DERIVED DEFORMATIONS OF ARTIN STACKS

J.P.PRIDHAM

Abstract. We generalise the techniques of [Pri5] to describe derived deformations in simplicial categories. This allows us to consider deformation problems with higher automorphisms, such as chain complexes (which have homotopies) and stacks (which have 2-automorphisms).

Introduction

This paper is motivated by the wish to describe derived deformations of an algebraic stack. In [Ols2] and [Aok], it was shown that deforming an algebraic stack can be regarded as a special case of deforming a simplicial algebraic space. The category of simplicial spaces has a natural simplicial structure (meaning that the Hom-sets can be enriched to give simplicial sets), and the 2-groupoid of deformations of an algebraic stack can be recovered from this simplicial structure.

After reviewing background material from [Pri4] in §1, we introduce derived deformation complexes (DDCs) In Section 2; these extend the SDCs of [Pri2] to simplicial categories. We then adapt the various constructions of [Pri4], showing how to associate derived deformation functors to DDCs, and how to compare them with derived deformation functors coming from SDCs.

Section 3 adapts the ideas of [Pri2], showing how to associate DDCs to bialgebraic deformation problems in simplicial categories. In §3.3, we show how deformations of morphisms and diagrams can be used to compare deformations of weakly equivalent objects.

Several simple examples of such problems are considered in Section 4: chain complexes (with more interesting variants in Remarks 4.7), simplicial complexes and simplicial algebras.

The motivating example of algebraic stacks is finally considered in Section 5. We first describe derived deformations of simplicial affine schemes (§5.1), then show in §5.2 how to adapt this to describe derived deformations of an algebraic stack 𝔛, with an indication in Remark 5.28 of how this approach also works for Artin n-stacks. The idea is to consider derived deformations of a suitable hypercovering X_{•} of 𝔛. To see that this does, indeed, extend the 2-groupoid of deformations of 𝔛, we establish comparisons with Olsson’s Extgroups of the cotangent complex (§5.2.1) and Aoki’s description of the deformation 2groupoid (§5.2.2).

Contents

Introduction 1 1. Derived deformation functors 2 1.1. Simplicial Artinian rings 2 1.2. Properties of morphisms 3 1.3. Derived deformation functors 3 1.4. Quotient spaces 4 1.5. Cohomology and obstructions 5 1.6. Model structures 6 2. Derived deformation complexes 7 2.1. Definitions 7 2.2. Comparison with SDCs 10 3. Constructing DDCs 12 3.1. Simplicial monadic adjunctions 12 3.2. The construction 13 3.3. Deformations of diagrams and invariance under weak equivalence 16 4. Examples 18 4.1. Chain complexes 18 4.2. Simplicial complexes 20 4.3. Simplicial algebras 21 5. Deformations of Artin stacks and simplicial schemes 24 5.1. Cosimplicial algebras 24 5.2. Quasi-compact, quasi-separated stacks 30 5.3. Arbitrary algebraic stacks 34 Bibliography 36 References 36

The author was supported during this research by Trinity College, Cambridge.

1. Derived deformation functors

With the exception of §1.4, the definitions and results in this section can all be found in [Pri4]. Fix a complete local Noetherian ring Λ, with maximal ideal μ and residue field k.

1.1. Simplicial Artinian rings.

Definition 1.1. Let C_{Λ} denote the category of local Artinian Λ-algebras with residue field k. We define sC_{Λ} to be the category of Artinian simplicial local Λ-algebras, with residue field k.

Definition 1.2. Given a simplicial complex V_{•}, recall that the normalised chain complex N^{s}(V)_{•} is given by N^{s}(V)_{n} := ⋂_{i>0} ker(∂_{i} : V_{n} → V_{n−1}), with differential ∂_{0}. The simplicial Dold-Kan correspondence says that N^{s} gives an equivalence of categories between simplicial complexes and non-negatively graded chain complexes in any abelian category. Where no ambiguity results, we will denote N^{s} by N.

Lemma 1.3. A simplicial complex A_{•} of local Λ-algebras with residue field k and maximal ideal 𝔪(A)_{•} is Artinian if and only if:

(1) the normalisation N(cot A) of the cotangent space cot A := 𝔪(A)/(𝔪(A)^{2}+μ𝔪(A)) is finite-dimensional (i.e. concentrated in finitely many degrees, and finitedimensional in each degree).

(2) For some n > 0, 𝔪(A)^{n} = 0.

Proof. [Pri4] Lemma 1.16 □

As in [Gro], we say that a functor is left exact if it preserves all finite limits. This is equivalent to saying that it preserves final objects and fibre products.

Definition 1.4. Define Sp to be the category of left-exact functors from C_{Λ} to Set. Define cSp to be the category of left-exact functors from sC_{Λ} to Set.

Definition 1.5. Given a functor F : C_{Λ} → Set, we write F : sC_{Λ} → Set to mean A ↦ F (A_{0}) (corresponding to the inclusion Sp ↪ cSp).

1.2. Properties of morphisms.

Definition 1.6. As in [Man], we say that a functor F : C_{Λ} → Set is smooth if for all surjections A → B in C_{Λ}, the map F (A) → F (B) is surjective.

Definition 1.7. We say that a map f : A → B in sĈ_{Λ} is acyclic if π_{i}(f) : π_{i}(A) → π_{i}(B) is an isomorphism of pro-Artinian Λ-modules for all i. f is said to be surjective if each f_{n} : A_{n} → B_{n} is surjective.

Note that for any simplicial abelian group A, the homotopy groups can be calculated by π_{i}A ≅ H_{i}(NA), the homology groups of the normalised chain complex. These in turn are isomorphic to the homology groups of the unnormalised chain complex associated to A.

Definition 1.8. We define a small extension e : I → A → B in sC_{Λ} to consist of a surjection A → B in sC_{Λ} with kernel I, such that 𝔪(A) · I = 0. Note that this implies that I is a simplicial complex of k-vector spaces.

Lemma 1.9. Every surjection in sC_{Λ} can be factorised as a composition of small extensions. Every acyclic surjection in sC_{Λ} can be factorised as a composition of acyclic small extensions.

Proof. [Pri4] Lemma 1.23. □

Definition 1.10. We say that a morphism α : F → G in cSp is smooth if for all small extensions A ↠ B in sC_{Λ}, the map F (A) → F (B) ×_{G(B)} G(A) is surjective.

Similarly, we call α quasi-smooth if for all acyclic small extensions A → B in sC_{Λ}, the map F (A) → F (B) ×_{G(B)} G(A) is surjective.

Lemma 1.11. A morphism α : F → G in Sp is smooth if and only if the induced morphism between the objects F, G ∈ cSp is quasi-smooth, if and only if it is smooth.

Proof. [Pri4] Lemma 1.31. □

1.3. Derived deformation functors.

Definition 1.12. Define the scSp to be the category of left-exact functors from sC_{Λ} to the category 𝕊 of simplicial sets. This is equivalent to the category of simplicial cosimplicial objects in Sp.

Define sSp to be the category of left-exact functors from C_{Λ} to 𝕊.

Definition 1.13. A morphism α : F → G in scSp is said to be smooth if

(S1) for every acyclic surjection A → B in sC_{Λ}, the map F (A) → F (B) ×_{G(B)} G(A) is

a trivial fibration in 𝕊; (S2) for every surjection A → B in sC_{Λ}, the map F (A) → F (B) ×_{G(B)} G(A) is a

surjective fibration in 𝕊. A morphism α : F → G in scSp is said to be quasi-smooth if it satisfies (S1) and (Q2) for every surjection A → B in sC_{Λ}, the map F (A) → F (B)×_{G(B)}G(A) is a fibration

in 𝕊.

Definition 1.14. Given A ∈ sC_{Λ} and a finite simplicial set K, define A^{K} ∈ C_{Λ} by

(A^{K})_{i} := Hom_{𝕊}(K × Δ^{i}, A) ×_{Hom_{Set}(π_{0}K,k)} k.

Definition 1.15. Given F ∈ scSp, define \underline{F} : sC_{Λ} → 𝕊 by

\underline{F} (A)_{n} := F_{n}(A^{Δ^{n}}).

For F ∈ cSp, we may regard F as an object of scSp (with the constant simplicial structure), and then define \underline{F} as above.

Lemma 1.16. A map α : F → G in cSp is smooth (resp. quasi-smooth) if and only if the induced map of functors \underline{α} : \underline{F} → \underline{G} is smooth (resp. quasi-smooth) in scSp.

Proof. [Pri4] Lemma 1.36. □

The following Lemma will provide many examples of functors which are quasi-smooth but not smooth.

Lemma 1.17. If F → G is a quasi-smooth map of functors F, G : sC_{Λ} → 𝕊, and K → L is a cofibration in 𝕊, then

F^{L} → F^{K} ×_{G^{K}} G^{L}

is quasi-smooth.

Proof. This is an immediate consequence of the fact that 𝕊 is a simplicial model category, following from axiom SM7, as given in [GJ] §II.3. □

The following lemma is a consequence of standard properties of fibrations and trivial fibrations in 𝕊.

Lemma 1.18. If F → G is a quasi-smooth map of functors F, G : sC_{Λ} → 𝕊, and H → G is any map of functors, then F ×_{G} H → H is quasi-smooth.

Definition 1.19. A map α : F → G of functors F, G : C_{Λ} → 𝕊 is said to be smooth (resp. quasi-smooth, resp. trivially smooth) if for all surjections A ↠ B in C_{Λ}, the maps

F (A) → F (B) ×_{G(B)} G(A)

are surjective fibrations (resp. fibrations, resp. trivial fibrations).

Proposition 1.20. A map α : F → G of left-exact functors F, G : C_{Λ} → 𝕊 is smooth if and only if the maps F_{n} ⟶^{α_{n}} G_{n} of functors F_{n}, G_{n} : C_{Λ} → Set are all smooth.

Proof. [Pri4] Proposition 1.39. □

Proposition 1.21. If a morphism F ⟶^{α} G of left-exact functors F, G : sC_{Λ} → 𝕊 is such that the maps

θ : F (A) → F (B) ×_{G(B)} G(A)

are surjective fibrations for all acyclic small extensions A → B, then \underline{α} : \underline{F} → \underline{G} is quasismooth (resp. smooth) if and only if θ is a fibration (resp. surjective fibration) for all small extensions A → B.

Proof. [Pri4] Proposition 1.63. □

Definition 1.22. We will say that a morphism α : F → G of quasi-smooth objects of scSp is a weak equivalence if, for all A ∈ sC_{Λ}, the maps π_{i}F (A) → π_{i}G(A) are isomorphisms for all i.

1.4. Quotient spaces.

Definition 1.23. Given functors X : sC_{Λ} → 𝕊 and G : sC_{Λ} → sGp, together with a right action of G on X, define the quotient space by

[X/G]_{n} = (X ×^{G} W G)_{n} = X_{n} × G_{n−1} × G_{n−2} × … G_{0},

with operations as standard for universal bundles (see [GJ] Ch. V). Explicitly:

∂_{i}(x, g_{n−1}, g_{n−2}, …, g_{0}) = \begin{cases}(∂_{0}x ∗ g_{n−1}, g_{n−2}, …, g_{0}) & i = 0; \\ (∂_{i}x, ∂_{i−1}g_{n−1}, …, (∂_{0}g_{n−i})g_{n−i−1}, g_{n−i−2}, …, g_{0}) & 0 < i < n; \\ (∂_{n}x, ∂_{n−1}g_{n−1}, …, ∂_{1}g_{1}) & i = n; \end{cases}
σ_{i}(x, g_{n−1}, g_{n−2}, …, g_{0}) = (σ_{i}x, σ_{i−1}g_{n−1}, …, σ_{0}g_{n−i}, e, g_{n−i−1}, g_{n−i−2}, …, g_{0}).

The space [•/G] is also denoted W̄ G, and is a model for the classifying space BG of G.

Lemma 1.24. If G : sC_{Λ} → sGp is smooth, then W̄ G is smooth.

Proof. For any surjection A → B, we have G(A) → G(B) fibrant and surjective on π_{0}, which by [GJ] Corollary V.6.9 implies that W̄ G(A) → W̄ G(B) is a fibration. If A → B is also acyclic, then everything is trivial by properties of W̄ and G. □

Remark 1.25. Observe that this is our first example of a quasi-smooth functor which is not a right Quillen functor for the simplicial model structure. The definitions of smoothness and quasi-smoothness were designed with W̄ G in mind.

Lemma 1.26. If X is quasi-smooth, then so is [X/G] → W̄ G.

Proof. This follows from the observation that for any fibration (resp. trivial fibration) Z → Y of G-spaces, [Z/G] → [Y/G] is a fibration (resp. trivial fibration). □

Corollary 1.27. If X is quasi-smooth and G smooth, then [X/G] is quasi-smooth.

Proof. Consider the fibration X → [X/G] → W̄ G. □

1.5. Cohomology and obstructions. Given a quasi-smooth morphism α : F → G in scSp, there exist k-vector spaces H^{i}(F/G) for all i ∈ ℤ.

By [Pri4] Corollary 1.46, these have the property that for any simplicial k-vector space V with finite-dimensional normalisation,

π_{m}(F (k ⊕ V) ×_{G(k⊕V)} {0}) ≅ H^{−m}(F/G ⊗ V),

where V^{2} = 0 and

H^{i}(F/G ⊗ V) := ⨁_{n≥0} H^{i+n}(F/G) ⊗ π_{n}(V).

If G = • (the one-point set), we write H^{j}(F) := H^{j}(F/•).

We now have the following characterisation of obstruction theory:

Theorem 1.28. If α : F → G in scSp is quasi-smooth, then for any small extension e : I → A ⟶^{f} B in sC_{Λ}, there is a sequence of sets

π_{0}(F A) ⟶^{f_{∗}} π_{0}(F B ×_{GB} GA) ⟶^{o_{e}} H^{1}(F/G ⊗ I)

exact in the sense that the fibre of o_{e} over 0 is the image of f_{∗}. Moreover, there is a group action of H^{0}(F/G ⊗ I) on π_{0}(F A) whose orbits are precisely the fibres of f_{∗}.

For any y ∈ F_{0}A, with x = f_{∗}y, the fibre of F A → F B ×_{GB} GA over x is isomorphic to ker(α : F I → GI), and the sequence above extends to a long exact sequence

\xymatrix{⋯ \ar[r]^{f_{∗}} & π_{n}(F B ×_{GB} GA, x) \ar[r]^{o_{e}} & H^{1−n}(F/G ⊗ I) \ar[r]^{∂_{e}} & π_{n−1}(F A, y) \ar[r]^{f_{∗}} & ⋯ \\ ⋯ \ar[r]^{f_{∗}} & π_{1}(F B ×_{GB} GA, x) \ar[r]^{o_{e}} & H^{0}(F/G ⊗ I) \ar[r]^{−∗y} & π_{0}(F A).}

Proof. [Pri4] Theorem 1.45. □

Corollary 1.29. A map α : F → G of quasi-smooth F, G ∈ scSp is a weak equivalence if and only if the maps H^{j}(α) : H^{j}(F) → H^{j}(G) are all isomorphisms.

Corollary 1.30. If α : F → G is quasi-smooth in scSp, then α is smooth if and only if H^{i}(F/G) = 0 for all i > 0.

Proposition 1.31. Let X, Y, Z : sC_{Λ} → 𝕊 be left-exact functors, with X ⟶^{α} Y and Y ⟶^{β} Z quasi-smooth. There is then a long exact sequence

… ⟶^{∂} H^{j}(X/Y) → H^{j}(X/Z) → H^{j}(Y/Z) ⟶^{∂} H^{j+1}(X/Y) → H^{j+1}(X/Z) → …

Proof. [Pri4] Proposition 1.61. □

1.6. Model structures.

Theorem 1.32. There is a simplicial model structure on scSp, for which the fibrations are quasi-smooth morphisms, and weak equivalences between quasi-smooth objects are those given in Definition 1.22.

Proof. This is [Pri4] Theorem 2.14. □

Thus the homotopy category Ho(scSp) is equivalent to the category of quasi-smooth objects in scSp, localised at the weak equivalences of Definition 1.22.

Definition 1.33. Given any morphism f : X → Z, we define H^{n}(X/Z) := H^{n}(X̂/Z), for X ⟶^{i} X̂ ⟶^{p} Z a factorisation of f with i a geometric trivial cofibration, and p a geometric fibration.

1.6.1. Homotopy representability.

Definition 1.34. Define the category S to consist of functors F : sC_{Λ} → 𝕊 satisfying the following conditions:

(A0) F (k) is contractible.

(A1) For all small extensions A ↠ B in sC_{Λ}, and maps C → B in sC_{Λ}, the map F (A×_{B} C) → F (A)×_{F (B)}^{h}F (C) is a weak equivalence, where ×^{h} denotes homotopy fibre product.

(A2) For all acyclic small extensions A ↠ B in sC_{Λ}, the map F (A) → F (B) is a weak

equivalence. Say that a natural transformation η : F → G between such functors is a weak equiva-

lence if the maps F (A) → G(A) are weak equivalences for all A ∈ sC_{Λ}, and let Ho(S) be the category obtained by formally inverting all weak equivalences in S.

Theorem 1.35. There is a canonical equivalence between the geometric homotopy category Ho(scSp) and the category Ho(S).

Proof. This is [Pri4] Theorem 2.30. □

1.6.2. Equivalent formulations. If k is a field of characteristic 0, then we may work with dg algebras rather than simplicial algebras.

Definition 1.36. Define dgC_{Λ} to be the category of Artinian local differential ℕ_{0}-graded graded-commutative Λ-algebras with residue field k.

Definition 1.37. Define a map A → B in dgC_{Λ} to be a small extension if it is surjective and the kernel I satisfies I · 𝔪(A) = 0.

