forming the nerve B[Y/H] (which is a bisimplicial set), giving [Y/^{h}H] ≃ diag B[Y/H].

Given C ∈ csGp equipped with a E^{∗}(ℤ)-coaction, we now define the cosimplicial diagram β(C) ∈ (csGp)^{Δ} by the property that \underline{Hom}_{csGp}(β(D), G) = B[\underline{Hom}_{csGp}(D, G)/G^{0}] ∈

(𝕊)^{Δ^{opp}}. Explicitly, β(D)^{n} = D ⋆ \overbrace{E^{∗}(ℤ) ⋆ E^{∗}(ℤ) ⋆ … ⋆ E^{∗}(ℤ)}^{n}.

Now E^{∗}(ℤ) is just the cosimplicial group E^{∗}(ℤ)^{n} = \overbrace{ℤ ⋆ ℤ ⋆ … ⋆ ℤ}^{n+1}, so E^{∗}(ℤ) is levelwise cofibrant. Since E^{∗}(ℤ), E^{∗}(Ξ), E^{∗}(ι•) and Φ are all levelwise cofibrant, the morphisms

β(E^{∗}(Ξ)) → β(E^{∗}(ι•)) ← β(Φ)

are (levelwise) weak equivalences.

In the Reedy model category (csGp)^{Δ}, we now choose a cofibrant replacement C for β(Φ), and a factorisation β(E^{∗}(Ξ)) → F → β(E^{∗}(ι•)) with β(E^{∗}(Ξ)) → F a trivial cofibration and F → β(E^{∗}(ι•)) a trivial fibration. Since C and F are weakly equivalent in the Reedy model category (csGp)^{Δ} ↓ β(E^{∗}(ι•)), there is an explicit weak equivalence f : C → F in this category. Now, Reedy cofibrations are a fortiori levelwise cofibrations, so the objects C^{n}, F^{n} are cofibrant in csGp. We therefore have levelwise weak equivalences

\underline{Hom}_{csGp}(β(Φ), G) → \underline{Hom}_{csGp}(C, G), and \underline{Hom}_{csGp}(F, G) → \underline{Hom}_{csGp}(β(E^{∗}(Ξ)), G)

in (𝕊)^{Δ^{opp}}, for all G ∈ csGp. Combining these with f gives weak equivalences

B[\underline{MC}(G)/G^{0}] → \underline{Hom}_{csGp}(C, G) ⟵^{f^{∗}} \underline{Hom}_{csGp}(F, G) → B[\underline{MC}(E(G))/G^{0}],

in (𝕊)^{Δ^{opp}}, and taking diagonals gives the required result that Def(G) ≃ Def(E(G)). □

6.2. Lie algebras.

6.2.1. Nilpotent DGLAs.

Definition 6.12. Let DG_{ℤ}N̂LA denote the category of pro-nilpotent differential graded Lie algebras (DGLAs) over k.

Explicitly, a DGLA is a graded vector space L = ⨁_{i∈ℤ} L^{i} over k, equipped with operators [,] : L × L → L bilinear and d : L → L linear, satisfying:

(1) [L^{i}, L^{j}] ⊂ L^{i+j}.

(2) [a, b] + (−1)^{āb̄}[b, a] = 0.

(3) (−1)^{c̄ā}[a, [b, c]] + (−1)^{āb̄}[b, [c, a]] + (−1)^{b̄c̄}[c, [a, b]] = 0.

(4) d(L^{i}) ⊂ L^{i+1}.

(5) d ∘ d = 0.

(6) d[a, b] = [da, b] + (−1)ā[a, db]

Here ā denotes the degree of a, mod 2, for a homogeneous.

A DGLA is said to be nilpotent if the lower central series Γ_{n}L (given by Γ_{1}L = L, Γ_{n+1}L = [L, Γ_{n}L]) vanishes for n ≫ 0.

Thus DG_{ℤ}N̂LA is the category of pro-objects in the category of nilpotent DGLAs.

Definition 6.13. Given a pro-nilpotent Lie algebra 𝔤, define Û(𝔤) to be the pro-unipotent completion of the universal enveloping algebra of 𝔤, regarded as a pro-object in the category of algebras. As in [Qui1] Appendix A, this is a pro-Hopf algebra, and we define exp(𝔤) ⊂ Û(𝔤) to consist of elements g with ε(g) = 1 and Δ(g) = g ⊗ g, for ε : Û(𝔤) → k the augmentation (sending 𝔤 to 0), and Δ : Û(𝔤) → Û(𝔤) ⊗ Û(𝔤) the comultiplication.

Since k is assumed to have characteristic 0, exponentiation gives an isomorphism from 𝔤 to exp(𝔤), so we may regard exp(𝔤) as having the same elements as 𝔤, but with multiplication given by the Campbell–Baker–Hausdorff formula.

Definition 6.14. Given a ℤ-graded pro-nilpotent DGLA L^{•}, define the Maurer-Cartan set by

MC(L) := \{ω ∈ L^{1} | dω + \frac{1}{2}[ω, ω] = 0 ∈ ⨁_{n} L^{2}\}

Define the gauge group Gg(L) by Gg(L) := exp(L^{0}), which acts on MC(L) by the gauge action

g(ω) = g · ω · g^{−1} − dg · g^{−1},

where · denotes multiplication in the universal enveloping algebra of L. That g(ω) ∈ MC(L) is a standard calculation (see [Kon] or [Man1]).

Let π^{1}(L) := MC(L)/Gg(L) be the quotient set.

Lemma 6.15. If e : L ↠ M has kernel K, with [K, L] = 0, then there is an obstruction map o_{e} : π^{1}(M) → H^{2}(K), with o_{e}^{−1}(0) being the image of π^{1}(L). Moreover, π^{1}(L) is a principal H^{1}(K)-bundle over o_{e}^{−1}(0)

Proof. This is well-known (see [Man1] §3, for instance). □

Definition 6.16. Let O(MC) be the pro-nilpotent DGLA representing MC, so Hom(O(MC), L) ≅ MC(L). Explicitly, O(MC) is the free pro-nilpotent graded Lie algebra on one generator x in degree 1, with differential determined by dx = \frac{1}{2}[x, x].