Definition 1.38. Define sDGSp to be the category of left exact functors from dgC_{Λ} to 𝕊.

Definition 1.39. Say a map X → Y in sDGSp is quasi-smooth if for all small extensions f : A → B in dgC_{Λ}, the morphism

X(A) → Y (A) ×_{Y (B)} X(B)

is a fibration in 𝕊, which is moreover a trivial fibration if f is acyclic.

Definition 1.40. We will say that a morphism α : F → G of quasi-smooth objects of sDGSp is a weak equivalence if, for all A ∈ sC_{Λ}, the maps π_{i}F (A) → π_{i}G(A) are isomorphisms for all i.

Proposition 1.41. There is a model structure on sDGSp, for which the fibrations are quasi-smooth morphisms, and weak equivalences between quasi-smooth objects are those given in Definition 1.40.

Proof. This is [Pri4] Proposition 4.12. □

Most of the constructions from sC_{Λ} carry over to dgC_{Λ}. However, there is no straightforward analogue of Definition 1.15.

Definition 1.42. Define the normalisation functor N : sC_{Λ} → dgC_{Λ} by mapping A to its associated normalised complex NA, equipped with the Eilenberg-Zilber shuffle product (as in [Qui]).

Definition 1.43. Define Spf N^{∗} : sDGSp → scSp by mapping X : dgC_{Λ} → 𝕊 to the composition X ∘ N : sC_{Λ} → 𝕊. Note that this is well-defined, since N is left exact.

Theorem 1.44. Spf N^{∗} : sDGSp → scSp is a right Quillen equivalence.

Proof. This is [Pri4] Theorem 4.18. □

In particular, this means that Spf N^{∗} maps quasi-smooth morphisms to quasi-smooth morphisms, and induces an equivalence RSpf N^{∗} : Ho(sDGSp) → Ho(scSp).

2. Derived deformation complexes

2.1. Definitions.

Definition 2.1. Define a pre-SDC to consist of homogeneous functors E^{n} : C_{Λ} → Set, for n ∈ ℕ_{0}, together with maps

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

an associative product ∗ : E^{m} × E^{n} → E^{m+n}, with identity 1 : • → E^{0}, such that: (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) ∂^{i}(e) ∗ f = ∂^{i}(e ∗ f).

(5) e ∗ ∂^{i}(f) = ∂^{i+m}(e ∗ f), for e ∈ E^{m}.

(6) σ^{i}(e) ∗ f = σ^{i}(e ∗ f).

(7) e ∗ σ^{i}(f) = σ^{i+m}(e ∗ f), for e ∈ E^{m}.

Remark 2.2. Note that a pre-SDC is an SDC (in the sense of [Pri2] if and only if the spaces E^{n} are smooth for all n.

Definition 2.3. Define a pre-derived deformation complex (pre-DDC) E to be a simplicial complex E_{•} of pre-SDCs.

Given K ∈ 𝕊, observe that E_{K}^{n} := Hom_{𝕊}(K, E^{n}) is a pre-SDC.

Remark 2.4. If each E_{m} is an SDC, then Lemma 1.20 implies that for all n, E^{n} : C_{Λ} → 𝕊 is smooth. For K ∈ 𝕊 contractible, this implies that E_{K} is an SDC.

Definition 2.5. Given a left-exact functor F : C_{Λ} → Set, define the tangent space t_{F} (or t(F)) by t_{F} := F (k[ϵ]/(ϵ^{2})). Since k[ϵ]/(ϵ^{2}) is an abelian group object in C_{Λ}, t_{F} is an abelian group. The endomorphisms ϵ ↦ λϵ of k[ϵ]/(ϵ^{2}) make t_{F} into a vector space over k.

Given a morphism α : F → G of such functors, define the relative tangent space tan(F/G) := ker(tan F → tan G).

Definition 2.6. Given a morphism f : E → F of pre-SDCs for which each f^{n} : E^{n} → F^{n} is smooth, we may define cohomology groups H^{∗}(E/F) as cohomology of the cosimplicial complex C^{•}(E/F) given by C^{n}(E/F) := tan(E/F), with cosimplicial structure defined as in [Pri2] §1.

Definition 2.7. Given a morphism E → F of pre-DDCs, levelwise smooth in the sense that each f_{n}^{i} : E_{n}^{i} → F_{n}^{i} is smooth, observe that the cohomology groups H^{i}(E/F) are simplicial vector spaces, and denote the corresponding normalised chain complexes by NH^{i}(E/F).

Definition 2.8. A morphism f : E → F of pre-DDCs is said to be quasi-smooth if: Q1. for all n, i ≥ 0, E_{n}^{i} → E_{∂Δ^{n}}^{i} ×_{F_{∂Δ^{n}}^{i}} F_{n}^{i} is smooth, and Q2. for all i > 0, H^{i}(E/F) is a constant simplicial complex, or equivalently Q2’. for all n > 0, i > 0, N_{n}H^{i}(E/F) = 0. Say that a pre-DDC E is a DDC if it is quasi-smooth, i.e. if E → • is quasi-smooth.

Definition 2.9. Given a levelwise smooth morphism f : E → F of pre-DDCs, define the tangent chain cochain complex by NC_{•}^{•}(E/F).

Definition 2.10. Say that a simplicial cosimplicial complex V ∈ scVect_{k} is quasi-smooth if H_{n}(NV^{i}) = 0 for all n, i ≥ 0 and H^{i}(NV)_{n} = 0 for all i > 0 and n > 0.

Definition 2.11. Given V ∈ scVect_{k} quasi-smooth, define a cochain complex ⌟V by:

(⌟V)^{n} := \begin{cases}V_{0}^{n} & n ≥ 0 \\ H^{0}(N_{−n}V) & n < 0, \end{cases}

with differential d_{c} in non-negative degrees, and d^{s} in negative degrees.

Given a levelwise smooth morphism f : E → F of pre-DDCs, define the cohomology groups H^{∗}(⌟E/F) := H^{∗}(⌟C^{•}(E/F)), noting that these are given by

H^{i}(⌟E/F) ≅ \begin{cases}H^{i}(C_{0}^{•}(E/F)) & i > 0 \\ H_{−i}H^{0}(NC_{•}^{•}(E/F)) & i ≤ 0. \end{cases}

Lemma 2.12. If V ∈ scVect_{k} is quasi-smooth, then the inclusion map

⌟V → Tot NV

is a quasi-isomorphism, and

H_{i}(NZ^{n}V) ≅ H^{n−i}(⌟V)

for all i, n ≥ 0.

Proof. Combine the proofs of [Pri4] Lemma 1.56 and [Pri4] Proposition 1.59. □

Lemma 2.13. A levelwise smooth morphism f : E → F of pre-DDCs is quasi-smooth if and only if C_{•}^{•}(E/F) is quasi-smooth (in the sense of Definition 2.10).

Proof. Since f is levelwise smooth, we know by Proposition 1.20 that each E^{i} → F^{i} is a smooth map of functors C_{Λ} → 𝕊. For a small extension A → B in C_{Λ} with kernel I, we thus deduce that E^{i}(A) → F^{i}(A) ×_{F^{i}(B)} E^{i}(B) is a fibration, with fibre C_{•}^{i}(E/F) ⊗ I.

Hence H_{∗}C_{•}^{i}(E/F) = 0 for all i if and only if we have E^{i} → F^{i} trivially smooth for all i, i.e. if Definition 2.8.(Q1) holds. The result now follows from the characterisation of Definition 2.10. □

Definition 2.14. A morphism f : E → F of DDCs is said to be a quasi-isomorphism if H^{∗}(⌟f) : H^{∗}(⌟E) → H^{∗}(⌟F) is an isomorphism.

Definition 2.15. Recall from [Pri5] Definition 4.1 that for any pre-SDC E, we define the Maurer-Cartan functor MC_{E} : sC_{Λ} → Set by

MC_{E}(A) ⊂ ∏_{n≥0} E^{n+1}(A^{I^{n}}),

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−1}ω_{n}(t_{1}, …, t_{n}) = ω_{n−1}(t_{1}, …, t_{n−1}),
σ^{0}ω_{0} = 1,

where I := Δ^{1}.

Definition 2.16. Given a pre-DDC E, define the derived Maurer-Cartan functor 𝔐ℭ(E) : sC_{Λ} → 𝕊 by Hom_{𝕊}(K, 𝔐ℭ(E)) := MC(E_{K}).

Proposition 2.17. If f : E → F is a quasi-smooth morphism of pre-DDCs, then

𝔐ℭ(f) : 𝔐ℭ(E) → 𝔐ℭ(F)

is quasi-smooth, with cohomology groups

H^{i}(𝔐ℭ(E)/𝔐ℭ(F)) ≅ H^{i+1}(⌟E/F).

In particular, if E is a DDC, then 𝔐ℭ(E) is quasi-smooth.

Proof. By construction, the simplicial matching maps are given by

MC(E_{n}) → MC(F_{n}) ×_{MC(F_{∂Δ^{n}})} MC(E_{∂Δ^{n}}) = MC(E_{∂Δ^{n}} ×_{F_{∂Δ^{n}}} F_{n}).

Condition (Q1) from Definition 2.8 for f implies that

E_{n} → E_{∂Δ^{n}} ×_{F_{∂Δ^{n}}} F_{n}

is a levelwise smooth map of SDCs, so [Pri5] Proposition 4.3 implies that 𝔐ℭ(f) satisfies condition (S1) from Definition 1.13.

We now need to check that the quasi-smooth partial matching maps

α : MC(E_{n}) → MC(E_{Λ_{k}^{n}}) ×_{MC(F_{Λ^{n}_{k}})} MC(F_{n})

are smooth. To do this, we verify the criterion of Corollary 1.30.

Taking the relative version of [Pri5] Proposition 4.7, we see that H^{∗}(α) is cohomology of the cochain complex

ker(N_{c}(E_{n}/F_{n})[1] → N_{c}(E_{Λ_{k}^{n}}/F_{Λ_{k}^{n}});

this is isomorphic to N_{n}^{s}N_{c}(E/F)[1], which gives isomorphisms

H^{0}(α) ≅ N_{n}Z^{1}C(E/F), H^{i}(α) ≅ N_{n}H^{i+1}(E/F) for i > 0.

Since N_{n}H^{i}(E/F) = 0 all i > 0 by condition (Q2’), we see that H^{i}(α) = 0 for all i > 0. This implies quasi-smoothness of 𝔐ℭ(f).

Now for i > 0, the calculations above combine with Lemma 2.12 to give

H^{i}(𝔐ℭ(E)/𝔐ℭ(F)) = H^{i}(MC(E_{0})/MC(F_{0})) = H^{i+1}(E_{0}/F_{0}) = H^{i+1}(⌟E/F).

For i ≤ 0,

H^{i}(𝔐ℭ(E)/𝔐ℭ(F)) = H_{−i}H^{0}(C(E/F)[1]) = H_{−i}Z^{1}(C(E/F)) = H^{i+1}(⌟E/F),

since C(E/F) is quasi-smooth (in the sense of Definition 2.10) □

Corollary 2.18. If f : E → F is a quasi-isomorphism of DDCs, then 𝔐ℭ(f) : 𝔐ℭ(E) → 𝔐ℭ(F) is a weak equivalence.

Definition 2.19. Given a pre-DDC E, note that E^{0} acts on 𝔐ℭ(E) by conjugation. Define the derived deformation functor 𝔇𝔢𝔣(E) : sC_{Λ} → 𝕊 by

𝔇𝔢𝔣(E) := [𝔐ℭ(E)/E^{0}],

the homotopy quotient (as in Definition 1.23).

Lemma 2.20. Given a simplicial group G, and a fibration X → Y of simplicial G-sets, if each X_{n} and Y_{n} is a free G_{n}-set, then X/G → Y/G is a fibration in 𝕊.

Proof. Combine [GJ] Corollary V.2.7 and [GJ] Lemma V.3.7. □

Corollary 2.21. If f : E → F is a quasi-smooth morphism of pre-DDCs, then

(f, q) : 𝔇𝔢𝔣(E) → 𝔇𝔢𝔣(F) ×_{W̄ F^{0}} W̄ E^{0},

is quasi-smooth, where W̄ G := G\W G is a model for the classifying space BG of G (as in [GJ] §V.4).

Thus

𝔇𝔢𝔣(f) : 𝔇𝔢𝔣(E) → 𝔇𝔢𝔣(F)

is quasi-smooth, and

𝔐ℭ(E) → 𝔇𝔢𝔣(E) ×_{𝔇𝔢𝔣(F)} 𝔐ℭ(F)

is a weak equivalence.

In particular, if E is a DDC, then 𝔐ℭ(E) → 𝔇𝔢𝔣(E) is a weak equivalence.

Proof. First observe that E^{0} → F^{0} is trivially smooth, so 𝔐ℭ(E)×W E^{0} → 𝔐ℭ(F)×W E^{0} is quasi-smooth.

Given a surjection A → B in sC_{Λ}, apply Lemma 2.20, taking

X = W E^{0}(A) × 𝔐ℭ(E)(A), Y = W E^{0}(A) × 𝔐ℭ(E)(B) ×_{𝔐ℭ(F)(B)} 𝔐ℭ(F)(A),

and G = E^{0}(A). This shows that (f, q) satisfies condition (S1) from Definition 1.13. Condition (Q2) follows similarly, so (f, q) is quasi-smooth.

That 𝔇𝔢𝔣(F) is quasi-smooth follows from the observation that W̄ E^{0} → W̄ F^{0} is trivially smooth.

For the final statements, note that 𝔐ℭ(E) = 𝔇𝔢𝔣(E) ×_{W̄ E^{0}} 1, and

Y := 𝔇𝔢𝔣(E) ×_{𝔇𝔢𝔣(F)} 𝔐ℭ(F) = 𝔇𝔢𝔣(E) ×_{W̄ F^{0}} 1

If Z := W̄ E^{0} ×_{W̄ F^{0}} 1, then Z is trivially fibrant, so 1 → Z is a weak equivalence. The map 𝔐ℭ(E) → Y is then just the pullback of 1 → Z along Y → Z. □

2.2. Comparison with SDCs.

Definition 2.22. Given a pre-DDC E, define a pre-DDC DE by (DE)_{n} := (E_{n})^{Δ^{n}}, in the notation of [Pri5] Definition 3.11, i.e.

(E^{X})^{n} = (E^{n})^{X_{n}}.

For x ∈ X_{n+1}, y ∈ Y_{n+1}, z ∈ X_{m+n}, 1 ≤ i ≤ n, 0 ≤ j < n, e ∈ (E^{X})^{n} and f ∈ (E^{X})^{m}, we define the operations by

∂^{i}(e)(x) := ∂^{i}(e(∂_{i}x))
σ^{j}(e)(y) := σ^{j}(e(σ_{i}y)),
(f ∗ e)(z) := f((∂_{m+1})^{n}z) ∗ e((∂_{0})^{m}z).

Proposition 2.23. If f : E → F is a map of pre-DDCs with (1) f^{i} : E^{i} → F^{i} smooth for all i, and

(2) H^{i}(E/F) a constant simplicial complex for all i > 0, then Df : DE → DF is quasi-smooth.

In particular, DE is a DDC for all SDCs E.

Proof. By smooth base change, we know that Df is levelwise smooth. We now verify the conditions of Lemma 2.13.

C^{i}(DE/DF)_{n} = C^{i}(E_{n}/F_{n})^{Δ_{i}^{n}} = C^{i}(E_{n}/F_{n}) ⊗ C^{i}(Δ^{n}, k).

Thus

H_{∗}(DE/DF)^{i} = H_{∗}(E^{i}/F^{i}) ⊗ H_{∗}(C^{i}(Δ^{•}, k)) = 0,

since the cosimplicial complex k ⊗ Δ_{i}^{•} is contractible. Moreover,

H^{∗}(DE/DF)_{n} = H^{∗}(E_{n}/F_{n}) ⊗ H^{∗}(Δ^{n}, k) = H^{∗}(E_{n}/F_{n}),

so H^{i}(DE/DF)_{n} is constant for i > 0. □

Lemma 2.24. If X in scSp is a levelwise quasi-smooth object for which the simplicial vector spaces H^{i}(tan X) have constant simplicial structure for i > 0, then

H^{∗}(X) ≅ H^{∗}(Tot N tan X),

as defined in Definitions 1.33 and 2.5 respectively.

Proof. By [Pri4] Lemma 2.26, there is a weak equivalence X → \underline{X} to a quasi-smooth object. Since this is also a levelwise weak equivalence, we find that Tot N tan X → Tot N tan \underline{X} is a quasi-isomorphism. Now Proposition 1.21 implies that \underline{X} is quasi-smooth, so H^{∗}(X) ≅ H^{∗}(\underline{X}) ≅ H^{∗}(Tot N tan X), the last isomorphism coming from [Pri4] Theorem 1.59. □

Proposition 2.25. If E is a pre-DDC for which E → • satisfies the conditions of Proposition 2.23, then α : 𝔇𝔢𝔣(E) → 𝔇𝔢𝔣(DE) is a quasi-smooth replacement for 𝔇𝔢𝔣(E).

Proof. By Proposition 2.23, we know that 𝔇𝔢𝔣(DE) is quasi-smooth, so we just need to show that α is a weak equivalence. Now, for i > 0,

H^{i}(tan 𝔇𝔢𝔣(E)) = H^{i}(tan 𝔐ℭ(E)) = H^{i+1}(E),

which has constant simplicial structure, so H^{i}(𝔇𝔢𝔣(E)) = H^{i}(Tot N tan 𝔇𝔢𝔣(E)) = H^{i+1}(Tot NC^{•}(E)).

Similarly, H^{i}(𝔇𝔢𝔣(DE)) = H^{i+1}(Tot NC^{•}(DE)) ≅ H^{i+1}(Tot NC^{•}(E)), so [Pri4] Corollary 2.16 ensures that α is a weak equivalence. □

Corollary 2.26. For an SDC E, the functors Def_{E} (from [Pri5] Definition 4.4) and 𝔇𝔢𝔣(DE) (and hence 𝔐ℭ(DE)) are weakly equivalent (in scSp).

Proof. Recall that Def_{E} is [\underline{MC}_{E}/\underline{E}^{0}], which is a quasi-smooth replacement of [MC_{E}/E^{0}] by [Pri4] Lemma 2.26, in the sense that there is a weak equivalence [MC_{E}/E^{0}] → Def_{E}. If we let E denote the constant pre-DDC E_{n} := E, then [MC_{E}/E^{0}] = 𝔇𝔢𝔣(E), and we may apply Proposition 2.25. □

Lemma 2.27. If E is a pre-DDC for which E → • satisfies the conditions of Proposition 2.23, then for all A ∈ C_{Λ}, the map α(A) : 𝔇𝔢𝔣(E)(A) → 𝔇𝔢𝔣(DE)(A) is a weak equivalence in 𝕊.

Proof. First observe that 𝔇𝔢𝔣(E)(A)_{0} = MC_{E_{0}}(A) = 𝔇𝔢𝔣(DE)(A), and write π^{0}F := F |_{C_{Λ}}. Now, tan π^{0}𝔇𝔢𝔣(E) is the mapping cone of C^{0}(E) ⟶^{d_{c}} Z^{1}(E), so (Q2) ensures that π_{n} tan π^{0}𝔇𝔢𝔣(E) = H^{1−n}(⌟E). Observe that π_{n} tan π^{0}𝔇𝔢𝔣(DE) = H^{1−n}(⌟E), similarly.

The proofs of Proposition 1.28 and Corollary 1.29 adapt to show that α(A) is a weak equivalence in 𝕊 for all A, by taking small extensions A → B with kernel I, and considering the long exact sequences

… ⟶ H^{1−n}(⌟E) ⊗ I ⟶ π_{n}(𝔇𝔢𝔣(E)(A), x) ⟶ π_{n}(𝔇𝔢𝔣(E)(B), x) ⟶ …
↓  ↓  ↓  ↓  ↓
… ⟶ H^{1−n}(⌟E) ⊗ I ⟶ π_{n}(𝔇𝔢𝔣(DE)(A), x) ⟶ π_{n}(𝔇𝔢𝔣(DE)(B), x) ⟶ ….

associated to the fibrations 𝔇𝔢𝔣(E)(A) → 𝔇𝔢𝔣(E)(B) and 𝔇𝔢𝔣(DE)(A) → 𝔇𝔢𝔣(DE)(B).

□

3. Constructing DDCs

3.1. Simplicial monadic adjunctions. A simplicial category C has a class Ob C of objects, and for all A, B ∈ Ob C, a simplicial set \underline{Hom}_{C}(A, B) of morphisms, with the usual multiplication and identity properties. For a simplicial category C, we denote by C_{n} the category with objects Ob C and morphisms Hom_{C_{n}}(A, B) := \underline{Hom}_{C}(A, B)_{n}.

Definition 3.1. Say that a functor F : C → D of simplicial categories is an equivalence if the functors F_{n} : C_{n} → D_{n} are all equivalences.

Definition 3.2. Given a simplicial category C, set Hom_{C}(A, B) := (\underline{Hom}_{C})_{0}(A, B).

Definition 3.3. For simplicial categories D, E, and a pair of functors

\xymatrix{D \ar[r]^{G} & E \ar[l]^{F}},

recall that an adjunction F ⊣ G is a natural isomorphism

\underline{Hom}_{D}(F A, B) ≅ \underline{Hom}_{E}(A, GB).

We say that F is left adjoint to G, or G is right adjoint to F. Let ⊥ = F G, and ⊤ = GF. To give an adjunction is equivalent to giving two natural transformations, the unit and co-unit

η : id_{E} → ⊤, ε : ⊥ → id_{D},

satisfying the triangle identities εF ∘ F η = id_{F}, Gε ∘ ηG = id_{G}.

Given an adjunction

\xymatrix{D \ar[r]^{U}_{⊤} & E \ar[l]^{F}}

with unit η : id → UF and co-unit ε : F U → id, we let ⊤ = UF, and define the simplicial category E^{⊤} of ⊤-algebras to have objects

⊤E ⟶^{θ} E,

for θ ∈ \underline{Hom}_{0}(⊤E, E), such that θ ∘ η_{E} = id and θ ∘ ⊤θ = θ ∘ Uε_{F E}. We define morphisms by setting

\underline{Hom}_{E^{⊤}}(⊤E_{1} ⟶^{θ} E_{1}, ⊤E_{2} ⟶^{ϕ} E_{2}) ⊂ \underline{Hom}_{E}(E_{1}, E_{2})

to be the equaliser of

\xymatrix{\underline{Hom}_{E}(E_{1}, E_{2}) \ar[r]^{ϕ_{∗}∘⊤} \ar[r]_{θ^{∗}} & \underline{Hom}_{E}(⊤E_{1}, E_{2}).}

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

B ↦ (UF UB ⟶^{Uε_{B}} UB)

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

Definition 3.4. An adjunction

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

of simplicial categories is said to be monadic if K : D → E^{⊤} is an equivalence.

Examples 3.5. 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 3.6. Given an adjunction

\xymatrix{D \ar[r]^{V} & E \ar[l]^{G}|{⊥}},

let ⊥ = V G, so ⊥^{opp} is a monad on E^{opp}. Define E_{⊥} := ((E^{opp})^{⊥^{opp}})^{opp}, with K = K^{opp} : D → E_{⊥}. The adjunction is said to be comonadic if K : D → E_{⊥} is an equivalence.

Example 3.7. 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 u^{−1} ⊣ u_{∗} on sheaves is comonadic, so the category of sheaves on X is equivalent u^{−1}u_{∗}-coalgebras in the category 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.

3.1.1. Bialgebras. As in [Van] §IV, take a category B equipped with both a monad (⊤, μ, η) and a comonad (⊥, Δ, γ), together with a distributivity transformation λ : ⊤⊥ ⟹ ⊥⊤ satisfying various additional conditions.

Definition 3.8. Given a distributive monad-comonad pair (⊤, ⊥) on a simplicial category B, define the category B_{⊥}^{⊤} of bialgebras as follows. The objects of B_{⊥}^{⊤} are triples (θ, B, β)

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

⊤B ⟶^{⊤β} ⊤⊥B ⟶^{λ} ⊥⊤B ⟶^{⊥θ} ⊥B.

Morphisms are then given by setting

\underline{Hom}_{B_{⊥}^{⊤}}(⊤B ⟶^{θ} B ⟶^{β} ⊥B, ⊤B′ ⟶^{θ′} B′ ⟶^{β′} ⊥B′) ⊂ \underline{Hom}_{B}(B, B′)

to be the equaliser of

\xymatrix{\underline{Hom}_{B}(B, B′) \ar[r]^{(θ′_{∗}∘⊤,β′_{∗})} \ar[r]_{(θ^{∗},β^{∗}∘⊥)} & \underline{Hom}_{B}(⊤B, B′) × \underline{Hom}_{B}(B, ⊥B′).}

Example 3.9. 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.

3.2. The construction. We let sCat denote the category of simplicial categories.

Definition 3.10. Given functors A ⟶^{f} B ⟵^{g} C of simplicial categories, define the fibre product A ×_{B} C by

Ob (A ×_{B} C) = {(A, β, C) : A ∈ Ob A, C ∈ Ob C, β ∈ Iso_{B_{0}}(fA, gC)},

with morphisms

\underline{Hom}_{A×_{B}C}((A, β, C), (A′, β′, C′)) = \underline{Hom}_{A}(A, A′) ×_{β′_{∗}f,\underline{Hom}_{B}(fA,gC′),β^{∗}g} \underline{Hom}_{C}(C, C′).

Definition 3.11. We say that a functor F : C_{Λ} → Set is homogeneous if for all small extensions A → B in C_{Λ},

F (A ×_{B} C) → F (A) ×_{F (B)} F (C)

is an isomorphism. Note that this is equivalent to being a disjoint union of left-exact functors.

Similarly, a functor D : C_{Λ} → sCat is said to be homogeneous if

F (A ×_{B} C) → F (A) ×_{F (B)} F (C)

is an equivalence for all small extensions A → B.

Definition 3.12. We say that a homogeneous functor B : C_{Λ} → sCat has uniformly trivial deformation theory if

(1) for all A ∈ C_{Λ} and all B_{1}, B_{2} ∈ Ob B(A), the functor \underline{Hom}_{B}(B_{1}, B_{2}) : C_{A} → Set of morphisms from B_{1} to B_{2} is trivially smooth (in the sense of Definition 1.19) and homogeneous;

(2) for A′ ↠ A in C_{Λ}, B_{0}(A′) → B_{0}(A) is essentially surjective.

Now, assume that we have 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}^{⊣}}

of adjunctions of homogeneous simplicial category-valued functors on C_{Λ}, with F ⊣ U monadic and G ⊢ V comonadic. Let

⊤_{h} = UF  ⊥_{h} = F U
⊥_{v} = V G  ⊤_{v} = GV,

with

η : 1 → ⊤_{h}, γ : ⊥_{v} → 1, ε : ⊥_{h} → 1 and α : 1 → ⊤_{v}.

Assume that these adjunctions satisfy the simplicial analogues of [Van] §IV or [Pri2] §2, in other words that U and V commute with everything (although G and F need not commute).

Fix D ∈ Ob D(k), such that we may lift UV D ∈ Ob B(k) to B ∈ Ob B(Λ), up to isomorphism (in B_{0}(k)).

Theorem 3.13. There is a natural pre-DDC E associated to this diagram, given by

E^{n} = \underline{Hom}_{B}(⊤_{h}^{n}B, ⊥_{v}^{n}B)_{UV (α_{D}^{n}∘ε_{D}^{n})}.

If E is levelwise smooth, satisfying Condition (Q2) of Definition 2.8, then the classifying space W̄ D_{D,id} is canonically weakly equivalent to the restriction π^{0}𝔇𝔢𝔣(E) (from Lemma 2.27) as a functor from C_{Λ} to 𝕊. Here D_{D,id}(A) is the simplicial groupoid given by the fibre product

D(A) ×_{D(k)} (D, id),

where (D, id) is the category with one object and one morphism.

Proof. For each m, E_{m} is the SDC defined in [Pri2] §2 associated to the monad ⊤_{h} and comonad ⊥_{v} over the category B_{m}.

Since the adjunctions are monadic or comonadic, the proof of [Pri2] Theorem 2.2 adapts to give functorial equivalences

K(A) : D(A) → B_{⊥_{v}}^{⊤_{h}}(A)

between D and the simplicial category of (⊤_{h}, ⊥_{v})-bialgebras.

Let D′ ∈ Ob B_{⊥_{v}}^{⊤_{h}}(k) be the bialgebra over B̄ ∈ Ob B(k), with bialgebraic structure coming from the isomorphism UV D ≅ B̄. Let G be the full subcategory of B_{⊥_{v}}^{⊤_{h}}(A) on objects

MC_{E_{0}}(A) = {ω ∈ Ob B^{⊤_{h},⊥_{v}}(A) : X̄ = D′ ∈ Ob B^{⊤_{h},⊥_{v}}(k).

Morphisms in G are just

\underline{Hom}_{G}(ω, ω′) = \{f ∈ E^{0} : f ∗ ω = ω′ ∗ f\},

from which we deduce that G is a simplicial groupoid. Moreover, observe that G → (B_{⊥_{v}}^{⊤_{h}})_{D′,id} is an isomorphism of simplicial categories, so G is equivalent to D_{D,id}. In particular, this implies that D_{D,id} is a simplicial groupoid. It therefore suffices to compare G with 𝔇𝔢𝔣(E).

Lemma 3.14. The functor G : C_{Λ} → sGpd is quasi-smooth, in the sense that it maps small extensions to fibrations (as defined in [GJ] §V.7).

Proof of lemma. Smoothness of E_{0}^{0} implies that the path-lifting property is satisfied. Given K ↪ L ∈ 𝕊, and a small extension A → B with kernel I, the obstruction to lifting the diagram

K ⟶ \underline{Hom}_{G}(ω, ω′)(A)
↓  ↓
L ⟶ \underline{Hom}_{G}(ω, ω′)(A)

lies in H^{1}(ker(C^{•}(E_{L}) → C^{•}(E_{K}))). If we write V^{•} = ker(C^{•}(E_{L}) → C^{•}(E_{K})), then we have an exact sequence

H^{0}(E_{L}) ⟶^{α} H^{0}(E_{K}) → H^{1}(V^{•}) → H^{1}(E_{L}) ⟶^{β} H^{1}(E_{K}).

If K ↪ L is a trivial cofibration, then Condition (Q2) of Definition 2.8 ensures that α is surjective and β an isomorphism, so the obstruction is zero and the lift exists, proving that \underline{Hom}_{G}(ω, ω′) is quasi-smooth, as required. □

Now, the inclusions MC_{E_{0}} ↪ MC(E) and \underline{Hom}_{G}(ω, ω′) ↪ E^{0} define a morphism

α : W̄ G → π^{0}𝔇𝔢𝔣(E)

of quasi-smooth functors C_{Λ} → Set. Note that W̄ G_{0} = MC_{E_{0}} = π^{0}𝔇𝔢𝔣(E)_{0}.

(Q2) also ensures that α is a weak equivalence on tangent spaces, with π_{n}W̄ G(k[ϵ]) = H^{1−n}(⌟E). As in the proof of Lemma 2.27, this implies that α(A) is a weak equivalence in 𝕊 for all A. □

Remark 3.15. If B has uniformly trivial deformation theory, then note that D̃ always exists, and that the pre-DDC E of Theorem 3.13 automatically satisfies Definition 2.8.(Q1).

However, if E just satisfies all the conditions of Theorem 3.13, then Propositions 2.23 and 2.25 then give a DDC DE, which by Lemma 2.27 also has W̄ D_{D,id} ∼ π^{0}𝔇𝔢𝔣(DE).

3.3. Deformations of diagrams and invariance under weak equivalence. In a similar vein, we may study deformations of a morphism, or even of a diagram.

Definition 3.16. Define Δ_{∗∗} to be the subcategory of the ordinal number category Δ containing only those morphisms f : m → n with f(0) = 0, f(m) = n. 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 3.17. Given a monoidal category C and a set O, recall from [Pri3] that a C-valued quasi-descent datum X on objects O consists 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 commute for all , 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 ∗.

Note that a pre-DDC over Λ is a quasi-descent datum (on one object) in the monoidal category (sSp, ×).

Definition 3.18. 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.

We say that D is an enrichment of a C-enriched category F if Ob F ≅ Ob D and F(x, y) ≅ D^{0}(x, y), compatible with the product and identities.

Proposition 3.19. For a diagram of simplicial category-valued functors as in §3.2, the sSp-enriched category B(Λ) has a natural enrichment in QDat(sSp). If the simplicial structure on B is constant, then this enrichment is in QDat(Sp).

Proof. This is just [Pri3] Proposition 2.12. The enriched Hom-set \underline{ℋ om} (B, B′) : C_{Λ} → 𝕊^{Δ_{∗∗}} is given by

\underline{ℋ om} ^{n}(B, B′) := \underline{Hom}_{B}(⊤_{h}^{n}B, ⊥_{v}^{n}B′).

If the simplicial structure on B is constant, then \underline{Hom}_{B} = Hom_{B}, so \underline{ℋ om} (B, B′) lies in Sp. □

Definition 3.20. Given a morphism f : D → D′ in D(k) for which UV D, UV D′ lift to B, B′ in B(Λ), define

E_{D/B}^{n}(f) := \underline{ℋ om} ^{n}(B, B′)_{UV (α_{D′}^{n}∘f∘ε_{D}^{n})} ∈ sSp

Write E_{D/B}^{∗}(D) := E_{D/B}^{∗}(id_{D}).

Definition 3.21. Given a morphism f : D → D′ in D(k) for which E_{D/B}^{∗}(f) ∈ (sSp_{k})^{Δ_{∗∗}} is levelwise smooth, define

C_{D/B}^{•}(f) := tan E_{D/B}^{∗}(f),

and note that that this becomes a cosimplicial complex (of simplicial complexes), by [Pri5] Lemma 3.10. Explicitly,

C_{D/B}^{n}(f) = tan \underline{Hom}_{D|_{C_{k}}} (⊥_{h}^{n+1}D, ⊤_{h}^{n+1}D′)_{α_{D′}^{n+1}∘f∘ε_{D}^{n}}.

Define

\underline{Ext}_{D/B}^{n}(f) := H^{n}(C_{D/B}^{•}(f)) ∈ sVect_{k}
Ext_{D/B}^{n}(f) := H^{n}(Tot NC_{D/B}^{•}(f)) ∈ Vect_{k}.

Definition 3.22. Say that a morphism f : D → D′ in D(k) is Q2 over B if (1) UV D, UV D′ lift to B(Λ),

(2) E_{D/B}^{∗}(f) is levelwise smooth, and

(3) \underline{Ext}_{D/B}^{i}(f) is a constant simplicial complex for i > 0.