Similarly, define O(Gg) to represent Gg; this is freely generated by y, dy, for y ∈ O(Gg)^{0}. Note that this has a cogroup structure in DG_{ℤ}N̂LA, coming from the group structure on Gg.

Define T and O(T) by T (L) = Hom(O(T), L) := exp(Z^{0}L); this is freely generated by z ∈ O(T)^{0}, with dz = 0. The embedding T ↪ Gg corresponds to the projection y ↦ z, dy ↦ 0, and O(T) is a quotient cogroup of O(Gg) in DG_{ℤ}N̂LA.

We may therefore regard MC, Gg, T as being objects of the opposite category (DG_{ℤ}N̂LA)^{opp}, which is a full subcategory of the category of presheaves on DG_{ℤ}N̂LA, via the Yoneda embedding.

Proposition 6.17. The map q : Gg → MC given by g ↦ g(0) gives an isomorphism between MC and the right quotient of Gg by T in the category (DG_{ℤ}N̂LA)^{opp}.

Proof. It is immediate that for h ∈ T (L) we have h(0) = 0, since dh = 0. This gives a morphism Gg(L)/T (L) → MC(L), functorial in L, and we need to show that MC is the universal representable presheaf with this property, in other words that

\xymatrix{Gg × T \ar[r]^{^{pr^{1}}} \ar[r]_{μ} & Gg \ar[r]^{q} & MC}

is a coequaliser in (DG_{ℤ}N̂LA)^{opp}, where μ(g, t) = g · t.

The forgetful functor U^{opp} from DG_{ℤ}N̂LA to the category G_{ℤ}N̂LA of pro-nilpotent ℤ-graded Lie algebras preserves and reflects all equalisers, so U : (DG_{ℤ}N̂LA)^{opp} → (G_{ℤ}N̂LA)^{opp} preserves and reflects all coequalisers. It therefore suffices to show that UMC = (UGg)/(UT).

Now, the forgetful functor U^{opp} has a right adjoint R, given by (RL)^{n} = L^{n} × L^{n+1}, with [(a, a′), (b, b′)] = ([a, b], [a′, b] + (−1)ā[a, b′]) and d(a, a′) = (a′, 0). This gives

(UMC)(L)/(UGg)(L) = MC(RL)/Gg(RL) = π^{1}(RL).

Applying R the tower … → L/[L, [L, L]] → L/[L, L] gives a tower of surjections satisfying the conditions of Lemma 6.15. Since H^{∗}(RM) = 0 for all M, this implies by induction that π^{1}(RL) = 0 for all L. Thus UGg acts transitively on UMC.

In particular, this means that the canonical element in (UMC)(U^{opp}O(MC)) is of the form Uq(s) for some s ∈ (UGg)(U^{opp}O(MC)). Via the Yoneda embedding, s is equivalent to the data of a section of Uq : UGg → UMC. Since the fibres of Uq are precisely the UT -orbits, the map UMC × UT → UGg given by (ω, t) ↦ s(ω) · t is an isomorphism. It follows immediately that the fork is a coequaliser, since (UMC × UT)/UT = UMC. □

6.2.2. Cosimplicial simplicial groups.

Definition 6.18. Given G ∈ csGp, define the gauge group Gg(G) to be the subgroup of ∏_{n} G_{n}^{n} consisting of those \underline{g} satisfying

∂_{i}g_{n} = ∂^{i}g_{n−1} ∀i > 0,
σ_{i}g_{n} = σ^{i}g_{n+1} ∀i,

similarly to [Pri3] Definition 3.5. Note that G_{0}^{0} can be regarded as a subgroup of Gg(G), setting g_{n} = (∂^{1})^{n}(σ_{0})^{n}g, for g ∈ G_{0}^{0}. The group T (G) := Tot (G)_{0} is the subgroup of Gg(G) consisting of those \underline{g} for which ∂_{0}g_{n} = ∂^{0}g_{n−1}.

The action of Definition 6.9 extends to an action of Gg(G) on MC(G), given by

(g ∗ ω)_{n} = (∂_{0}g_{n+1}) · ω_{n} · (∂^{0}g_{n}^{−1}),

as in [Pri3] Definition 3.8.

Remark 6.19. If G ∈ csGp is of the form G = C^{•}(X, H), for X ∈ 𝕊 and H ∈ sGp, then [Pri3] (Lemma 3.3 and Proposition 3.9) gives a canonical isomorphism Gg(G) ≅ Hom_{𝕊}(X, W H) for W the canonical G-torsor on W̄ G, as in [GJ] Ch.V. Moreover, T (H) ≅ Hom_{𝕊}(X, H), and the map q : Gg(G) → MC(G) corresponding to the gauge action on 1 comes from the identification W̄ G = G\W G of [GJ] Ch.V.

Lemma 6.20. The functors MC, Gg and T are all representable in csGp.

Proof. MC is represented by the object Φ defined in Proposition 6.7, which we will now denote by O(MC), so Hom_{csGp}(O(MC), G) ≅ MC(G).

Define O(T) by O(T)^{n} = Fr(Δ^{n}) (where Fr denotes the free group functor), with cosimplicial operations coming from those on the cosimplicial space Δ^{•}; this gives Hom_{csGp}(O(T), G) ≅ T (G).

Similarly, define O(Gg) to represent Gg; this is given by O(Gg)^{n} = Fr(∐_{j≤n} Δ^{j}), with operations given on x ∈ Δ^{j} ⊂ O(Gg)^{n} by

∂_{O(Gg)}^{i}(x) = \begin{cases}∂_{Δ}^{i−n+j}(x) & i > n − j \\ x & i ≤ n − j. \end{cases}
σ_{O(Gg)}^{i}(x) = \begin{cases}σ_{Δ}^{i−n+j}(x) & i > n − j \\ x & i ≤ n − j. \end{cases}

The isomorphism Gg(G) ≅ Hom_{csGp}(O(Gg), G) is given by \underline{g} ∈ Gg(G) mapping Δ^{j} ⊂ O(Gg)^{n} to G^{n} via the element (∂_{0})^{n−j}g_{n} ∈ G_{j}^{n}.

Note that O(Gg) has a cogroup structure in csGp, coming from the group structure on Gg. Explicitly, this is the map O(Gg) → O(Gg) ⋆ O(Gg) given by x ↦ x ⋆ x for x ∈ Δ^{j} ⊂ O(Gg)^{n}. The embedding T ↪ Gg corresponds to the quotient map i : O(Gg) → O(T) given by mapping Δ^{j} ⊂ O(Gg)^{n} to Δ^{n} ⊂ O(T)^{n} via (∂_{Δ}^{0})^{n−j}. Thus O(T) is a quotient cogroup of O(Gg) in csGp. □

We may therefore regard MC, Gg, and T as being objects of the opposite category (csGp)^{opp}.

Proposition 6.21. The map q : Gg → MC given by g ↦ g ∗ 1 gives an isomorphism between MC and the right quotient of Gg by T in the category (csGp)^{opp}.

Proof. Define P ∈ csGp by

P^{n} := Fr(∐_{j≤n} Δ^{j}).

We give this the operations dual to those on W G in [GJ] Ch.V, i.e. for x ∈ Δ^{j} ⊂ P^{n}, we set

∂_{P}^{i} (x) = \begin{cases}∂_{Δ}^{i−n+j}(x) & i > n − j \\ (∂_{Δ}^{0}x) · x & i = n − j \\ x & i < n − j. \end{cases}

σ_{P}^{i} (x) = \begin{cases}σ_{Δ}^{i−n+j}(x) & i ≥ n − j \\ 1 & i = n − j \\ x & i < n − j. \end{cases}

In particular, O(MC)^{n} is the simplicial subgroup of P^{n} on generators ∐_{j<n} Δ^{j}, making O(MC) a subobject of P.

Now, [Pri3] Proposition 3.9 adapts to give an isomorphism ψ : P → O(Gg), given by mapping x ∈ Δ^{j} ⊂ P^{n} to (∂_{Δ}^{0}x) · x^{−1} when j < n, and x^{−1} when j = n.

The right action of T on Gg corresponds to the coaction μ : O(Gg) → O(Gg) ⋆ O(T) given by mapping x ∈ Δ^{j} ⊂ O(Gg)^{n} to x_{Gg} · ((∂_{Δ}^{0})^{n−j}x)_{T}, where y_{Gg} and y_{T} denote the copies of y in O(Gg)^{n} and in O(T)^{n}, respectively. There is an obvious isomorphism P^{n} ≅ O(MC)^{n} ⋆ O(T)^{n}, since O(T)^{n} = Fr(Δ^{n}), and this is equivariant for the O(T)-coaction (with trivial coaction on O(MC)), since x ∈ Δ^{j} ⊂ P^{n} has μ(ψ(x)) = ψ(x) ∈ O(Gg) for j < n, and μ(ψ(x)) = ψ(x) · i(x) when j = n.

Therefore

\xymatrix{Gg × T \ar[r]^{^{pr^{1}}} \ar[r]_{μ} & Gg \ar[r]^{q} & MC}

is a coequaliser in csGp^{opp}, as required. □

6.2.3. Lie algebras to groups.

Definition 6.22. Given an ℕ_{0}-graded DGLA L, let DL be its cosimplicial denormalisation. Explicitly,

D^{n}L := ⨁_{\begin{matrix}m+s=n \\ 1≤j_{1}<...<j_{s}≤n \end{matrix}} ∂^{j_{s}} … ∂^{j_{1}}L^{m}.

We then define operations ∂^{j} and σ^{i} using the cosimplicial identities, subject to the conditions that σ^{i}L = 0 and ∂^{0}v = dv − ∑_{i=1}^{n+1}(−1)^{i}∂^{i}v for all v ∈ L^{n}.

We now have to define the Lie bracket ⟦, ⟧ from D^{n}L ⊗ D^{n}L to D^{n}L. Given a finite set I of distinct strictly positive integers, write ∂^{I} = ∂^{i_{s}} … ∂^{i_{1}}, for I = {i_{1}, … i_{s}}, with i_{1} < … < i_{s}. The Lie bracket is then defined on the basis by

⟦∂^{I}v, ∂^{J}w⟧ := \begin{cases}∂^{I∩J}(−1)^{(J\I,I\J)}[v, w] & v ∈ L^{|J\I|}, w ∈ L^{|I\J|}, \\ 0 & otherwise, \end{cases}

where for disjoint sets S, T of integers, (−1)^{(S,T)} is the sign of the shuffle permutation of S ⊔ T which sends the first |S| elements to S (in order), and the remaining |T | elements to T (in order). Beware that this formula cannot be used to calculate ⟦∂^{I}v, ∂^{J}w⟧ when 0 ∈ I ∪ J (for the obvious generalisation of ∂^{I} to finite sets I of distinct non-negative integers).

Theorem 6.23. Given a simplicial pro-nilpotent ℕ_{0}-graded DGLA L_{•}^{•}, there are canonical isomorphisms

Gg(exp(DL)) ≅ Gg(Tot^{Π}N^{s}L), MC(exp(DL)) ≅ MC(Tot ΠN^{s}L),

compatible with the respective gauge actions. Here, D is cosimplicial denormalisation and N^{s} is simplicial normalisation (as in Definition 4.6). This isomorphism acts as the identity on the subgroups exp(L_{0}^{0}) ≤ Gg(exp(DL)) and exp(L_{0}^{0}) ≤ Gg(Tot^{Π}N^{s}L)

Proof. In order to keep track of the various gradings in these categories, we will write DG_{ℤ} for ℤ-graded cochain complexes, DG for cochain complexes in non-negative degrees, and dg for chain complexes in non-negative degrees.