We say that f is quasi-smooth over B if in addition H_{∗}C_{D/B}^{n}(f) = 0 for all n. Remark 3.23. Note that if f is Q2 over B, then we have Ext_{D/B}^{∗}(f) = H^{∗}(⌟NC_{D/B}^{•}(f)), by Lemma 2.12. Definition 3.24. Given a small category 𝕀, and an 𝕀-diagram 𝔻 : 𝕀 → D(k) with objects UV 𝔻(i) lifting to B(Λ), define the pre-DDC E_{D/B}^{•}(𝔻) by

E_{D/B}^{n}(𝔻) = ∏_{\begin{matrix}i_{0}⟶^{f_{1}}i_{1}⟶^{f_{2}}...⟶^{fn}i_{n} \\ in 𝕀 \end{matrix}} E^{n}(𝔻(f_{n} ∘ f_{n−1} ∘ … f_{0})) = ∏_{x∈B𝕀_{n}} E^{n}(𝔻(∂_{1}^{n−1}x)),

where B𝕀 is the nerve of 𝕀 (so B𝕀_{0} = Ob (𝕀), B𝕀_{1} = Mor (𝕀)), and ∂_{1}^{−1} := σ_{0}. The operations are defined as in Definition 2.22. Lemma 3.25. Given an 𝕀-diagram 𝔻 : 𝕀 → D(k) with all morphisms 𝔻(f) quasi-smooth (resp. Q2) over B, the pre-DDC E_{D/B}^{•}(𝔻) is quasi-smooth (resp. is levelwise smooth and satisfies Definition 2.8.(Q2)). Proposition 3.26. Given an 𝕀-diagram 𝔻 : 𝕀 → D(k) with all morphisms Q2, the classifying space W̄ (D^{𝕀})_{𝔻,id} and π^{0}𝔇𝔢𝔣(E_{D/B}^{•}(𝔻)) are canonically weakly equivalent as functors from C_{Λ} to 𝕊. Proof. This is just [Pri3] Lemma 1.36. □ Definition 3.27. Say that a morphism f : D → D′ in D(k) is an Ext_{D/B}-equivalence if UV D, UV D′ lift to B, B′ in B(Λ), with E^{∗}(f) levelwise smooth, and the maps

Ext_{D/B}^{∗}(id_{D}) ⟶^{f_{∗}} Ext_{D/B}^{∗}(f) ⟵^{f^{∗}} Ext_{D/B}^{∗}(id_{D′})

are isomorphisms. Proposition 3.28. If a morphism f : D → D′ in D(k) is an Ext_{D/B}-equivalence, with the morphisms f, id_{D}, id_{D′} all Q2, then the DDCs DE_{D/B}^{•}(D) and DE_{D/B}^{•}(D′) are quasiisomorphic. Proof. Let 𝕀 := (• → •) be the category with two objects and one non-identity morphism, and consider the diagram 𝔻 : 𝕀 → D(k) given by D ⟶^{f} D′. By Lemma 3.25 and Proposition 2.23, we know that the pre-DDCs DE_{D/B}^{•}(D), DE_{D/B}^{•}(D′) and DE_{D/B}^{•}(𝔻) are all DDCs.

The inclusions of objects into 𝕀 give morphisms E_{D/B}^{•}(D) ← E_{D/B}^{•}(𝔻) → E_{D/B}^{•}(D′). We just need to describe the cohomology groups H^{∗}(⌟E_{D/B}^{•}(𝔻)) to show that these induce quasi-isomorphisms.

The tangent space C_{D/B}^{•}(𝔻) is the diagonal cosimplicial complex associated to the bicosimplicial complex

∏_{x∈B𝕀_{m}} C_{D/B}^{n}(𝔻(∂_{1}^{m−1}x)),

whose horizontal normalisation is the cochain complex

C_{D/B}^{•}(id_{D}) × C_{D/B}^{•}(id_{D′}) ⟶^{(f_{∗},−f^{∗})} C_{D/B}^{•}(f)

in degrees 0 and 1.

Thus C_{D/B}^{•}(𝔻) is the mapping cone of the morphism (f_{∗}, −f^{∗}). Since f is an Ext_{D/B}- equivalence, we deduce that the maps Ext_{D/B}^{∗}(id_{D}) ← H^{∗}(⌟E_{D/B}^{•}(𝔻)) → Ext_{D/B}^{∗}(id_{D′}) are indeed isomorphisms. □

3.3.1. Constrained deformations. We now consider a generalisation, by taking a small diagram

𝔻 : 𝕀 → D(k),

a subcategory 𝕁 ⊂ 𝕀, and 𝔻̃|_{𝕁} : 𝕁 → D(Λ) lifting 𝔻|_{𝕁}. We wish to describe deformations of 𝔻 which agree with 𝔻̃|_{𝕁} on 𝕁. Note that when 𝕀 = (0 → 1) and 𝕁 = {1}, this is the type of problem considered in [FM] and [Ran].

Proposition 3.29. Given an 𝕀-diagram 𝔻 : 𝕀 → D(k) with all morphisms Q2, and with 𝔻̃|_{𝕁} as above, the simplicial groupoid of deformations of 𝔻 fixing 𝔻̃|_{𝕁} is governed by the pre-DDC

E_{D/B}^{•}(𝔻) ×_{E_{D/B}^{•}(𝔻|_{𝕁})} •,

where • → E_{D/B}^{•}(𝔻|_{𝕁}) is defined by the object of MC(E_{D/B}^{•}(𝔻|_{𝕁}))_{0} corresponding to 𝔻̃|_{𝕁}.

Proof. We need to show that the classifying space

W̄ (D^{𝕀} ×_{D^{𝕁}}^{h} 𝔻̃|_{𝕁})_{𝔻,id}

of the homotopy fibre of simplicial categories is canonically weakly equivalent to

π^{0}𝔇𝔢𝔣(E_{D/B}^{•}(𝔻) ×_{E_{D/B}^{•}(𝔻|_{𝕁})} •)

as a functor from C_{Λ} to 𝕊.

We know that the functor 𝔇𝔢𝔣 preserves inverse limits, so

π^{0}𝔇𝔢𝔣(E^{•}(𝔻) ×_{E^{•}(𝔻|_{𝕁})} •) = π^{0}𝔇𝔢𝔣(E_{D/B}^{•}(𝔻)) ×_{π^{0}𝔇𝔢𝔣(E^{•}(𝔻|_{𝕁}))} •

By Lemma 2.27, Lemma 3.25 and Corollary 2.21, we know that π^{0}𝔇𝔢𝔣(E_{D/B}^{•}(𝔻))(A) → π^{0}𝔇𝔢𝔣(E^{•}(𝔻|_{𝕁}))(A) is a fibration in 𝕊, so the fibre over any point is the homotopy fibre. Proposition 3.26 now shows that this is equivalent to the homotopy fibre of W̄ (D(A)^{𝕀}) → W̄ (D(A)^{𝕁}) over 𝔻̃|_{𝕁}, as required. □

4. Examples

We now show how to apply Theorem 3.13, combining it with Definitions 2.16 or 2.19 to obtain derived deformation functors. This gives many new examples coming from categories with non-trivial simplicial structure.

4.1. Chain complexes. We will denote chain complexes by V_{•}, and their underlying graded modules by V_{∗}.

Definition 4.1. Define dgFMod(A) to be the category of chain complexes of flat modules over A. We make this into a simplicial category by defining the simplicial normalisation N^{s}\underline{Hom}(U_{•}, V_{•}) to be the chain complex

N_{n}^{s}\underline{Hom}(U_{•}, V_{•}) := \begin{cases}Hom(U_{•}, V_{•}) & n = 0 \\ ∏_{i≥0} Hom(U_{i}, V_{i+n}) & n > 0 \end{cases},

with boundary map ad_{d}(f) := d ∘ f ± f ∘ d. This determines the simplicial module \underline{Hom}(U_{•}, V_{•}) := (N^{s})^{−1}N^{s}\underline{Hom}(U_{•}, V_{•}) by the Dold-Kan correspondence.

Definition 4.2. Define gFMod(A) to be the category of flat ℕ_{0}-graded modules over A, with the simplicial structure

N_{n}^{s}\underline{Hom}(U_{∗}, V_{∗}) := \begin{cases}Hom(U_{∗}, V_{∗}) & n = 0 \\ Hom(U_{∗}, V_{∗}[n]) × Hom(U_{∗}, V_{∗}[n − 1]) & n > 0 \end{cases},

where the boundary map is given by d(f, g) := (g, 0).

Lemma 4.3. The functor gFMod : C_{Λ} → sCat has uniformly trivial deformation theory.

Proof. Since flat A-modules are free, it follows that objects lift. The other properties from Definition 3.12 now follow by a simple calculation. □

Definition 4.4. Let the forgetful functor dgFMod → gFMod be given by V_{•} ↦ V_{∗}, and defined on simplicial morphisms by mapping f ∈ N_{n}^{s}\underline{Hom}(U_{•}, V_{•}) to (f, ad_{d}(f)) ∈ N_{n}^{s}\underline{Hom}(U_{∗}, V_{∗}).

Lemma 4.5. The forgetful functor dgFMod → gFMod of simplicial categories has a right adjoint G, and the resulting adjunction is comonadic.

Proof. Define (GV_{∗})_{n} := V_{n} ⊕V_{n−1}, with d(v, w) = (w, 0). The unit α : U_{•} → G(U_{∗}) of the adjunction is α(u) = (u, du), for any chain complex U_{•}, and the co-unit γ : G(V_{∗}) → V_{∗} is the map γ(v, w) = v of graded modules. □

Let ⊥ = V G and ⊤ = GV, for V the forgetful functor.

Proposition 4.6. For U_{•} ∈ dgFMod(k), the pre-DDC

E^{n}(A) := \underline{Hom}_{gFMod_{A}}(Ũ_{∗} ⊗ A, ⊥^{n}Ũ_{∗} ⊗ A)_{V (α_{U}^{n})}

of Theorem 3.13 is quasi-smooth, with cohomology

H^{∗}(⌟E) = H^{∗}(… ⟶^{ad_{d}} Hom_{g_{ℤ}Mod_{k}}(U_{∗}, U_{∗}[−n]) ⟶^{ad_{d}} Hom_{g_{ℤ}Mod_{k}}(U_{∗}, U_{∗}[−n − 1]) ⟶^{ad_{d}} …)
= 𝔼xt_{dg_{ℤ}Vect_{k}}^{∗}(U_{•}, U_{•}),

for dg_{ℤ}Vect_{k} the category of ℤ-graded chain complexes over k, and 𝔼xt the hyperext functor of [Wei] §10.7.

Proof. Observe that H^{∗}(E_{n}) is cohomology of the complex

\underline{Hom}_{n}(U_{•}, ⊤U_{•}) → \underline{Hom}_{n}(U_{•}, ⊤^{2}U_{•}) → …

associated to the monad ⊤ (as in [Wei] §8.7), so for n > 0, N_{n}^{s}H^{∗}(E) = H^{∗}(N_{n}^{s}E) is cohomology of the complex

Hom(U_{∗}, (⊤U)_{∗}[n]) → Hom(U_{∗}, (⊤^{2}U)_{∗}[n]) → ….
Now, (⊤U)_{∗}  =  ⊥(U_{∗}), and the augmented cosimplicial complex

\xymatrix{U_{∗} \ar[r] & ⊥(U_{∗}) \ar[r] \ar[r] & ⊥^{2}(U_{∗}) \ar[r] & …} is canonically contractible (in the sense of [Wei] 8.4.6), giving

N_{n}^{s}H^{i}(E) = \begin{cases}Hom(U_{∗}, U_{∗}[n]) & i = 0 \\ 0 & i > 0, \end{cases}

for n > 0, so E is quasi-smooth, and H^{−n}(⌟E) = 𝔼xt_{dg_{ℤ}Vect_{k}}^{−n}(U_{•}, U_{•}) for n ≥ 0.

Since ker(γ_{U} : ⊥U_{∗} → U_{∗}) = U_{∗}[−1], the cosimplicial normalisation N_{c}^{n}(⊥^{•}U_{∗}) = N_{c}^{n−1}(⊥^{•}U_{∗})[−1], so N_{c}^{n}(⊥^{•}U_{∗}) = U_{∗}[−n]. Thus N_{c}C^{•}(E_{0}) is just

Hom(U_{∗}, U_{∗}) ⟶^{ad_{d}} Hom(U_{∗}, U_{∗}[−1])) ⟶^{ad_{d}} Hom(U_{∗}, U_{∗}[−2])) ⟶^{ad_{d}} …,

so H^{i}(⌟E) = 𝔼xt_{dg_{ℤ}Vect_{k}}^{i}(U_{•}, U_{•}) for i > 0. □

Remarks 4.7. (1) Dually, we may consider deformations of (non-negatively graded) cochain complexes. This simplicial category is monadic over graded modules.

(2) We may incorporate the constructions of this section into more interesting examples. For instance, deformations of a complex of 𝒪_{X}-modules on an algebraic space X are given by considering the diagram

\xymatrix{dg𝒪_{X}Mod(X) \ar[r]_{⊤} \ar[d]_{u^{−1}} & dgMod(X) \ar[l]^{𝒪_{X}⊗−} \ar[d]_{u^{−1}} \\ g(u^{−1}𝒪_{X})Mod(X′) \ar[u]_{u_{∗}G}^{⊣} \ar[r]_{⊤} & gMod(X′), \ar[l]^{(u^{−1}𝒪_{X})⊗−} \ar[u]_{u_{∗}G}^{⊣}}

of simplicial categories, where u : X′ → X is the map to X from its set of geometric points. The resulting pre-DDC will be quasi-smooth whenever Ext_{𝒪_{X}}^{i} (M_{m}, M_{n}) = 0 for all i > 0 and n > m.

4.2. Simplicial complexes.

Definition 4.8. Define sFMod(A) to be the category of simplicial flat modules over A. We make this into a simplicial category by setting

\underline{Hom}(U_{•}, V_{•})_{n} := Hom(Δ^{n} ⊗ U_{•}, V_{•}),

where, for a set X and module U, we set U ⊗ X := ⨁_{x∈X} U.

Definition 4.9. Define Δ_{∗} to be the subcategory of the ordinal number category Δ containing only those morphisms fixing 0. Given a category C, define the category s_{+}C of almost simplicial complexes (resp.the category c_{+}C of almost cosimplicial complexes) in C to consist of functors (Δ_{∗})^{opp} → C (resp. Δ_{∗} → C). Thus an almost simplicial object X_{∗} consists of objects X_{n} ∈ C, with all of the operations ∂_{i}, σ_{i} of a simplicial complex except ∂_{0}, satisfying the usual relations. Similarly, an almost cosimplicial complex has all of the coface and coboundary operations except ∂^{0}.

From now on, we will denote simplicial sets by X_{•}, and their underlying almost simplicial complexes by X_{∗}.

Definition 4.10. Define a simplicial structure on the category s_{+}FMod(A) by setting

\underline{Hom}(U_{∗}, V_{∗})_{n} := Hom(Δ_{∗}^{n} ⊗ U_{∗}, V_{∗}).

Remark 4.11. Recall that the Dold-Kan correspondence gives an equivalence N : sMod → dgMod of categories, by the formula N(V)_{n} = ⋂_{i=1}^{n} ker(∂_{i} : V_{n} → V_{n−1}), with d := ∂_{0}. Observe that this extends to an equivalence N : s_{+}Mod → gMod of categories, given by the same formula. This is only a weak equivalence of simplicial categories, not an equivalence.

Lemma 4.12. The forgetful functor sFMod → s_{+}FMod of simplicial categories has a right adjoint G_{∂}, and the resulting adjunction is comonadic.

Proof. Let (G_{∂}V_{∗})_{n} := V_{n} ⊕ V_{n−1} ⊕ … ⊕ V_{0}, with operations

∂_{i}(v_{n}, …, v_{0}) = (∂_{i}v_{n}, ∂_{i−1}v_{n−1}, …, ∂_{1}v_{n−i+1}, v_{n−i−1}, …, v_{1}, v_{0})
σ_{i}(v_{n}, …, v_{0}) = (σ_{i}v_{n}, σ_{i−1}v_{n−1}, …, σ_{0}v_{n−i}, v_{n−i}, …, v_{1}, v_{0}).

The unit α : U_{•} → (G_{∂}U_{∗})_{•} of the adjunction is

α(u) = (u, ∂_{0}u, ∂_{0}^{2}u, …, ∂_{0}^{n}u),

for any simplicial complex U_{•}, and u ∈ U_{n}. The co-unit γ : G_{∂}V_{∗} → V_{∗} is the map γ(v_{n}, …, v_{0}) = v_{n} of almost simplicial complexes. □

Remark 4.13. The forgetful functor V_{∂} also has a left adjoint, which does not respect the simplicial structure of the categories. It is given by L_{∂}(V_{∗})_{n} = V_{n+1}, with ∂_{i}^{L_{∂}V} = ∂_{i+1}^{V}, σ_{i}^{L_{∂}V} = σ_{i+1}^{V}, and unit σ_{0} : V_{∗} → V_{∂}L_{∂}(V_{∗}) = L_{∂}(V)_{∗}. Note that L_{∂}V_{∂} is the functor DEC^{opp} defined on simplicial sets in [Gle].

Definition 4.14. Define objects Ξ^{n} ∈ s_{+}Set by Ξ_{m}^{n} = Hom_{Δ_{∗}}([m], [n]), and let ∂Ξ^{n} be the boundary of Ξ^{n} (i.e. the union of the images of all maps Ξ^{n−1} → Ξ^{n}). Note that L(Ξ^{n}) = Δ^{n} and, for n > 0, L(∂Ξ^{n}) = Λ_{0}^{n}, the 0th horn.

Lemma 4.15. For X ∈ s_{+}Set, X_{0} → L(X) is a weak equivalence.

Proof. Any injective map f : Z → X in s_{+}Set is an inductive limit of pushouts of maps ∂Ξ^{n} → Ξ^{n} for n ≥ 0. If f_{0} : Z_{0} → X_{0} is an isomorphism, we may take n > 0 only. Then L(f) : L(Z) → L(X) is an inductive limit of pushouts of maps Λ_{0}^{n} → Δ^{n}, so is a trivial cofibration. Taking Z = X_{0} gives the required result. □

Let ⊥ = V_{∂}G_{∂} and ⊤ = G_{∂}V_{∂}, for V_{∂} : sFMod(A) → s_{+}FMod(A) the forgetful functor.

Proposition 4.16. s_{∗}FMod has uniformly trivial deformation theory, so given U_{•} ∈ sFMod(k), we may lift U_{∗} ∈ s_{+}FMod(k) to Ũ_{∗} ∈ s_{+}FMod(Λ). The conditions of Theorem 3.13 are satisfied, and the pre-DDC

E^{n}(A) := \underline{Hom}_{s_{+}FMod(A)}(Ũ_{∗}, ⊥^{n}Ũ_{∗})_{V (α_{U}^{n})}

is then a DDC, with cohomology H^{∗}(⌟E) given by the complex

… ⟶^{ad_{d}} Hom_{g_{ℤ}Mod_{k}}(NU_{∗}, NU_{∗}[−n]) ⟶^{ad_{d}} Hom_{g_{ℤ}Mod_{k}}(NU_{∗}, NU_{∗}[−n − 1]) ⟶^{ad_{d}} …,

i.e. 𝔼xt_{dg_{ℤ}Vect_{k}}^{∗}(NU_{•}, NU_{•}).

Proof. This is similar to Proposition 4.6. The only difficulty lies in establishing Definition 2.8.(Q2’):

N_{n}C^{i}(E) = N_{n}\underline{Hom}_{sFMod(k)}(U_{•}, ⊤^{i+1}U_{•}) = Hom_{sFMod(k)}(U_{•} ⊗ (Δ^{n}/Λ_{0}^{n}), ⊤^{i+1}U_{•}).

Now, since k ⊗ Λ_{0}^{n} → k ⊗ Δ^{n} is a weak equivalence admitting a retraction, P := U_{•} ⊗ (Δ^{n}/Λ_{0}^{n}) is trivially cofibrant, so is a projective object in the category sFMod(k). The cosimplicial complex U^{•} in sFMod(k) given by U^{i} := ⊤^{i+1}U_{•} is a resolution of U := U_{•} (since the augmented cosimplicial complex V_{∂}U → V_{∂}U^{•} is contractible). Thus

H^{∗}N_{n}C^{•}(E) = 𝔼xt_{sFMod(k)}^{∗}(P, U^{•}) = Ext_{sFMod(k)}^{∗}(P, U) = Hom_{sFMod(k)}(P, U).

We have therefore shown that for n > 0,

N_{n}H^{i}(E) = \begin{cases}N_{n}\underline{Hom}_{sFMod(k)}(U_{•}, U_{•}) & i = 0 \\ 0 & i > 0. \end{cases}

□

4.3. Simplicial algebras. Although the results in this section are expressed for commutative algebras, they will hold for any category equipped with a suitable forgetful functor to flat modules, and in particular algebras over any operad.

Definition 4.17. Let FAlg(A) be the category of flat (commutative) A-algebras, with sFAlg(A) := FAlg(A)^{Δ^{opp}} and s_{+}FAlg(A) := FAlg(A)^{Δ_{∗}^{opp}} . Recall that, for K ∈ 𝕊 and R ∈ sFAlg(A), we define R ⊗ K ∈ sFAlg(A) by

(R ⊗ K)_{n} := R_{n}^{⊗K_{n}} = \overbrace{R_{n} ⊗ R_{n} ⊗ … ⊗ R_{n}}^{|K_{n}|} .

Define ⊗K : s_{+}FAlg → s_{+}FAlg by the same formula.

Now, we make sFAlg(A), s_{+}FAlg(A) into simplicial categories by setting

\underline{Hom}_{sFAlg}(R_{•}, S_{•})_{n} := Hom_{sFAlg}(Δ^{n} ⊗ R_{•}, S_{•}),
\underline{Hom}_{s_{+}FAlg}(R_{∗}, S_{∗})_{n} := Hom_{s_{+}FAlg}(Δ_{∗}^{n} ⊗ R_{∗}, S_{∗}).

We now consider the commutative diagram

\xymatrix{sFAlg \ar[r]_{⊤}^{U} \ar[d]_{V_{∂}} & sFMod \ar[l]^{Symm} \ar[d]_{V_{∂}} \\ s_{+}FAlg \ar[u]_{G_{∂}}^{⊣} \ar[r]_{⊤}^{U} & s_{+}FMod \ar[l]^{Symm} \ar[u]_{G_{∂}}^{⊣}}

of adjunctions of homogeneous simplicial category-valued functors on C_{Λ}.

Definition 4.18. Given an almost simplicial A-algebra R_{∗}, define the category s_{+}Mod(R_{∗}) to consist of almost simplicial A-modules M_{∗} equipped with an associative multiplication R_{∗} ⊗ M_{∗} → M_{∗}, respecting the almost simplicial structures. This has a simplicial model structure (by applying [Hir] Theorem 11.3.2 to the forgetful functor s_{+}Mod(R_{∗}) → s_{+}Mod(A)).

All objects of s_{+}Mod(R_{∗}) are fibrant. Since A ⊗ (∂Δ^{n})_{∗} → A ⊗ (Δ^{n})_{∗} has a retraction in s_{+}Mod(A), we see that for all cofibrant C_{∗} ∈ s_{+}Mod(R_{∗}),

\underline{Hom}_{s_{+}Mod(R_{∗})}(C_{∗}, M_{∗})

is trivially fibrant.

Proposition 4.19. Fix R_{•} ∈ sFAlg(k), set M_{∗} := UV_{∂}R_{•}, and choose a lift M̃_{∗} ∈ s_{+}FMod(Λ).

Theorem 3.13 gives a pre-DDC

E^{n} := \underline{Hom}_{s_{+}FMod}((USymm)^{n}M̃_{∗}, (V_{∂}G_{∂})^{n}R̃_{∗})_{UV_{∂}(α_{R}^{n}∘ε_{R}^{n})}.

Moreover, if R_{•} ∈ sFAlg(k) is cofibrant, then the pre-DDC E of Theorem 3.13 is a DDC. The almost simplicial k-algebra R_{∗} then lifts to R̃_{∗} ∈ s_{+}FAlg(Λ), and E is quasiisomorphic to the DDC defined by

(E′)_{n} = \underline{Hom}_{s_{+}FAlg}(R̃_{∗}, (V_{∂}G_{∂})^{n}R̃_{∗})_{V_{∂}(α_{R}^{n})}

coming from the comonadic adjunction \xymatrix{sFAlg \ar[r]_{⊥}^{V_{∂}} & s_{+}FAlg \ar[l]^{G_{∂}}}.

Proof. It is straightforward to verify [Pri2] equations 1–4, since all our constructions commute with forgetful functors, so E is a pre-DDC. Since s_{+}FMod is uniformly of trivial deformation theory, Definition 2.8.(Q1) is satisfied by E.

To establish quasi-smoothness, we must compute cohomology groups. Given a k-algebra S, recall that the cotangent complex is given by 𝕃_{n}(S/k) = J_{n}/(J_{n})^{2}, where J_{n} is the kernel of the diagonal map (SymmU)^{n+1}(S) ⊗_{k} S → S. The cosimplicial complex C^{•}(E_{n}) is then given by C^{m}(E_{n}) = Hom_{s_{+}Mod(R_{∗})}(𝕃_{m}(R/k)_{∗} ⊗ Δ^{n}, G_{∂}^{m}R_{∗}). Thus H^{∗}(E_{n}) is the total cohomology of the double complex

C^{ij} = Hom_{s_{+}Mod(R_{∗})}(𝕃_{i}(R/k)_{∗} ⊗ Δ^{n}, G_{∂}^{j}R_{∗}) = Hom_{s_{+}Mod(R_{∗})}(𝕃_{i}(R/k)_{∗}, (G_{∂}^{j}R_{∗})^{Δ^{n}}).

Now, if R_{•} is cofibrant, the augmented complex 𝕃_{•}(R_{∗}) → Ω(R_{∗}/k) is a levelwise cofibrant resolution in s_{+}Mod(R_{∗}). Since all maps in s_{+}Mod(R_{∗}) are weak equivalences, cofibrant modules are projective, so the complex is contractible.

Define André-Quillen cohomology on s_{+}Mod(R_{∗}) by D^{q}(R_{∗}/k, M_{∗}) := H^{q}Hom_{s_{+}Mod(R_{∗})}(𝕃_{•}(R/k)_{∗}, M_{∗}). Given a small extension A → B with kernel I, and a flat almost simplicial B-algebra S_{∗}, note that the obstruction to lifting S_{∗} to a flat A-algebra lies in D^{2}(S_{∗}/B, S_{∗} ⊗_{B} I) = D^{2}((S_{∗} ⊗_{B} k)/k, S_{∗} ⊗_{B} I), applying [Pri2] Theorem 2.2 to the adjunction

\xymatrix{s_{+}FAlg \ar[r]_{⊤} & s_{+}FMod \ar[l]^{Symm}}.

This ensures that R_{∗} lifts to some R̃_{∗} ∈ s_{+}FAlg(Λ), so E′ can be defined.

Similarly to [Pri2] §3.2.2, we see that E′ is a levelwise smooth DDC, and that H^{∗}(E′_{n}) is cohomology of the complex (C′)^{m} = Hom_{s_{+}Mod(R_{∗})}(Ω(R/k)_{∗} ⊗ Δ^{n}, G_{∂}^{m}R_{∗}).

Now, the canonical map E′ → E gives quasi-isomorphisms E′_{n} → E_{n} for all n. We know that E automatically satisfies (Q1). Since Ω(R_{∗}/k) is a cofibrant R_{∗}-module, the tangent space C(E′)^{n} = \underline{Hom}_{s_{+}Mod(R_{∗})}(Ω(R_{∗}/k), G_{∂}^{n}R_{∗}) is trivially fibrant, so E′ also satisfies (Q1).

It only remains to show that E′ satisfies (Q2’); the proof of Proposition 4.16 adapts. □

Remark 4.20. We may weaken the condition that R_{•} be cofibrant to requiring that the cotangent complex diag 𝕃_{•}(R_{•}/k) of R_{•} is equivalent in Ho(sMod(R_{•})) to Ω(R_{•}/k), and that the latter is cofibrant. If a k-algebra R (with constant simplicial structure) is smooth,

[Ill] Proposition III.3.1.2 implies that this holds.

Definition 4.21. Given a simplicial k-algebra R_{•}, and a simplicial R_{•}-module M_{•}, define the simplicial vector space \underline{Der}(R, M) of derived derivations by

\underline{Der}(R_{•}, M_{•})_{n} := Der(R_{•} ⊗ Δ^{n}, M_{•}),

the set of simplicial k-algebra morphisms f : R_{•} ⊗Δ^{n} → R_{•} ⊕M_{•}ϵ extending the canonical map R_{•} ⊗ Δ^{n} → R_{•}, where ϵ^{2} = 0.

Remark 4.22. For R_{•} ∈ sAlg(k) cofibrant, and E as in Proposition 4.19, H^{n}(E) = π_{−n}\underline{Der}_{sAlg}(R_{•}, R_{•}) for n ≤ 0. For n > 0, H^{n}(E) is the nth cohomology of the cosimplicial complex

C^{n} := Der_{sAlg}(L_{∂}^{n+1}R_{•}, R_{•})

associated to the comonad L_{∂} of Definition 4.13.

Proposition 4.23. If R_{•} → R is a cofibrant resolution of a k-algebra R, then the DDC E of Proposition 4.19 is quasi-isomorphic to DF, for F the SDC

F^{n} = Hom_{Mod}(Symm^{n}M̃, M̃)

from [Pri1] §1.2.1, and M̃ ∈ FMod(Λ) lifting the k-module M underlying R.

Proof. By Proposition 3.28, we may assume that R_{•} is the standard resolution R_{n} = ⊥^{n+1}R, with R̃_{n} = Symm^{n+1}(M̃).

Then we have E quasi-isomorphic to the DDC E′ given by

(E′)_{K}^{n} = Hom_{s_{+}Alg}(K_{∗} ⊗ R̃_{∗}, G_{∂}^{n}R̃_{∗}) = Hom_{s_{+}Mod}(K_{∗} ⊗ Symm^{∗}R̃, G_{∂}^{n}R̃_{∗}).

The augmentation ε : R̃_{∗} → R̃ in s_{+}Mod(Λ) gives us a map

χ : (E′)_{K}^{n} → Hom_{s_{+}Mod}(K_{∗} ⊗ Symm^{∗}R̃, G_{∂}^{n}R̃) = Hom_{s_{+}Mod}(L^{n}(K_{∗} ⊗ Symm^{∗}R̃), R̃)
= Hom_{Mod}(K_{n} ⊗ Symm^{n}R̃, R̃).

But this is just (DF)_{K}^{n}, and it is straightforward to check that χ : E′ → DF respects all the SDC operations.

Since R_{•} → R is a resolution, we get a weak equivalence

Z^{n}C^{∗}(E) = Z^{n}\underline{Der}_{sAlg}(R_{•}, G^{∗+1}R_{•}) → Z^{n}\underline{Der}_{sAlg}(R_{•}, G^{∗+1}R)

Now by Lemma 4.15,

\underline{Der}_{sAlg}(R_{•}, G^{n+1}R)_{m} = Der_{Alg}(π_{0}L^{n+1}(Δ^{m} ⊗ R_{•}), R)
= Der_{Alg}(R_{n}, R)^{Δ_{n}^{m}}
= Der_{Alg}(Symm^{n}R, R)^{Δ_{n}^{m}}
= C^{n}(DF)_{m}.

Since C^{•}(E) is quasi-smooth, H^{n−i}(⌟E) = π_{i}Z^{n}C^{∗}(E), so

H^{∗}(⌟E) ≅ H^{∗}(⌟DF) = H^{∗}(F),

which is just André-Quillen cohomology 𝔻_{k}^{∗}(R, R), and so χ is a quasi-isomorphism. □

Remark 4.24. Propositions 4.19 and 4.23 together imply that derived deformations of a k-algebra R are equivalent to derived deformations of the operation ∂_{0} on any cofibrant resolution R_{•} → R.

5. Deformations of Artin stacks and simplicial schemes

The problem we now wish consider is that of deforming of an algebraic stack 𝔛. We may take a smooth simplicial hypercovering X_{•} → 𝔛, with each X_{n} a disjoint union of affine schemes (similarly to the proof of [Ols1] Theorem 11.1), and our first step will be to consider derived deformations of X_{•}.

5.1. Cosimplicial algebras. Let X_{•} be a simplicial affine scheme. Equivalently, we may consider the cosimplicial algebra [n] ↦ Γ(X_{n}, 𝒪_{X_{n}}).

Definition 5.1. The categories cFMod(A), cFAlg(A), c_{+}FMod(A), and c_{+}FAlg(A) (as given in Definition 4.9) can be made into simplicial categories (i.e. enriched in simplicial sets) by setting (S^{K})^{n} := (S^{n})^{K_{n}} for K ∈ 𝕊, with structure maps (S^{K})(f) = S(f)^{K_{n}} ∘ K(f)^{∗} : (S^{m})^{K_{m}} → (S^{n})^{K_{n}}, for morphisms f in Δ. We then define the simplicial \underline{Hom} functor by

\underline{Hom}(R, S)_{n} := Hom(R, S^{Δ^{n}}).

There is the following diagram of monadic adjunctions of functors C_{Λ} → sCat:

\xymatrix{cFAlg(A) \ar[r]^{U_{alg}}_{⊤} \ar[d]_{U_{∂}}^{⊢} & cFMod(A) \ar[l]^{Symm} \ar[d]_{U_{∂}}^{⊢} \\ c_{+}FAlg(A) \ar[u]_{F_{∂}} \ar[r]^{U_{alg}}_{⊤} & c_{+}FMod(A), \ar[l]^{Symm} \ar[u]_{F_{∂}}}

where F_{∂} : c_{+}FMod(A) → cFMod(A) is left adjoint to the forgetful functor U_{∂}, given by

(F_{∂}V^{∗})^{n} = V^{n} ⊕ V^{n−1} ⊕ … ⊕ V^{0},

with operations dual to those in Lemma 4.12. Similarly, F_{∂} : c_{+}FAlg(A) → cFAlg(A) is the left adjoint given by

(F_{∂}R^{∗})^{n} = R^{n} ⊗ R^{n−1} ⊗ … ⊗ R^{0}.

The diagram satisfies the following commutativity conditions:

U_{∂}U_{alg} = U_{alg}U_{∂} SymmF_{∂} = F_{∂}Symm, U_{∂}Symm = SymmU_{∂}.

These adjunctions combine to give a monadic adjunction

\xymatrix{cFAlg(A) \ar[r]^{U_{∂}U_{alg}}_{⊤} & c_{+}FMod(A) \ar[l]^{SymmF_{∂}}}.

Lemma 5.2. c_{+}FMod(A) has uniformly trivial deformation theory.

Proof. This is essentially the same as Proposition 4.16. □

Proposition 5.3. In the scenario above, Theorem 3.13 gives a pre-DDC

E^{n} := \underline{Hom}_{c_{+}FMod}(⊤^{n}{ \atop U_{∂}U_{alg}R}, { \atop U_{∂}U_{alg}R}),

satisfying Definition 2.8.(Q1), where ⊤ = U_{∂}U_{alg}SymmF_{∂}.