On the category sDGN̂LA of simplicial pro-nilpotent ℕ_{0}-graded DGLA, these functors are all representable, since exp, D, Tot^{Π} and N^{s} are all right adjoints, so

Hom_{sDGN̂LA}(D^{∗} exp^{∗} O(Gg), L) ≅ Gg(exp(DL))
Hom_{sDGN̂LA}(D^{∗} exp^{∗} O(MC), L) ≅ MC(exp(DL))
Hom_{sDGN̂LA}((N^{s})^{∗}(Tot^{Π})^{∗}O(Gg), L) ≅ Gg(Tot^{Π}N^{s}L)
Hom_{sDGN̂LA}((N^{s})^{∗}(Tot^{Π})^{∗}O(MC), L) ≅ MC(Tot ΠN^{s}L),

for exp^{∗} : csGp → csN̂LA, D^{∗} : csN̂LA → sDGN̂LA, (Tot^{Π})^{∗} : DG_{ℤ}N̂LA → dgDGN̂LA, and (N^{s})^{∗} : dgDGN̂LA → sDGN̂LA the corresponding left adjoints.

We may therefore regard the functors D^{∗} exp^{∗} Gg, D^{∗} exp^{∗} MC, (N^{s})^{∗}(Tot^{Π})^{∗}Gg and (N^{s})^{∗}(Tot^{Π})^{∗}MC as objects of (sDGN̂LA)^{opp}, and likewise for D^{∗} exp^{∗} T and (N^{s})^{∗}(Tot^{Π})^{∗}T.

Now, the isomorphism Gg(exp(DL)) ≅ Gg(Tot^{Π}N^{s}L) simply follows from the proof of the Dold-Kan correspondence, which generalises to give an equivalence of categories between ℕ_{0}-graded complexes and “simplicial abelian groups without ∂_{0}”. This isomorphism maps the subgroup T (exp(DL)) to T (Tot^{Π}N^{s}L) isomorphically, by the usual Dold-Kan correspondence.

The isomorphism D^{∗} exp^{∗} MC ≅ (N^{s})^{∗}(Tot^{Π})^{∗}MC follows by taking the right quotients

(D^{∗} exp^{∗} Gg)/(D^{∗} exp^{∗} T) ≅ ((N^{s})^{∗}(Tot^{Π})^{∗}Gg)/((N^{s})^{∗}(Tot^{Π})^{∗}T)

in (sDGN̂LA)^{opp}, by Propositions 6.21 and 6.17. □

Remark 6.24. Note that this gives a shorter and more conceptual proof of [Pri3] Theorems 4.39 and 4.44, and that in that context we may shorten the arguments, replacing Proposition 6.21 with [Pri3] Proposition 3.9.

Definition 6.25. Define \underline{MCTot ΠN^{s}} : sDGN̂LA → 𝕊 by

\underline{MCTot ΠN^{s}}(L)_{n} := MC(Tot ΠN^{s}(L^{Δ^{n}})),

and define Del : sDGN̂LA → 𝕊 to be the homotopy quotient Del := [\underline{MCTot ΠN^{s}}/^{h} exp(L^{0})], i.e.

Del(L) := \underline{MCTot ΠN^{s}}(L) ×_{exp(L^{0})} W exp(L^{0}),

where exp(L^{0})_{n} ⊂ Gg(L^{Δ^{n}}) acts via the gauge action.

Remark 6.26. This is essentially the functor used to construct derived formal stacks in [Hin].

Corollary 6.27. There are canonical isomorphisms

Del(L) ≅ Del(exp(DL))

in 𝕊, functorial in L ∈ sDGN̂LA.

Proof. By Theorem 6.23, \underline{MCTot ΠN^{s}}(L) ≅ \underline{MC}(L). This isomorphism is equivariant with respect to the action of the simplicial group exp(L^{0})_{n} = exp((L^{Δ^{n}})_{0}^{0} ≤ Gg(L^{Δ^{n}}), so gives the required isomorphism on taking homotopy quotients. □

Appendix A. Quasi-monads, quasi-comonads and quasi-distributivity

To date, the main context in which s.h. ⊤-algebras have been studied is when the monad ⊤ comes from an operad. This is partly because these were the only case for which a satisfactory theory of morphisms was developed in [CLM]. Since this difficulty was resolved for general monads (and even distributive monad-comonad pairs) in §3.5, we now look into how related constructions adapt to this generality.

In [KS], free resolutions of operads were exploited to study deformations of algebras. Since an objectwise weak equivalence ⊤′ → ⊤ of arbitrary monads on a simplicial category gives a weak equivalence of the associated simplicial quasi-descent data (from Proposition 2.9), the respective Segal spaces of s.h. ⊤-algebras are weakly equivalent. This means that many of the ideas from [KS] carry over to arbitrary monads.

However, in some settings, such as [vdL], operads are too restrictive, and homotopy operads have to be used instead. We now introduce quasi-monads and quasi-comonads, which give a context sufficiently general to be analogous to homotopy operads, while providing a natural generalisation of the constructions of §2.

A.1. Quasi-monads and quasi-comonads.

Definition A.1. Define a quasi-monad (resp. a quasi-comonad) on a category B to be a quasi-monoid (resp. quasi-comonoid), as in Definition 1.4, in the monoidal category (End(B), ∘) of endofunctors of B.

Substituting the monoidal category (End(B), ∘) into Lemma 1.5 yields the following two lemmas.

Lemma A.2. A quasi-monad consists of functors ⊤_{n} : B → B, together with the following data:

\begin{matrix}μ_{i} : ⊤_{n+1} ⟹ ⊤_{n} & 1 ≤ i ≤ n \\ η_{i} : ⊤_{n−1} ⟹ ⊤_{n} & 0 ≤ i < n, \end{matrix}

an associative coproduct ξ_{mn} : ⊤_{m+n} ⟹ ⊤_{m} ∘ ⊤_{n}, with coidentity ξ_{0} : ⊤_{0} ⟹ id, satisfying:

(1) μ_{i}μ_{j} = μ_{j−1}μ_{i} i < j.