We now seek conditions under which the pre-DDC E (or similarly a pre-DDC E(𝔻) associated to a diagram as in §3.3) is quasi-smooth.

Definition 5.4. Given a cosimplicial (resp. almost cosimplicial) A-algebra R, define the category cMod(R) (resp. c_{+}Mod(R)) to consist of cosimplicial (resp. almost cosimplicial) A-modules M equipped with an associative multiplication R ⊗ M → M, respecting the cosimplicial (resp. almost cosimplicial) structures. These categories have simplicial structures, with (M^{K})^{n} := (M^{n})^{K_{n}}, for K ∈ 𝕊, the R-module structure on M^{K} coming from the map R → R^{K}. As usual, denote the left adjoint to M ↦ M^{K} by N ↦ N ⊗ K.

Given M ∈ cMod(R) and an injective map K ↪ L in 𝕊, set M ⊗ (L/K) := coker (M ⊗ K → M ⊗ L) and M^{L/K} := ker(M^{L} → M^{K}).

Definition 5.5. Let ⊥_{alg} = SymmU_{alg}, ⊥_{∂} = F_{∂}U_{∂} and ⊥ = SymmF_{∂}U_{∂}U_{alg} = ⊥_{∂}⊥_{alg}. Given R ∈ cAlg(k), define 𝕃_{n}^{⊥}(R) ∈ cMod(R) by the property that

Hom_{cMod(R)}(𝕃_{n}^{⊥}(R), M^{•}) ≅ Der_{k}(⊥^{n+1}R, M^{•})

functorial in M^{•} ∈ cMod(R). Here, Der_{k}(S^{•}, M^{•}) is the set of morphisms f : S^{•} → S^{•} ⊕ M^{•}ϵ in cAlg_{k} extending the identity, where ϵ^{2} = 0.

Define 𝕃_{n}(R) ∈ cMod(R) by

Hom_{cMod(R)}(𝕃_{n}(R), M^{•}) ≅ Der_{k}(⊥_{alg}^{n+1}R, M^{•}).

Observe that 𝕃_{•}(R) and 𝕃_{•}^{⊥}(R) both form simplicial complexes in cMod(R).

Definition 5.6. Given an object R of cFAlg(A) (resp. c_{+}FAlg(A)), we may extend R uniquely to a cocontinuous functor R : 𝕊 → FAlg(A) (resp. R : s_{+}Set → FAlg(A)) extending the functor R : Δ → FAlg(A) (resp. R : Δ_{+} → FAlg(A)) given by R(Δ^{n}) = R^{n} (resp. R(Ξ^{n}) = R^{n}, for Ξ as in Definition 4.14).

Lemma 5.7. For all m, the simplicial complex 𝕃_{•}^{⊥}(R)^{m} is a model for the cotangent complex of R^{m}.

Proof. Write ⊥_{n}R := ⊥^{n+1}R; these form a simplicial complex ⊥_{•}R in cAlg(k). We need to show that (⊥_{•}R)^{m} is a cofibrant resolution of R^{m} in sAlg(k). If we apply the forgetful functor U_{∂}U_{alg} to the augmented simplicial complex ⊥_{•}R → R, we see that it becomes contractible. In particular, this implies that ⊥_{•}R → R is contractible as an augmented complex of k-vector spaces, so it is a resolution. □

Lemma 5.8. 𝕃_{n}^{⊥}(R) is a projective object of cMod(R), and U_{∂}𝕃_{n}(R), U_{∂}𝕃_{n}^{⊥}(R) are both projective objects of c_{+}Mod(R^{∗}).

Proof. By adjointness,

Der_{k}(⊥^{n+1}R, M^{•}) ≅ Hom_{c_{+}Mod_{k}}(U_{∂}U_{alg}⊥^{n}R, M^{∗}),

so Der_{k}(⊥^{n+1}R, −) defines a right exact functor, hence 𝕃_{n}^{⊥}(R) is projective. The other results follow similarly. □

Lemma 5.9. There is a natural transformation F_{∂}U_{alg} → U_{alg}F_{∂}, giving transformations ⊥_{∂}⊥_{alg} → ⊥_{alg}⊥_{∂}.

Proof. The transformation is given on level n by

R^{0} ⊕ R^{1} ⊕ … ⊕ R^{n} ∋ ∑_{i=0}^{n} r_{i} ↦ ∑_{i=0}^{n} 1 ⊗ … ⊗ 1 ⊗ r_{i} ⊗ 1 ⊗ … 1 ∈ R^{0} ⊗ R^{1} ⊗ … ⊗ R^{n}.

□

Definition 5.10. A morphism f : X_{•} → Y_{•} of simplicial schemes over A is said to be quasi-smooth (resp. trivially smooth) if the morphism

Hom_{𝕊}(L, X_{•}) → Hom_{𝕊}(K, X_{•}) ×_{Hom_{𝕊}(K,Y_{•})} Hom_{𝕊}(L, Y_{•})

of affine schemes is smooth for all trivial cofibrations (resp. all cofibrations) K → L of finite simplicial sets. The map f is said to be smooth if it is quasi-smooth and f_{0} : X_{0} → Y_{0} is smooth.

We say that a morphism R^{•} → S^{•} in cAlg(A) is quasi-smooth (resp. trivially smooth, resp. smooth) if Spec S^{•} → Spec R^{•} is so.

Lemma 5.11. In Definition 5.10, we may replace cofibrations (resp. trivial cofibrations) K → L by generating cofibrations ∂Δ^{n} → Δ^{n} (resp. generating trivial cofibrations Λ_{k}^{n} → Δ^{n}).

Proof. This follows because every cofibration (resp. trivial cofibration) is a composition of pushouts of generating cofibrations (resp. generating trivial cofibrations), and the fact that smooth morphisms are closed under pullback and finite composition. □

Lemma 5.12. A morphism f : X_{•} → Y_{•} of simplicial schemes is quasi-smooth (resp. trivially smooth, resp. smooth) if and only if the following conditions hold:

(1) for all square-zero extensions A ↠ B of k-algebras, the map X_{•}(A) → X_{•}(B) ×_{Y_{•}(B)} Y_{•}(A) is a fibration (resp. a trivial fibration, resp. a surjective fibration) in 𝕊.

(2) for all all vertices v ∈ Δ_{0}^{n} the maps v^{∗} : X_{n} → X_{0} (resp. the schemes X_{n}, resp. the schemes X_{n}) are locally of finite presentation.

Proof. This follows from the fact that a morphism is smooth if and only if it is quasismooth and locally of finite presentation, and that U → V is locally of finite presentation if and only if the map U(A_{α}) → U(\underrightarrow{lim} A_{α}) ×_{V (⟶^{lim} A_{α})} \underrightarrow{lim} V (A_{α}) is an isomorphism. We also use the result that if g ∘ f is locally of finite presentation, then f must also be so. □

Corollary 5.13. For all cofibrations i : K → L of finite simplicial sets, and f : X → Y a quasi-smooth morphism of simplicial affine schemes, the map

g : X^{L} → X^{K} ×_{Y^{K}} Y^{L}

is quasi-smooth. Moreover, if either i or f is trivial, then so is g.

Lemma 5.14. If R → S is a trivially smooth map in cAlg, then Ω(S/R) is projective in cMod(S).

Proof. By definition, we know that R(L)⊗_{R(K)} S(K) → S(L) is smooth for all cofibrations K ↪ L of finite simplicial sets.

Given M^{•} ∈ cMod(S) and K ∈ 𝕊, define M(K) ∈ Mod(S(K)) by S(K) ⊕ M(K)ϵ = (S ⊕ Mϵ)(K). Note that Ω(S/R)(K) = Ω(S(K)/R(K)).

Take a surjection L^{•} → N^{•} in cMod(S) and a morphism f : Ω(S/R) → N^{•}. We will construct a lifting f̃ of f inductively. Assume that we have R^{m}-linear maps f̃^{m} : Ω(S/R)^{m} → L^{m} lifting f compatibly with the cosimplicial operations, for all m < n. If M^{n}L denotes the mth matching object (as in [GJ] Lemma VII.4.9), then extending f̃ compatibly to Ω(S/R)^{n} amounts to finding a lift

\xymatrix{Ω(S/R)(∂Δ^{n}) ⊗_{S(∂Δ^{n})} S^{n} \ar[r] \ar[d] & L^{n} \ar[d]^{α} \\ Ω(S/R)^{n} \ar[r] \ar@{-->}[ur] & N^{n} ×_{M^{n−1}N} M^{n−1}L.}

Now, T := R^{n} ⊗_{R(∂Δ^{n})} S(∂Δ^{n}) → S^{n} is smooth, as is R(K) → S(K) for all K. Thus the sequence

0 → Ω(S(∂Δ^{n})/R(∂Δ^{n})) ⊗_{S(∂Δ^{n})} S^{n} → Ω(S^{n}/R^{n}) → Ω(S^{n}/T) → 0

is exact, with all terms projective. Since α is surjective, projectivity of Ω(S^{n}/T) gives the required lift. □

Lemma 5.15. If R is quasi-smooth, then for all trivial cofibrations K → L of finite simplicial sets, Ω(R) ⊗ (L/K) is projective in cMod(R).

Proof. By Corollary 5.13, R ⊗ K → R ⊗ L is trivially smooth. Since Ω(R) ⊗ X = Ω(R ⊗ X) ⊗_{R⊗X} R, projectivity follows from Lemma 5.14. □

Definition 5.16. Let L^{∨} : c_{+}Mod → cMod be right adjoint to U_{∂}.

Lemma 5.17. For R ∈ cAlg(k) and an injective map f : Z → X in s_{+}Set, there is an isomorphism

H^{i}Hom_{c_{+}Mod(R^{∗})}(𝕃_{•}(R^{∗}), N ⊗ k^{X/Z}) ≅ 𝔼xt_{cMod(R)}^{i}(𝕃_{•}^{⊥}(R) ⊗ (LX/LZ), L^{∨}N)

for all N ∈ c_{+}Mod(R^{∗}).

Proof. By Lemma 5.8, 𝕃_{n}^{⊥}(R) is projective, so 𝕃_{n}^{⊥}(R) ⊗ (LX/LZ) must also be projective, as M ↦ M^{K} sends surjections to surjections. Thus

𝔼xt_{cMod(R)}^{∗}(𝕃_{•}^{⊥}(R) ⊗ (LX/LZ), L^{∨}N) = H^{∗}Hom_{cMod(R)}(𝕃_{•}^{⊥}(R) ⊗ (LX/LZ), L^{∨}N)
= H^{∗}Hom_{cMod(R)}(𝕃_{•}^{⊥}(R), L^{∨}N ⊗ L^{∨}k^{X/Z})
= H^{∗}Hom_{c_{+}Mod(R^{∗})}(U_{∂}𝕃_{•}^{⊥}(R), N ⊗ k^{X/Z})
= 𝔼xt_{c_{+}Mod(R^{∗})}^{∗}(U_{∂}𝕃_{•}^{⊥}(R), N ⊗ k^{X/Z}).

Now, Lemma 5.9 gives compatible transformations ⊥_{∂}^{n+1}⊥_{alg}^{n+1}(R) → ⊥^{N+1}R. The unit of the adjunction F_{∂} ⊣ U_{∂} gives compatible transformations U_{∂} → U_{∂}⊥_{∂}^{n+1}, so there is a map 𝕃_{•}(R^{∗}) = U_{∂}𝕃_{•}(R^{•}) → U_{∂}𝕃_{•}^{⊥}(R), which is an equivalence in the derived category by Lemma 5.7. Hence

𝔼xt_{c_{+}Mod(R^{∗})}^{∗}(U_{∂}𝕃_{•}^{⊥}(R), N ⊗ k^{X/Z}) ≅ 𝔼xt_{c_{+}Mod(R^{∗})}^{∗}(𝕃_{•}(R^{∗}), N ⊗ k^{X/Z})
= H^{∗}Hom_{c_{+}Mod(R^{∗})}(𝕃_{•}(R^{∗}), N ⊗ k^{X/Z}).
□

Lemma 5.18. For R ∈ cAlg(k) quasi-smooth, there is an exact sequence

0 → 𝕃_{•}(R^{0}) ⊗_{R^{0}} R^{∗} → 𝕃_{•}(R^{∗}) → Ω(R^{∗}/R^{0}) → 0

in the derived category of projective complexes in c_{+}Mod(R^{∗}), where the morphism R^{0} → R^{∗} is given in level n by (∂^{1})^{n}.

Proof. There is an exact sequence

0 → 𝕃_{•}(R^{0}) ⊗_{R^{0}} R^{∗} → 𝕃_{•}(R^{∗}) → 𝕃(R^{∗}/R^{0}) → 0

in the derived category. Since R^{•} is quasi-smooth, the maps (∂^{1})^{n} : R^{0} → R^{n} are all smooth, giving 𝕃(R^{∗}/R^{0}) ∼ Ω(R^{∗}/R^{0}) □

Proposition 5.19. If R is a quasi-smooth object of cAlg(k), then every morphism ρ : R → S in cAlg is quasi-smooth over c_{+}Mod, in the sense of Definition 3.22. The Ext-groups are then given by

Ext_{cAlg/c_{+}Mod}^{i}(ρ) := \begin{cases}𝔼xt_{cMod(R)}^{i}(𝕃_{•}^{R/k}, S) & i > 0 \\ π_{−i}Hom_{cMod(R)}(Ω_{(R⊗Δ^{•})/k} ⊗_{R⊗Δ^{•}} R, S) & i ≤ 0, \end{cases}

where Hom_{cMod(R)}(Ω_{(R⊗Δ^{•})/k} ⊗_{R⊗Δ^{•}} R, S) is the simplicial complex given in level n by Hom_{cMod(R)}(Ω_{(R⊗Δ^{n})/k} ⊗_{R⊗Δ^{n}} R, S).

Proof. First observe that since c_{+}Mod has uniformly trivial deformation theory, ρ is quasismooth over c_{+}Mod whenever it is Q2 over c_{+}Mod. Now,

C_{cAlg/c_{+}Mod}^{•}(ρ) = \underline{Hom}_{cMod(R)}(𝕃_{•}^{⊥}(R), S),

so for K ∈ 𝕊,

\underline{Ext}_{cAlg/c_{+}Mod}^{∗}(ρ) = H^{∗}\underline{Hom}_{cMod(R)}(𝕃_{•}^{⊥}(R), S),
\underline{Ext}_{cAlg/c_{+}Mod}^{∗}(ρ)_{K} = H^{∗}Hom_{cMod(R)}(𝕃_{•}^{⊥}(R) ⊗ K, S),
≅ 𝔼xt_{cMod(R)}^{∗}(𝕃_{•}^{⊥}(R) ⊗ K, S),

the latter isomorphism following since 𝕃_{n}^{⊥}(R) ⊗ K is projective.

Now, consider the monad ⊤_{∂} := L^{∨}U_{∂} on cMod(R), and observe that the augmented cosimplicial complex ⊤_{∂}^{•+1}M given in level n by ⊤_{∂}^{n+1}M is a resolution in cMod(R), since it becomes contractible on applying U_{∂}. Thus

\underline{Ext}_{cAlg/c_{+}Mod}^{∗}(ρ)_{K} ≅ 𝔼xt_{cMod(R)}^{∗}(𝕃_{•}^{⊥}(R) ⊗ K, ⊤_{∂}^{•+1}S).

Given f : Z → X in s_{+}Set with f_{0} an isomorphism, by Lemmas 5.17 and 5.18, we have

𝔼xt_{cMod(R)}^{i}(𝕃_{•}^{⊥}(R) ⊗ (LX/LZ), L^{∨}N) = H^{i}Hom_{c_{+}Mod(R^{∗})}(𝕃_{•}(R^{0}) ⊗_{R^{0}} R^{∗}, N ⊗ k^{X/Z})

for all i > 0. However,

Hom_{c_{+}Mod(R^{∗})}(𝕃_{•}(R^{0}) ⊗_{R^{0}} R^{∗}, N ⊗ k^{X/Z}) = Hom_{Mod(R^{0})}(𝕃_{•}(R^{0}), (N ⊗ k^{X/Z})^{0}) = 0,

since (k^{X/Z})^{0} = 0.

Hence 𝔼xt_{cMod(R)}^{i}(𝕃_{•}^{⊥}(R) ⊗ (LX/LZ), L^{∨}N) = 0 for all i > 0, so the spectral sequence associated to ⊤_{∂}^{•+1} gives

𝔼xt_{cMod(R)}^{∗}(𝕃_{•}^{⊥}(R) ⊗ (LX/LZ), M) ≅ H^{∗}Hom_{cMod(R)}(Ω(R) ⊗ (LX/LZ), ⊤_{∂}^{•+1}M).

Since Ω(R) ⊗ (LX/LZ) is projective (by Lemma 5.14), this is just

𝔼xt_{cMod(R)}^{∗}(Ω(R) ⊗ (LX/LZ), ⊤_{∂}^{•+1}M) ≅ Ext_{cMod(R)}^{∗}(Ω(R) ⊗ (LX/LZ), M)
= Hom_{cMod(R)}(Ω(R ⊗ LX/R ⊗ LZ) ⊗_{R⊗LX} R, M).

Taking Z = ∂Ξ^{n}, X = Ξ^{n}, we have LZ = Λ_{0}^{n}, LX = Δ^{n}, and

N_{n}\underline{Ext}_{cAlg/c_{+}Mod}^{∗}(ρ) = Hom_{cMod(R)}(Ω(R ⊗ Δ^{n}/R ⊗ Λ_{0}^{n}) ⊗_{R⊗Δ^{n}} R, S),

so \underline{Ext}_{cAlg/c_{+}Mod}^{i}(ρ) is constant for i > 0.

Thus ρ is Q2 over c_{+}Mod, as required. The description of positive Ext-groups follows from Lemma 5.17, while that of non-positive Ext-groups follows from the definition of π_{−i}H^{0}C_{cAlg/c_{+}Mod}^{•}(ρ). □

Corollary 5.20. For any diagram in cAlg with quasi-smooth objects, the associated pre-DDC given by Definition 3.24 and Proposition 3.19 applied to the adjunction

\xymatrix{cFAlg(A) \ar[r]^{U_{∂}U_{alg}}_{⊤} & c_{+}FMod(A) \ar[l]^{SymmF_{∂}}}

is a DDC by Lemma 3.25, and governs deformations in the simplicial category cAlg (by Proposition 3.26).

5.1.1. Comparison with deformations of schemes. In [Pri2] §3.2.1, an SDC was constructed to describe deformations of a separated scheme X, and we now wish to compare it with the DDC above.

Take an open affine cover (X_{α})_{α∈I} of X, and set X̌ := ∐_{α∈I} X_{α}. Define the simplicial scheme Z_{•} by Z = cosk_{0}(X̌/X), i.e.

Z_{n} = \overbrace{X̌ ×_{X} X̌ ×_{X} … ×_{X} X̌}^{n+1},

with r_{n} : Z_{n} → X, and s_{n} : Z_{n} → X̌ given by projection onto the first factor.

The map v : X̌ → X gives adjoint functors v^{−1} ⊣ v_{∗} on sheaves. This yields the following diagram of Cat-valued functors:

\xymatrix{FAlg_{A}(X) \ar[r]_{⊤} \ar[d]_{v^{−1}} & FMod_{A}(X) \ar[l]^{_{Symm_{A}}} \ar[d]_{v^{−1}} \\ FAlg_{A}(X̌) \ar[u]_{v_{∗}}^{⊣} \ar[r]_{⊤} & FMod_{A}(X̌), \ar[l]^{_{Symm_{A}}} \ar[u]_{v_{∗}}^{⊣}}

where FMod_{A}(Y) and FAlg_{A}(Y) denote sheaves of flat A-modules and of flat A-algebras on Y.

Definition 5.21. The SDC Ě^{•} of [Pri2] §3.2.1 was then given by

Ě^{n}(A) = Hom_{FMod_{A}(X̌)}((Symm_{A})^{n}𝒩 ⊗ A, (v^{−1}v_{∗})^{n}𝒩 ⊗ A)_{v^{−1}(α^{n}∘ε^{n})},

for 𝒩 a flat μ-adic Λ-module on X̌ lifting v^{−1}𝒪_{X}, with α^{n} : 𝒪_{X} → (v_{∗}v^{−1})^{n}𝒪_{X} coming from the unit of the adjunction, and similarly ε^{n} : (Symm_{k})^{n}𝒪_{X} → 𝒪_{X}.

Definition 5.22. Define functors Č^{•} : FMod_{A}(X) → cFMod(A), Č^{∗} : FMod_{A}(X̌) → c_{+}FMod(A) by

Č^{n}(ℱ) := Γ(Z_{n}, r_{n}^{−1}ℱ), Č^{n}(𝒢) := Γ(Z_{n}, s_{n}^{−1}𝒢),

with the standard cosimplicial operations.

Lemma 5.23. There are canonical isomorphisms

C^{∗}(v^{−1}ℱ) = U_{∂}C^{•}(ℱ) C^{•}(v_{∗}𝒢) ≅ L^{∨}C^{∗}(𝒢).

Lemma 5.24. There is a canonical natural transformation Symm ∘ Č^{•} → Č^{•} ∘ Symm.

Proposition 5.25. The SDC Ě is quasi-isomorphic to the DDC E of Proposition 5.3, in the sense that 𝔇𝔢𝔣(Ě) and 𝔇𝔢𝔣(E) are weakly equivalent (equivalently, DĚ and E are quasi-isomorphic DDCs).

Proof. We have maps

Hom_{Mod(X̌)}(⊤_{alg}^{n}ℳ , (v^{−1}v_{∗})^{n}ℳ)  → Hom_{c_{+}Mod}(Č^{∗}(⊤_{alg}^{n}ℳ), Č^{∗}(v^{−1}v_{∗})^{n}ℳ)
= Hom_{c_{+}Mod}(Č^{∗}(⊤_{alg}^{n}ℳ), (L^{∨})^{n}Č^{∗}ℳ)
= Hom_{c_{+}Mod}(⊤_{∂}^{n}Č^{∗}(⊤_{alg}^{n}ℳ), Č^{∗}ℳ)
→ Hom_{c_{+}Mod}(⊤_{alg}^{n}⊤_{∂}^{n}Č^{∗}(ℳ), Č^{∗}ℳ)
→ Hom_{c_{+}Mod}(⊤^{n}Č^{∗}(ℳ), Č^{∗}ℳ).

These are compatible with the SDC operations, giving a morphism Ě → E_{0} of SDCs. Now, as in Proposition 5.19,

H^{∗}(E_{0}) = 𝔼xt_{𝒪_{Z•}}^{∗} (𝕃_{•}^{Z/k}, 𝒪_{Z}).

However, since the maps r_{n} : Z_{n} → X are all open, and hence étale, 𝕃_{•}^{Z/k} is quasiisomorphic to r^{∗}𝕃_{•}^{X}. Thus

𝔼xt_{Z_{•}}^{∗}(𝕃_{•}^{Z/k}, 𝒪_{Z}) = 𝔼xt_{𝒪_{X}}^{∗} (𝕃_{•}^{X/k}, r_{∗}𝒪_{Z_{•}}) = 𝔼xt_{𝒪_{X}}^{∗} (𝕃_{•}^{X/k}, 𝒪_{X}),

since r_{∗}𝒪_{Z_{•}} = r_{∗}r^{−1}𝒪_{X} is a resolution of 𝒪_{X}. This means that Ě → E_{0} is a quasiisomorphism of SDCs.

Finally, to see that 𝔇𝔢𝔣(E_{0}) → 𝔇𝔢𝔣(E) is a quasi-isomorphism, apply Lemma 2.24, noting that the strictly positive cohomology groups automatically agree. For n ≤ 0,

H^{n}E = H_{−n}Hom_{𝒪_{Z}} (i^{∗}Ω_{Z^{Δ^{•}}} , 𝒪_{Z}),

for i : Z → Z^{Δ^{n}}. However, Z is quasi-étale (the analogous notion to quasi-smooth), so the vertex maps a : Z^{Δ^{n}} → Z are trivially étale, and thus Ω_{Z^{Δ^{n}}} ≅ a^{∗}Ω_{Z}, so i^{∗}Ω_{Z^{Δ^{•}}} = Ω_{Z}. Therefore H^{0}E = H^{0}E_{0} = Hom_{𝒪_{X}} (Ω_{X}, 𝒪_{X}), and H^{n}E = 0 for n < 0. □

5.2. Quasi-compact, quasi-separated stacks. Let 𝔛 be a quasi-compact, quasiseparated stack, with presentation P : X_{0} → 𝔛, for X_{0} affine, giving a simplicial algebraic space cosk_{0}^{𝔛}(X_{0}) (as considered in [Aok] §3). We may then take an étale hypercovering X_{•} → cosk_{0}^{𝔛}(X_{0}), for X_{•} a simplicial affine scheme, and denote the composition by P_{•} : X_{•} → 𝔛.

Lemma 5.26. Every smooth simplicial hypercovering is trivially smooth.

Proof. For a map U_{•} → V_{•} to be a smooth hypercovering says that the matching maps U_{n} → V_{n} ×_{M_{n}V} M_{n}U are all smooth surjections. □

Lemma 5.27. The simplicial affine scheme X_{•} is quasi-smooth.

Proof. Write Z_{•} := cosk_{0}^{𝔛}(X_{0}). Since Z_{•} = B𝔊, for 𝔊 the groupoid space \xymatrix{X_{0} ×_{𝔛} X_{0} \ar[r] \ar[r] & X_{0}} , all higher partial matching maps of Z_{•} are isomorphisms. In other words, for any trivial cofibration i : K → L in 𝕊 with i_{0} : K_{0} → L_{0} an isomorphism, the map

i^{∗} : M_{L}(Z) → M_{K}(Z)

is an isomorphism.

By [Aok] Theorem 2.1.5, 𝔊 has SQCS structure so the maps X_{1} → Hom_{𝕊}(Λ_{k}^{1}, X_{•}) are smooth surjections for both k. Thus Z_{•} is quasi-smooth, by Lemma 5.11. Since X_{•} → Z_{•} is trivially smooth, the result follows. □

Remark 5.28. Similarly, every strongly quasi-compact n-geometric Artin stack 𝔛 gives rise to a quasi-smooth simplicial affine scheme X_{•}, by [Pri6] Theorem 4.7. The statement of Proposition 5.29 will then carry over to this generality, taking 𝕃_{𝔛} to be the cotangent complex of [Pri6] §7.1.

5.2.1. Cohomology and the cotangent complex. Given any morphism f : 𝔜 → 𝔛 of quasicompact, quasi-separated stacks, lifting to a morphism f : Y_{•} → X_{•} of simplicial affine resolutions, in this section we will describe the Ext-groups

Ext_{cAlg/c_{+}Mod}^{∗}(f^{♯})

of Proposition 5.19 in terms of the cotangent complex of [Ols1] §8. Ext-groups of the cotangent complex are defined in [Ols2] §2.11.

Let 𝔛, X_{•} be as above, and let 𝒥 be a quasi-coherent sheaf on 𝔛. Since the cotangent complex 𝕃_{𝔛} is in degrees ≥ −1, we have Ext^{i}(𝕃_{𝔛}, 𝒥) = 0 for all i < −1. Since r : X_{•} → cosk_{0}^{𝔛}(X_{0}) is a hypercovering, the maps

H^{∗}(cosk_{0}^{𝔛}(X_{0}), ℱ) → H^{∗}(X_{•}, r^{∗}ℱ)

on cohomology are isomorphisms for all quasi-coherent sheaves ℱ.

By [Aok] Proposition 3.4.2,

(1) Ext^{i}(𝕃_{𝔛}, 𝒥) ≅ Ext^{i}(𝕃_{X_{•}}, P_{•}^{∗}𝒥) for i > 0.

(2) Ext^{i}(𝕃_{𝔛}, 𝒥) = H^{i}(Hom(Ω_{X_{0}/𝔛}, P^{∗}𝒥)[1] → Hom(Ω_{X_{•}}, P_{•}^{∗}𝒥)), for i ≤ 0.

Proposition 5.29.

Ext_{cAlg/c_{+}Mod}^{i}(f^{♯}) ≅ Ext^{i}(𝕃_{𝔛}, f_{∗}𝒪_{𝔜})

for all i ∈ ℤ.

Proof. For i > 0, this is just the observation that Ext^{i}(𝕃_{X_{•}}, P_{•}^{∗}𝒥) = Ext_{cAlg/c_{+}Mod}^{i}(f^{♯}) when 𝒥 = f_{∗}𝒪_{𝔜}.

Accordingly, we need to describe the non-positive Ext groups

H_{−i}Hom_{X}(c^{∗}Ω_{X^{Δ^{•}}} , 𝒥)

in terms of 𝔛, X_{0}, where c : X → X^{K} is the constant map.

Let U_{•} denote the simplicial complex Hom_{X}(c^{∗}Ω_{X^{Δ^{•}}} , 𝒥), and write Z_{•} := cosk_{0}^{𝔛}(X_{0}), with V_{•} : Hom_{Z}(c^{∗}Ω_{Z^{Δ^{•}}} , 𝒥). Since X → Z is trivially smooth, observe that the canonical map U → V is a trivial fibration, so H_{∗}(U) ≅ H_{∗}(V).

In general, if K is contractible, then

M_{K}Z = \overbrace{X_{0} ×_{𝔛} X_{0} ×_{𝔛} … ×_{𝔛} X_{0}}^{K_{0}},

so

Ω(M_{K}Z/𝔛) = ⨁_{v∈K_{0}} v^{∗}Ω(X_{0}/𝔛).

We therefore conclude that for a trivial cofibration K ↪ L,

Ω(M_{L}Z/M_{K}Z) = ⨁_{v∈L_{0}−K_{0}} v^{∗}Ω(X_{0}/𝔛),

so for 0 : • → I,

Ω(Z^{I}/Z)^{n} = Ω(M_{I×Δ^{n}}Z/M_{Δ^{n}}Z) = ⨁_{v∈Δ_{0}^{n}}(v × 1)^{∗}Ω(X_{0}/𝔛),

so

i^{∗}Ω(Z^{I}/Z)^{n} = ⨁_{v∈Δ_{0}^{n}} v^{∗}Ω(X_{0}/𝔛),

and

Hom_{𝒪(X)} (i^{∗}Ω(X^{I}/X), ℱ) = Hom_{𝒪(X_{0})} (Ω(X_{0}/𝔛), ℱ^{0}),

giving

N_{1}V = Hom_{𝒪(X)} (i^{∗}Ω(X^{I}/X), f_{∗}𝒪(Y)) = Hom_{𝒪(X_{0})} (Ω(X_{0}/𝔛), f_{∗}𝒪(Y_{0})).

Moreover, for n ≥ 2, Λ_{0}^{n} → Δ_{0}^{n} is an isomorphism, so N_{n}V = 0.

Thus

NV = (Hom(Ω_{X_{0}/𝔛}, P^{∗}𝒥)[−1] → Hom(Ω_{X_{•}}, P_{•}^{∗}𝒥)),

as required. □

5.2.2. Comparing deformation groupoids.

Definition 5.30. Given a small 2-category C, define a simplicial category B^{1}C by setting Ob (B^{1}C) = Ob (C), and \underline{Hom}_{B^{1}C}(x, y) = Bℋ om_{C}(x, y), where ℋ om_{C}(x, y) is the 1category of homomorphisms from x to y, and B is the nerve functor.

Lemma 5.31. Given x, y ∈ Ob C with ℋ om_{C}(x, y) a groupoid, π_{0}\underline{Hom}_{B^{1}C}(x, y) is the set of isomorphism classes in ℋ om_{C}(x, y), with π_{1}(\underline{Hom}_{B^{1}C}(x, y), f) the set of 2automorphisms of f, and π_{i}(\underline{Hom}_{B^{1}C}(x, y), f) = 0 for i > 1.

Definition 5.32. Define a 2-category structure on the category AlgGpdSp of algebraic groupoid spaces (as in [Aok]) by defining a 2-morphism η between morphisms f, f′ : G → H by analogy with natural transformations. Explicitly, let Ob G be the space of objects of G, with Mor G → (Ob G × Ob G) the space of isomorphisms, and similarly for H. We must have η : Ob G → Mor H, with s ∘ η = f, t ∘ η = f′, and the following diagram commuting

Mor (G)  ⟶^{(η∘t,Mor f)} Mor (H) ×_{s,Ob H,t} Mor (H)
(f′,η∘s)↓  ↓^{m}
Mor (H) ×_{s,Ob H,t} Mor (H) ⟶^{m}  Mor (H).

Definition 5.33. Given a 2-groupoid G, define Π_{0}G to be the groupoid with objects Ob G, and morphisms Hom_{Π_{0}G}(X, Y) = π_{0}ℋ om_{G}(X, Y). Similarly, for a simplicial groupoid G_{•}, define Π_{0}G_{•} to be the groupoid with objects Ob G, and morphisms Hom_{Π_{0}𝒢_{•}}(X, Y) = π_{0}\underline{Hom}_{G}(X, Y).

Lemma 5.34. Given G ∈ AlgGpdSp associated to an algebraic stack over k, the nerve functor B : AlgGpdSp → sAlgSp to the category of simplicial algebraic spaces gives an isomorphism

B^{1}𝔇𝔢𝔣_{AlgGpdSp}^{2}(G) ≅ \underline{𝔇𝔢𝔣}_{sAlgSp}(BG),

between the 2-groupoid of deformations in AlgGpdSp, and the simplicial groupoid of deformations in sAlgSp.

Proof. By [Aok] Corollary 3.1.5, we know that Π_{0}𝔇𝔢𝔣_{AlgGpdSp}^{2}(G) ≅ Π_{0}\underline{𝔇𝔢𝔣}_{sAlgSp}(BG), so we just need to show that, for algebraic groupoid spaces H, G,

\underline{Hom}_{sAlgSp}(BH, BG) = Bℋ om_{AlgGpdSp}(H, G).

Now, \underline{Hom}_{sAlgSp}(X, BG)_{n} = Hom_{sAlgSp}(X × Δ^{n}, BG) = Hom_{AlgGpdSp}(π_{f}X × π_{f}Δ^{n}, G), where we define the fundamental groupoid π_{f} : sAlgSp → AlgGpdSp to be left adjoint to B, noting that π_{f}BH = H. However, π_{f}Δ^{n} is the groupoid with n + 1 objects, and unique isomorphisms between them. Thus

Hom_{AlgGpdSp}(π_{f}X × π_{f}Δ^{n}, G) = B_{n}ℋ om_{AlgGpdSp}(π_{f}X, G),

as required. □

Lemma 5.35. The functor C defined in [Aok] §3.2 gives an equivalence between 𝔇𝔢𝔣_{AlgGpdSp}^{2}(G) and 𝔇𝔢𝔣^{2}(CG), the 2-groupoid of deformations of the algebraic stack CG.

Proof. First observe that C maps the 2-isomorphisms of Definition 5.32 to 2-isomorphisms of stacks, so C is well-defined.