(2) η_{i}η_{j} = η_{j+1}η_{i} i ≤ j.

(3) μ_{i}η_{j} = \begin{cases}η_{j−1}μ_{i} & i < j \\ id & i = j, i = j + 1 \\ η_{j}μ_{i−1} & i > j + 1 \end{cases} .

(4) (μ_{i}⊤_{n})ξ_{m+1,n} = ξ_{mn}μ_{i}.

(5) (⊤_{m}μ_{i})ξ_{m,n+1} = ξ_{mn}μ_{i+m}.

(6) (η_{i}⊤_{n})ξ_{m−1,n} = ξ_{mn}η_{i}.

(7) (⊤_{m}η_{i})ξ_{m,n−1} = ξ_{mn}η_{i+m}.

Lemma A.3. A quasi-comonad consists of functors ⊥^{n} : B → B, together with the following data:

\begin{matrix}Δ^{i} : ⊥^{n} ⟹ ⊥^{n+1} & 1 ≤ i ≤ n \\ ε^{i} : ⊥^{n} ⟹ ⊥^{n−1} & 0 ≤ i < n, \end{matrix}

an associative product ζ^{mn} : ⊥^{m} ∘⊥^{n} ⟹ ⊥^{m+n}, with identity ζ^{0} : id ⟹ ⊥^{0}, satisfying:

(1) Δ^{j}Δ^{i} = Δ^{i}Δ^{j−1} i < j.

(2) ε^{j}ε^{i} = ε^{i}ε^{j+1} i ≤ j.

(3) ε^{j}Δ^{i} = \begin{cases}Δ^{i}ε^{j−1} & i < j \\ id & i = j, i = j + 1 \\ Δ^{i−1}ε^{j} & i > j + 1 \end{cases} .

(4) ζ^{m+1,n}(Δ^{i}⊥^{n}) = Δ^{i}ζ^{mn}.

(5) ζ^{m,n+1}(⊥^{m}Δ^{i}) = Δ^{i+m}ζ^{mn}.

(6) ζ^{m−1,n}(ε^{i}⊥^{n}) = ε^{i}ζ^{mn}.

(7) ζ^{m,n−1}(⊥^{m}ε^{i}) = ε^{i+m}ζ^{mn}.

Lemma A.4. A quasi-monad on B gives rise to a quasi-descent datum enriching B, given by

Hom(B, B′)^{n} := Hom_{B}(⊤_{n}B, B′).

In particular, his allows us to define Maurer-Cartan sets. We also have a notion of homotopy monad on a simplicial category (when the ξ are all weak equivalences). Dually, we have the same constructions for quasi-comonads.

A.2. Distributivity for quasi-monads and quasi-comonads. We now need to describe a distributivity transformation λ for a quasi-monad ⊤ and a quasi-comonad ⊥. We wish to enrich B to a quasi-descent datum by setting

Hom_{B}(B, B′)^{n} := Hom(⊤_{n}B, ⊥^{n}B′),

so we need natural transformations

λ_{m}^{n} : ⊤_{m}⊥^{n} ⟹ ⊥^{n}⊤_{m}.

with the following diagrams commuting:

\xymatrix{⊤_{m}⊥^{n} \ar@{=>}[r]^{λ_{m}^{n}} \ar@{=>}[d]_{⊤_{m}Δ^{i}} & ⊥^{n}⊤_{m} \ar@{=>}[d]_{Δ^{i}⊤_{m}} & ⊤_{m}⊥^{n} \ar@{=>}[r]^{λ_{m}^{n}} & ⊥^{n}⊤_{m} \\ ⊤_{m}⊥^{n+1} \ar@{=>}[r]^{λ_{m}^{n+1}} & ⊥^{n+1}⊤_{m} & ⊤_{m+1}⊥^{n} \ar@{=>}[u]_{μ_{i}⊥^{n}} \ar@{=>}[r]^{λ_{m+1}^{n}} & ⊥^{n}⊤_{m+1} \ar@{=>}[u]_{⊥^{n}μ_{i}}}

\xymatrix{⊤_{m}⊥^{n} \ar@{=>}[r]^{λ_{m}^{n}} \ar@{=>}[d]_{⊤_{m}ε^{i}} & ⊥^{n}⊤_{m} \ar@{=>}[d]_{ε^{i}⊤_{m}} & ⊤_{m}⊥^{n} \ar@{=>}[r]^{λ_{m}^{n}} & ⊥^{n}⊤_{m} \\ ⊤_{m}⊥^{n−1} \ar@{=>}[r]^{λ_{m}^{n−1}} & ⊥^{n−1}⊤_{m} & ⊤_{m−1}⊥^{n} \ar@{=>}[u]_{η_{i}⊥^{n}} \ar@{=>}[r]^{λ_{m−1}^{n}} & ⊥^{n}⊤_{m−1}, \ar@{=>}[u]_{⊥^{n}η_{i}}}

together with unit rules

(ξ_{0}⊥^{n}) = (⊥^{n}ξ_{0}) ∘ λ_{0}^{n} ζ^{0}⊤_{m} = λ_{m}^{0} ∘ (⊤_{m}ζ^{0})

and associativity rules

(λ_{p}^{n}⊤_{q}) ∘ (⊤_{p}λ_{q}^{n}) ∘ (ξ_{pq}⊥^{n}) = (⊥^{n}ξ_{pq}) ∘ λ_{p+q}^{n},
(⊥^{p}λ_{m}^{q}) ∘ (λ_{m}^{p}⊥^{q}) ∘ (⊤_{m}ζ^{pq}) = (ζ^{pq}⊤_{m}) ∘ λ_{m}^{p+q}.

Lemma A.5. Given a category B, a quasi-monad ⊤ and a quasi-comonad ⊥ on B, together with a quasi-distributivity transformation λ satisfying the conditions above, there is a quasidescent datum on B given by setting

Hom_{B}(B, B′)^{n} := Hom(⊤_{n}B, ⊥^{n}B′).

Proof. We define the structures on Hom_{B} by

∂^{i}x = Δ^{i} ∘ x ∘ μ_{i},
σ^{i}x = ε^{i} ∘ x ∘ η_{i},
x ∗ y = ζ^{nm} ∘ (⊥^{n}x) ∘ λ_{m}^{n} ∘ (⊤_{m}y) ∘ ξ_{mn},

for x ∈ Hom_{B}(B′, B”)^{m} and y ∈ Hom_{B}(B, B′)^{n}, with identity ζ^{0} ∘ ξ_{0}.

It follows immediately from Lemmas A.2 and A.3 that Hom_{B}(B, B′) ∈ Set^{Δ_{∗∗}}. The first four diagrams above ensure that ∗ defines a map

Hom_{B}(B′, B”) ⊗ Hom_{B}(B, B′) → Hom_{B}(B, B”)

in Set^{Δ_{∗∗}}. The unit rules then ensure that ζ^{0} ∘ ξ_{0} is the multiplicative identity, and the associativity rules make this product associative. □

Appendix B. A_{∞}-algebras and homotopy operads

A_{∞}-algebras are designed to model deformation retracts of differential graded associative algebras without unit (DGAAs). They are thus an alternative candidate for the task performed in general by s.h. algebras, and are indeed often known as strong homotopy associative algebras (SHAAs). That both concepts are essentially equivalent seems to be folklore (though, if necessary, it could be inferred from the results of §6.2).

Now, a DGAA is just a semigroup in the category of cochain complexes (i.e. it satisfies all the requirements of a monoid, except the unit axiom). A third candidate to model deformation retracts of DGAAs is therefore a homotopy semigroup in cochain complexes (defined analogously to a homotopy monoid). These were studied in [Lei1], where it was conjectured that they give rise to A_{∞}-algebras.

B.1. Homotopy semigroups and semicogroups.

Definition B.1. Define Δ_{∗∗}^{+} to be the subcategory of Δ_{∗∗} on objects n ≠ 0 and containing only injective morphisms.

Definition B.2. Define a semigroupal category to be a category C equipped with a bifunctor C × C ⟶^{⊗} C satisfying the axioms of a monoidal category (but without a unit object).

Definition B.3. Given a semigroupal category C (in particular if C is monoidal), define a quasi-semigroup X in C to be a colax semigroupal functor X : (Δ_{∗∗}^{+})^{opp} → C. This means that we have maps

ξ_{mn} : X_{m+n} → X_{m} ⊗ X_{n},

satisfying naturality and coherence. If C is a model category, we say that X is a homotopy semigroup whenever the maps ξ_{mn} are all weak equivalences.

Lemma B.4. Giving a quasi-semigroup X in C is equivalent to giving objects X_{n} ∈ C for n ∈ ℕ_{1}, together with morphisms

∂_{i} : X_{n+1} → X_{n} 1 ≤ i ≤ n,

and an associative coproduct ξ_{mn} : X_{m+n} → X_{m} ⊗ X_{n}, satisfying:

(1) ∂_{i}∂_{j} = ∂_{j−1}∂_{i} i < j.

(2) (∂_{i} ⊗ id)ξ_{m+1,n} = ξ_{mn}∂_{i}.

(3) (id ⊗ ∂_{i})ξ_{m,n+1} = ξ_{mn}∂_{i+m}.

Definition B.5. Given a semigroupal category C, define a quasi-semicogroup X in C to be a lax semigroupal functor X : Δ_{∗∗}^{+} → C. This means that we have maps

ζ^{mn} : X^{m} ⊗ X^{n} → X^{m+n},

satisfying naturality and coherence. If C is a model category, we say that X is a homotopy semicogroup whenever the maps ζ^{mn} are all weak equivalences.

Lemma B.6. Giving a quasi-semicogroup X in C is equivalent to giving objects X^{n} ∈ C for n ∈ ℕ_{1}, together with morphisms

∂^{i} : X^{n} → X^{n+1} 1 ≤ i ≤ n

an associative product ζ^{mn} : X^{m} ⊗ X^{n} → X^{m+n}, satisfying:

(1) ∂^{j}∂^{i} = ∂^{i}∂^{j−1} i < j.

(2) ζ^{m+1,n}(∂^{i} ⊗ id) = ∂^{i}ζ^{mn}.

(3) ζ^{m,n+1}(id ⊗ ∂^{i}) = ∂^{i+m}ζ^{mn}.

B.2. Homotopy semigroups in abelian categories. .

Definition B.7. Given a quasi-semigroup V in an abelian semigroupal category V, define the chain complex C(V) ∈ Ch(V) by

C(V)_{n} := \begin{cases}V_{n} & n > 0 \\ 0 & n ≤ 0, \end{cases}

with differential d = ∑_{i}(−1)^{i}∂_{i}.

Definition B.8. A coalgebra C is said to be conilpotent if the iterated coproduct Δ^{n} : C → C^{⊗n} is 0 for n sufficiently large. A coalgebra C is said to be ind-conilpotent if it is a filtered colimit of conilpotent coalgebras.

Lemma B.9. C(V) has the natural structure of an ind-conilpotent coassociative coalgebra without counit in Ch(V).

Proof. We define the coproduct Δ : C(V) → C(V) ⊗ C(V) by ⨁_{i+j=n} ξ_{ij} : C(V)_{n} → (C(V) ⊗ C(V))_{n}. It is straightforward to verify that this is coassociative and a chain map, so C(V) is a coassociative coalgebra without counit.

Observe that the brutal truncations σ_{≤m}C(V) of C(V) form conilpotent subcoalgebras of C(V), since (C(V)^{⊗n})_{i} = 0 for all i < n. Thus C(V) is ind-conilpotent, since C(V) = \underrightarrow{lim} σ_{≤m}C(V). □

B.2.1. DG coalgebras. Now assume that V is the category of cochain complexes of vector spaces over a field k. If V is a quasi-semigroup in V, then Lemma B.9 implies that the cochain complex Tot C(V) (given by (Tot C(V))^{n} := ⨁_{i} C(V)_{i}^{i+n}) is an ind-conilpotent coassociative DG coalgebra without counit over k.