By [Aok] Proposition 3.2.5, we know that C induces a bijection on isomorphism classes of objects. We need to show that for G ∈ 𝔇𝔢𝔣_{AlgGpdSp}^{2}(G)(A),

C : ℋ om_{𝔇𝔢𝔣_{AlgGpdSp}^{2}(G)(A)}(G, G) → ℋ om_{𝔇𝔢𝔣^{2}(CG)(A)}(CG, CG)

is an equivalence of groupoids. By [ibid.] Proposition 3.3.2, it is essentially surjective. Given f ∈ Ob ℋ om_{𝔇𝔢𝔣_{AlgGpdSp}^{2}(G)(A)}(G, G), we thus need to show that

θ : Aut_{𝔇𝔢𝔣_{AlgGpdSp}^{2}(G)(A)}^{2}(f) → Aut_{𝔇𝔢𝔣^{2}(CG)(A)}^{2}(Cf)

is an isomorphism of 2-automorphism groups. Multiplication by f^{−1} allows us to assume that f = id_{G}.

By [ibid.] Proposition 3.3.2, we have an exact sequence

0 → Aut_{𝔇𝔢𝔣^{2}(CG)}^{2}(id_{CG}) → Aut(X_{0}/CG)_{P} ⟶^{A} Aut_{𝔇𝔢𝔣_{AlgGpdSp}^{2}(G)}(G).

Since Aut(X_{0}/CG)_{P} is smooth, the homogeneous functor Aut_{𝔇𝔢𝔣^{2}(CG)}^{2}(id_{CG}) has tangent space ker(tan A) and obstruction space coker (tan A). By [ibid.] Proposition 3.4.2, these are Ext^{−1}(𝕃_{𝔛}, 𝒪_{fX}) and Ext^{0}(𝕃_{𝔛}, 𝒪_{fX}), respectively. Thus §5.2.1, Lemma 5.31 and Theorem 1.28 imply that θ gives isomorphisms on tangent and obstruction spaces, so must be an isomorphism of homogeneous functors by the standard smoothness criterion. □

Taking G = \xymatrix{X_{0} ×_{𝔛} X_{0} \ar[r] \ar[r] & X_{0}} , we have therefore shown that the deformation 2groupoid of 𝔛 is equivalent to the simplicial deformation groupoid of cosk_{0}^{𝔛}(X_{0}). We still need to compare this with the simplicial affine scheme X_{•} defined at the beginning of the section.

Proposition 5.36. The simplicial deformation groupoids of cosk_{0}^{𝔛}(X_{0}) and X_{•} are equivalent.

Proof. Let Z_{•} := cosk_{0}^{𝔛}(X_{0}). As in §3.3, we will consider three simplicial deformation problems F_{X}, F_{Z}, F_{r}: deformations of X_{•}, deformations of Z_{•}, and deformations of the diagram r : X_{•} → Z_{•}. Note that these all define quasi-smooth functors F : C_{Λ} → sGpd ⟶^{W̄} 𝕊, so we just need to compare tangent and obstruction spaces.

The calculations of Proposition 5.29 show that the loop spaces ΩF_{X}, ΩF_{Z} have tangent spaces

\underline{Hom}_{𝒪_{X}} (Ω_{X}, 𝒪_{X}), \underline{Hom}_{𝒪_{Z}} (Ω_{Z}, 𝒪_{Z}).

Similarly, there is a fibration F_{r} → F_{X} × F_{Z}, whose fibre has tangent space

\underline{Hom}_{𝒪_{Z}} (Ω_{Z}, r_{∗}𝒪_{X}).

Since the maps

\underline{Hom}_{𝒪_{X}} (Ω_{X}, 𝒪_{X}) ⟶^{r_{∗}} \underline{Hom}_{𝒪_{Z}} (Ω_{Z}, r_{∗}𝒪_{X}) ⟵^{r^{∗}} \underline{Hom}_{𝒪_{Z}} (Ω_{Z}, 𝒪_{Z})

are isomorphisms, we deduce that the maps F_{Z} ← F_{r} → F_{X} induce isomorphisms on tangent spaces of positive homotopy groups.

It only remains to show that the deformation functors π_{0}F_{X}, π_{0}F_{Z}, π_{0}F_{r} have isomorphic tangent and obstruction spaces. By adapting [Pri2] §1.3.1, we may deduce that these are (respectively)

Ext_{𝒪_{X}}^{i} (𝕃_{X}, 𝒪_{X}), Ext_{𝒪_{Z}}^{i} (𝕃_{Z}, 𝒪_{Z}),

and the groups T^{i} fitting into the long exact sequence

… → T^{1} → Ext_{𝒪_{X}}^{1} (𝕃_{X}, 𝒪_{X}) × Ext_{𝒪_{Z}}^{1} (𝕃_{Z}, 𝒪_{Z}) → Ext_{𝒪_{Z}}^{1} (𝕃_{Z}, r_{∗}𝒪_{X}) → T^{2} → ….

Now, since the maps

Ext_{𝒪_{X}}^{i} (𝕃_{X}, 𝒪_{X}) ⟶^{r_{∗}} Ext_{𝒪_{Z}}^{i} (𝕃_{Z}, r_{∗}𝒪_{X}) ⟵^{r^{∗}} Ext_{𝒪_{Z}}^{1} (𝕃_{Z}, 𝒪_{Z})

are isomorphisms for i ≥ 1, with r_{∗} surjective for i = 0, we see that the functors F_{r}, F_{X}, F_{Z} are all equivalent. □

5.3. Arbitrary algebraic stacks. We now wish to describe derived deformations of a simplicial scheme X_{•} over k, with each X_{n} a disjoint union of affine schemes.

Definition 5.37. For any scheme Y, let π(Y) be the set of connected components of Y, and π : Y → π(Y) the map of associated topological spaces.

Now, deformations of X_{n} are equivalent to deformations of the algebra π_{∗}𝒪_{X_{n}} over π(X_{n}).

Definition 5.38. Recall that the ordinal number categories Δ, Δ^{∗} can be regarded as subcategories of 𝕊, by identifying [n] with Δ^{n}.

Given a category C and K ∈ 𝕊, define cC^{K} (resp. c_{+}C^{K}) to be the category of functors from Δ ↓ K (resp. Δ_{∗} ↓ K) to C. Thus an object C ∈ cC^{K} consists of objects C_{a} for all n ∈ ℕ_{0}, a ∈ K_{n}, together with compatible maps ∂^{i} : M_{∂_{i}a} → M_{a}, σ^{i} : M_{σ_{i}a} → M_{a}, and similarly for c_{+}C^{K}.

Now, observe that π_{∗}𝒪_{X} defines an object of cAlg(k)^{π(X)}, with

(π_{∗}𝒪_{X})_{a} = Γ(π^{−1}(a), 𝒪_{X_{n}}),

for a ∈ π(X_{n}). Since any deformation of X_{•} will not change π(X), deformations of X_{•} are equivalent to deformations of π_{∗}𝒪_{X}.

The categories cFAlg^{π(X)}, c_{+}FAlg^{π(X)}, cFMod^{π(X)}, c_{+}FMod^{π(X)} can all be given simplicial structures as in Definition 5.1, setting (C^{K})_{a} = (C_{a})^{K_{n}} for a ∈ π(X_{n}).

Remark 5.39. Observe that for any category C and any map f : K → L in 𝕊, there are maps f^{−1} : cC^{L} → cC^{K}, f^{−1} : c_{+}C^{L} → c_{+}C^{K} given by (f^{−1}C)_{a} = C_{f(a)}. If C contains products, then f^{−1} has a right adjoint f_{∗}, given by (f_{∗}C)_{b} = ∏_{a∈f^{−1}(b)} C_{a}. For f : K × L → L, C^{K} = f_{∗}f^{−1}C.

If f : π(X) → • denotes the constant map, then we write Γ := f_{∗}, with the constant functor f^{−1} denoted by Γ^{∗}.

We then have a diagram of adjunctions of functors C_{Λ} → sCat:

\xymatrix{cFAlg(A)^{π(X)} \ar[r]^{U_{∂}U_{alg}}_{⊤} \ar[d]_{Γ}^{⊢} & c_{+}FMod(A)^{π(X)} \ar[l]^{SymmF_{∂}} \ar[d]_{Γ}^{⊢} \\ cFAlg(A) \ar[u]_{Γ^{∗}} \ar[r]^{U_{∂}U_{alg}}_{⊤} & c_{+}FMod(A), \ar[l]^{SymmF_{∂}} \ar[u]_{Γ^{∗}}}

where F_{∂} : c_{+}C^{π(X)} → cC^{π(X)} is left adjoint to the forgetful functor U_{∂}, given by

(F_{∂}C^{∗})_{a} = C_{a} ⊔ C_{∂_{0}a} ⊔ … ⊔ C_{(∂_{0})^{n}a},

for a ∈ π(X_{n}), with operations dual to those in Lemma 4.12.

We must check that Γ^{∗} ⊣ Γ is monadic. For this, we verify Beck’s Theorem (e.g. [Mac] Ch. VI.7 Ex. 6), observing that Γ commutes with coequalisers — this is effectively the observation that taking arbitrary products is an exact functor.

Writing U := U_{∂}U_{alg} and F := SymmF_{∂}, we also have the following commutativity conditions:

ΓU = UΓ Γ^{∗}F = F Γ^{∗}, Γ^{∗}U = UΓ^{∗},

and a natural transformation

F Γ → ΓF.

These adjunctions combine to give a monadic adjunction

\xymatrix{cFAlg(A)^{π(X)} \ar[r]^{ΓU_{∂}U_{alg}}_{⊤} & c_{+}FMod(A) \ar[l]^{Γ^{∗}SymmF_{∂}}}.

Definition 5.40. Given a simplicial scheme X_{•}, with each X_{n} a disjoint union of affine schemes, define cMod(X) to be the category of π_{∗}(𝒪_{X})-modules over cMod^{π(X)}.

Lemma 5.41. If X → Y is a trivially smooth map of simplicial schemes, with each X_{n}, Y_{n} a disjoint union of affine schemes, then π_{∗}Ω(X/Y) is projective in cMod(X).

Proof. This is similar to Lemma 5.14. We may define matching objects of L ∈ cMod(X) by letting M^{n}L on π(X_{n+1}) be the equaliser

\xymatrix{M^{n}L \ar[r] & ∏_{i=0}^{n} σ_{i∗}L^{n} \ar[r]^{a} \ar[r]_{b} & ∏_{0≤i<j≤n} σ_{i∗}σ_{j∗}L^{n−1}} ,

where pr_{ij} ∘ a = σ^{i} ∘ pr_{j}, pr_{ij} ∘ b = σ^{j−1} ∘ pr_{i}. Note that Γ(M^{n}L) = M^{n}(ΓL). Since Γ reflects isomorphisms, this means that for all surjections L ↠ N, the relative matching map L^{n} → M^{n−1}L ×_{M^{n−1}N} N^{n} is surjective.

In order to construct latching maps, note that any cocontinuous functor S : (Δ ↓ π(X)) → Alg extends to a cocontinuous functor S : (𝕊 ↓ π(X)) → Alg. Given M^{•} ∈ cMod(X) and a : K → π(X) in 𝕊, define M(a) ∈ Mod(π_{∗}(𝒪_{X})(a)) by

π_{∗}(𝒪_{X})(a) ⊕ M(a)ϵ = (π_{∗}(𝒪_{X}) ⊕ Mϵ)(a).

Note that if we set X(a) := Hom_{𝕊↓π(X)}(K, X), then Ω(X/Y)(a) = Ω(X(a)/Y (π(f)_{∗}a)).

The latching object of Ω(a), for a ∈ π(X)_{n}, is Ω(∂a), for ∂ : ∂Δ^{n} → Δ^{n}. It therefore suffices to show that (X^{∂})^{∗}Ω(∂a) → Ω(a) is projective in Mod(X(a)) for all such a. By adapting the proof of Lemma 5.14, it suffices to show that

X(a) → Y (π(f)_{∗}a) ×_{Y (π(f)_{∗}∂a)} X(∂a)

is smooth.

Set Y′ := X ×_{π(X)} π(Y), and observe that Lemma 5.12 implies that X → Y′ is trivially smooth. Thus the matching map X_{n} → M_{n}X ×_{M_{n}Y′} Y′_{n} is smooth. The required result is then obtained by taking the fibre over a ∈ π(X)_{n}. □

Lemma 5.42. If we set ⊥′ = Γ^{∗}F UΓ, and

𝕃_{•}^{⊥′}(X) := Ω((⊥′)^{n+1}π_{∗}(𝒪_{X})) ⊗_{(⊥′)^{n+1}π_{∗}(𝒪_{X})} π_{∗}(𝒪_{X}),

then for all m, the simplicial complex 𝕃_{•}^{⊥′}(X)^{m} is a model for the cotangent complex of X_{m}.

Proof. This is essentially the same as Lemma 5.7, making use of the observation that Γ is exact and reflects isomorphisms, so it suffices to prove that UΓ(⊥′)^{•+1}π_{∗}(𝒪_{X}) → UΓπ_{∗}(𝒪_{X}) is a resolution. □

Definition 5.43. Define D to be the simplicial category of pairs (K, R), for K ∈ 𝕊, R ∈ (cFAlg^{K})^{opp}, with a morphism f ∈ \underline{Hom}_{D}((K, R), (L, S))_{n} consisting of f : K → L in 𝕊, together with f^{♯} ∈ \underline{Hom}_{cFAlg^{K}}(f^{−1}S, R)_{n}.

Define B := 𝕊 × (c_{+}Mod)^{opp}, with simplicial structure coming from (c_{+}Mod)^{opp}.

Now, observe that we have a forgetful functor V : D → B, given by (K, R) ↦ (K, ΓU_{∂}U_{alg}R), with right adjoint G : B → D given by (K, M) ↦ (K, Γ^{∗}SymmF_{∂}M). We have already seen that this adjunction is comonadic (by fixing K).

Proposition 5.44. If X, Y are simplicial schemes over k, with each X_{n}, Y_{n} a disjoint union of affine schemes, and X quasi-smooth, then every morphism ρ : Y → X is quasismooth over B, in the sense of Definition 3.22.

Proof. The proof of Proposition 5.19 carries over, using Lemmas 5.41 and 5.42 instead of Lemmas 5.14 and 5.7. □

Corollary 5.45. For any diagram in D(k) with quasi-smooth objects, the associated pre-DDC given by Definition 3.24 and Proposition 3.19 applied to the adjunction G ⊢ V is a DDC by Lemma 3.25, and governs deformations in the simplicial category D (by Proposition 3.26).

References

[Aok] Masao Aoki. Deformation theory of algebraic stacks. Compos. Math., 141(1):19–34, 2005.

[FM] D. Fiorenza and M. Manetti. L_{∞} structures on mapping cones. Algebra Number Theory, 1(3):301– 330, 2007. arXiv math.QA/0601312.

[GJ] Paul G. Goerss and John F. Jardine. Simplicial homotopy theory, volume 174 of Progress in Mathematics. Birkhäuser Verlag, Basel, 1999.

[Gle] Paul G. Glenn. Realization of cohomology classes in arbitrary exact categories. J. Pure Appl. Algebra, 25(1):33–105, 1982.

[Gro] Alexander Grothendieck. Technique de descente et théorèmes d’existence en géométrie algébrique. II. Le théorème d’existence en théorie formelle des modules. In Séminaire Bourbaki, Vol. 5, pages Exp. No. 195, 369–390. Soc. Math. France, Paris, 1995.

[Hir] Philip S. Hirschhorn. Model categories and their localizations, volume 99 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2003.

[Ill] Luc Illusie. Complexe cotangent et déformations. I. Springer-Verlag, Berlin, 1971. Lecture Notes in Mathematics, Vol. 239.

[Mac] Saunders MacLane. Categories for the working mathematician. Springer-Verlag, New York, 1971. Graduate Texts in Mathematics, Vol. 5.

[Man] Marco Manetti. Deformation theory via differential graded Lie algebras. In Algebraic Geometry Seminars, 1998–1999 (Italian) (Pisa), pages 21–48. Scuola Norm. Sup., Pisa, 1999. arXiv math.AG/0507284.

[Ols1] Martin Olsson. Sheaves on Artin stacks. J. Reine Angew. Math., 603:55–112, 2007.

[Ols2] Martin C. Olsson. Deformation theory of representable morphisms of algebraic stacks. Math. Z., 253(1):25–62, 2006.

[Pri1] J. P. Pridham. Deforming l-adic representations of the fundamental group of a smooth variety. J. Algebraic Geom., 15(3):415–442, 2006.

[Pri2] J. P. Pridham. Deformations of schemes and other bialgebraic structures. Trans. Amer. Math. Soc., 360(3):1601–1629, 2008.

[Pri3] J. P. Pridham. The homotopy theory of strong homotopy algebras and bialgebras. Homology, Homotopy Appl., 12(2):39–108, 2010. arXiv:0908.0116v2 [math.AG].

[Pri4] J. P. Pridham. Unifying derived deformation theories. Adv. Math., 224(3):772–826, 2010. arXiv:0705.0344v5 [math.AG].

[Pri5] J. P. Pridham. Derived deformations of schemes. Comm. Anal. Geom., 20(3):529–563, 2012. arXiv:0908.1963v1 [math.AG].

[Pri6] J. P. Pridham. Presenting higher stacks as simplicial schemes. Adv. Math., 238:184–245, 2013. arXiv:0905.4044v3 [math.AG].

[Qui] Daniel Quillen. Rational homotopy theory. Ann. of Math. (2), 90:205–295, 1969.

[Ran] Ziv Ran. Lie atoms and their deformations. Geom. Funct. Anal., 18(1):184–221, 2008. arXiv:math/0412204v7.

[Van] Donovan H. Van Osdol. Bicohomology theory. Trans. Amer. Math. Soc., 183:449–476, 1973.

[Wei] Charles A. Weibel. An introduction to homological algebra. Cambridge University Press, Cambridge,