Definition B.10. Let DGC_{k} be the category of all ind-conilpotent coassociative DG coalgebras without counit over k. Let DGA_{k} be the category of associative DG algebras without unit over k.

Definition B.11. Define the cobar functor β^{∗} : DGC_{k} → DGA_{k} by letting β^{∗}C be the free graded associative k-algebra ⨁_{n>0} C[−1]^{⊗n} on generators C[−1], with differential defined on generators by d_{C} + Δ, for Δ : C[−1] → (C ⊗ C)[−1] = (C[−1] ⊗ C[−1])[1] the comultiplication.

It has right adjoint β : DGA_{k} → DGC_{k} given by letting βA be the cofree graded coassociative ind-conilpotent k-algebra ⨁_{n>0} A[1]^{⊗n} on cogenerators A[1], with differential defined on cogenerators by d_{A} + m : A ⊕ (A ⊗ A)[1] → A[1], for m : A ⊗ A → A the multiplication.

Definition B.12. Define the tangent space tan C of C ∈ DGC_{k} to be ker(Δ : C → C ⊗C).

Theorem B.13. There is a cofibrantly generated model structure on DGC_{k}, for which a morphism f is

(1) a cofibration if it is injective;

(2) a weak equivalence if either of the following equivalent conditions holds: (a) β^{∗}f is a quasi-isomorphism.

(b) f can be expressed as a filtered colimit of quasi-isomorphisms f_{α} : C_{α} → D_{α} between finite-dimensional objects of DGC_{k} (note that this is a stronger than requiring that f be a quasi-isomorphism);

(3) a fibration if f is cofree as a morphism of ind-conilpotent coassociative graded

coalgebras without counit. Moreover, for a fibrant object C there is a natural isomorphism H^{n}(tan C) ≅ H^{n+1}(β^{∗}C).

With respect to this model structure, β^{∗} is a left Quillen equivalence, when DGA_{k} is given its standard model structure.

Proof. Existence of such a model structure is given in [Pri5] Proposition 4.36 for the analogous case of cocommutative coassociative coalgebras and Lie algebras, but the proof carries over to any Koszul-dual pair of quadratic operads, so it adapts to our context (coassociative coalgebras and associative algebras). The generating cofibrations are injective morphisms f : C → D between finite-dimensional objects, satisfying the additional property that the coproduct coker f → D ⊗ coker f is zero. Generating trivial cofibrations have the additional property that H^{∗}(coker f) = 0.

Characterisation of the weak equivalences follows from [Pri5] Proposition 4.42. That β^{∗} is a Quillen equivalence follows from [Pri5] Theorem 4.55. □

Remark B.14. Note that fibrant objects of DGC_{k} are those whose underlying coalgebras are cofree. These are precisely strong homotopy associative algebras (SHAAs), as in [Kon], and weak equivalences between these are tangent quasi-isomorphisms. A choice of cogenerators on an SHAA is precisely an A_{∞}-algebra. This means that every A_{∞}-algebra has a weakly equivalent DG associative algebra.

Corollary B.15. There is a canonical equivalence class of A_{∞}-algebras associated to any quasi-semigroup V in the category of cochain complexes over k.

Proof. The A_{∞}-algebra is just a choice of cogenerators on a fibrant replacement for Tot C(V) □

When V is a homotopy semigroup, we wish to relate this A_{∞}-algebra to V.

Proposition B.16. Assume that C ∈ DGC_{k} is equipped with an exhaustive increasing filtration 0 = F_{0}C ⊂ F_{1}C ⊂ …, comultiplicative in the sense that

Δ(F_{n}) ⊂ ∑_{i+j=n} F_{i} ⊗ F_{j} ⊂ C ⊗ C,

and for which the resulting maps Δ_{n} : gr_{n}^{F} C → (F_{1}C)^{⊗n} are quasi-isomorphisms. Then there are canonical isomorphisms

H^{n+1}(β^{∗}C) ≅ H^{n}(F_{1}C).

Proof. The filtration F_{n}C induces a filtration on β^{∗}C by

F_{p}(C[−1]^{⊗n}) := ∑_{p_{1}+...+p_{n}=p}(F_{p_{1}}C[−1]) ⊗ (F_{p_{2}}C[−1]) ⊗ … ⊗ (F_{p_{n}}C[−1]).

Since this filtration is exhaustive and bounded below, the associated spectral sequence

E_{0}^{pq}(F) = (gr_{−p}^{F}β^{∗}C)^{p+q} ⟹ H^{p+q}(β^{∗}C)

converges.

Now, note that gr^{F} β^{∗}C ≅ β^{∗}(gr^{F} C), and that the quasi-isomorphisms Δ_{n} : gr_{n}^{F} C → (F_{1}C)^{⊗n} induce a graded quasi-isomorphism

δ : gr^{F} C → ⨁_{n>0}(F_{1}C)^{⊗n}

in DGC_{k}.

We now define a filtration W on β^{∗}C by W^{m}β^{∗}C = ⨁_{n≥m} C[−1]^{⊗n}, noting that F_{p}(β^{∗}C) ∩ W^{p+1}(β^{∗}C) = 0, since F_{n−1}(C^{⊗n}) = 0. Thus W induces a finite filtration on gr_{−p}^{F}β^{∗}C, so the associated spectral sequence

E_{0}^{nq}(W) = (gr_{W}^{n} gr_{−p}^{F}β^{∗}C)^{n+q} ⟹ H^{p+q}(gr_{−p}^{F}β^{∗}C)

converges. Now, the left-hand side is just gr_{−p}^{F}(C[−1]^{⊗n}), so E_{1}^{nq}(W) ≅ H^{n+q}(gr_{−p}^{F}(C[−1]^{⊗n})).

Therefore δ induces an isomorphism of E_{1} spectral sequences, making

gr_{−p}^{F}β^{∗}(δ) : gr_{−p}^{F}β^{∗}C → gr_{−p}^{F}β^{∗}(⨁_{n>0}(F_{1}C)^{⊗n})

a quasi-isomorphism. The right-hand side is just gr_{−p}^{F}β^{∗}β(F_{1}C[−1]), where F_{1}C[−1] is regarded as an object of DGA_{k} with zero multiplication. Therefore

E_{1}^{pq}(F) = H^{p+q}(gr_{−p}^{F}β^{∗}C) ≅ H^{p+q}(gr_{−p}^{F}β^{∗}β(F_{1}C[−1])).

Now, since β^{∗} ⊣ β are a pair of Quillen equivalences, the map β^{∗}β(F_{1}C[−1]) → F_{1}C[−1] is a quasi-isomorphism, so

E_{1}^{pq}(F) ≅ \begin{cases}H^{p+q}(F_{1}C[−1]) & p = −1, \\ 0 & otherwise, \end{cases}

and therefore the spectral sequence collapses at E_{1}, giving

H^{q−1}(β^{∗}C) ≅ H^{q−1}(F_{1}C[−1]) = H^{q−2}(F_{1}C),

as required. □

The following result confirms a conjecture from [Lei1] §3.5, although not with the proof envisaged there.

Proposition B.17. If V is a homotopy semigroup, then we may choose a representative A_{∞}-algebra with underlying cochain complex V_{1}.

Proof. We adapt the proof of [Pri3] Lemma 1.16. Take a trivial cofibration Tot C(V) → E, with E fibrant. The filtration F_{n}C := ⨁_{i≤n} V_{n} of C(V) satisfies the conditions of Proposition B.16, since the maps V_{n} → (V_{1})^{⊗n} are quasi-isomorphisms by hypothesis. Thus the map V_{1} → tan E is a quasi-isomorphism. Let the quotient be Q, and note that this is a contractible cochain complex.

Let B be the cofree coalgebra on generators Q; this is trivially fibrant. Since the morphism tan E → E is a cofibration in DGC_{k}, the composite map tan E → Q → B must extend to a morphism f : E → B in DGC_{k}. Let A be the coequaliser of f and the zero map; this is again a cofree object in DGC_{k}, hence fibrant, and tan A = ker(tan E → Q) ≅ V_{1}.

Therefore A is an SHAA with cogenerating space V_{1}, so defines an A_{∞}-structure on V_{1}. □

B.3. DG co-operads. We now show the relation between A_{∞}-algebras and homotopy semigroups of cochain complexes has an analogue for operads. Roughly speaking, we will show that a homotopy monad (in the sense of §A) with suitable operadic structure is related to the homotopy operads of [vdL].

Definition B.18. Given an additive cocomplete monoidal category C, we now define a full subcategory E(C) of the category End(C) of endofunctors on C. Objects of E correspond to collections {P_{n}}_{n≥0}, with P_{n} a C-representation of the symmetric group S_{n}, with the associated endofunctor given by

V ↦ ⨁_{n} P_{n} ⊗^{S_{n}} V^{⊗n}.

E forms a monoidal category under composition of functors, and an additive category under ⊕.

Definition B.19. An operad (resp. pseudo operad) on C is a monoid (resp. semigroup) in E(C), and a co-operad (resp. pseudo co-operad) on C is a comonoid (resp. semicogroup) in E(C).

Note that since E(C) is an additive category, there is a natural retraction (F ∘ X) ⊕ (F ∘ Y) → F ∘(X ⊕Y) → (F ∘X)⊕(F ∘Y), for any F, X, Y ∈ E(C). Thus, augmented operads ⊤ on abelian categories C correspond to pseudo operads S on C, by setting S := ker(⊤ → 1) and ⊤ = S ⊕ 1.

Let dgVect be the category of chain complexes, and gVect the category of graded vector spaces, both over a field of characteristic 0.

Lemma B.20. If C = dgVect or gVect, the forgetful functor from pseudo operads on C to E(C) has a left adjoint, denoted by T. Likewise, the forgetful functor from ind-conilpotent pseudo co-operads on C to E(C) has a right adjoint, denoted by T′.

Proof. These are described in [vdL] §2. In both cases, the underlying functor is F ↦

⨁_{n>0} \overbrace{F ∘ F ∘ … ∘ F}^{n}. □

Definition B.21. Recall from [vdL] Definition 3.1 that an operad up to homotopy on dgVect is defined to be a collection P ∈ E(gVect), together with a square-zero differential δ on the cofree pseudo co-operad

T′(P [1]).

The notion of operad up to homotopy in [vdL] generalises pseudo operads (or, equivalently, augmented operads) rather than operads. This motivates the following comparison, noting that the quasi-monads of Appendix A are a fortiori quasi-semigroups in End(C), and that pseudo co-operads can be replaced by homotopy operads, similarly to Corollary B.15.

Lemma B.22. Every quasi-semigroup Q in E(dgVect) (in the sense of Definition B.3) naturally gives rise to a pseudo co-operad β(Q) on dgVect.

Proof. Given a quasi-semigroup Q = {Q_{n}}_{n>0} in E(dgVect) , we may set β(Q) := ⨁_{n>0} Q_{n}[n] in E(gVect), with differential δ : d_{Q} ± ∑_{i}(−1)^{i}∂_{i}. Here, ∂_{i} : Q_{n}[n]_{j} → Q_{n+1}[n + 1]_{j−1} is the structural map ∂_{i} : (Q_{n})_{n+j} → (Q_{n+1})_{n+j} of the quasi-semigroup.

The functor β(Q) has the natural structure of a pseudo co-operad, with coproduct β(Q) → β(Q) ∘ β(Q) given on Q_{n}[n] ⊂ β(Q) by ∑_{i+j=n} ξ_{ij} : Q_{n}[n] → Q_{i}[i] ∘ Q_{j}[j], making use of the natural retraction (F ∘ X) ⊕ (F ∘ Y) → F ∘ (X ⊕ Y) → (F ∘ X) ⊕ (F ∘ Y). □

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