NOTES CHARACTERISING HIGHER AND DERIVED STACKS CONCRETELY

J.P.PRIDHAM

Abstract. This is an informal summary of the main concepts in [Pri], based on notes of various seminars. It gives constructions of higher and derived stacks without recourse to the extensive theory developed by Toën, Vezzosi and Lurie. Explicitly, higher stacks are described in terms of simplicial diagrams of affine schemes, which are analogous to atlases for manifolds. We also describe quasi-coherent sheaves and complexes on such objects.

Introduction

The need for simplicial objects. Any scheme X gives rise to a functor from rings to sets, sending A to X(A) = Hom(Spec A, X). Likewise, any algebraic stack gives a functor from rings to groupoids. When X is a moduli space or stack, the points of X have a geometric meaning. For instance

Hilb_{Y} (A) = {closed subschemes of Y × Spec A, flat over A}.

In derived algebraic geometry, the basic building blocks are simplicial rings, or equivalently in characteristic 0, dg rings (collectively referred to as derived rings). Thus derived moduli spaces and stacks have to give rise to functors on derived rings dAlg.To understand what such a functor must be, we start with a derived ring A and look at possible functors associated to Spec A.

(1) The obvious candidate is the functor Hom_{dAlg}(A, −): dAlg → Set. This is clearly no good, as it does not map quasi-isomorphisms to isomorphisms, even when A is a polynomial ring.

(2) This suggests the functor Hom_{Ho(dAlg)}(P, −): dAlg → Set, where Ho(dAlg) is the homotopy category (given by formally inverting quasi-isomorphisms). This works well for infinitesimal derived deformation theory (as in [Man]), but is not left-exact. Thus it cannot sheafify, so will not give a good global theory.

(3) The solution is to look at the derived Hom-functor R\underline{Hom}, which takes values in simplicial sets (to be defined in the sequel). This maps quasi-isomorphisms to weak equivalences, and has good exactness properties. Thus even if we start with a moduli problem without automorphisms, the derived problem leads us to consider simplicial sets.

Where do simplicial objects come from? Simplicial resolutions of schemes will be familiar to anyone who has computed Čech cohomology. Given a quasi-compact semiseparated scheme Y, we may take a finite affine cover U = ∐_{i} U_{i} of Y, and define the simplicial scheme Y̌ to be the Čech nerve Y̌ := cosk_{0}(U/Y). Explicitly,

Y̌_{n} = \overbrace{U ×_{Y} U ×_{Y} … ×_{Y} U}^{n+1} = ∐_{i_{0},...,i_{n}} U_{i_{0}} ∩ … ∩ U_{i_{n}},

so Y̌_{n} is an affine scheme, and Y̌ is the unnormalised Čech resolution of Y.

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

Given a quasi-coherent sheaf ℱ on Y, we can then form a cosimplicial complex Č^{n}(Y, ℱ) := Γ(Y̌_{n}, ℱ), and of course Zariski cohomology is given by

H^{i}(Y, ℱ) ≅ H^{i}Č^{•}(Y, ℱ).

Likewise, if 𝔜 is a semi-separated Artin stack, we can choose a presentation U → 𝔜 with U affine, and set Y̌ := cosk_{0}(U/𝔜), so

Y̌_{n} = \overbrace{U ×_{𝔜} U ×_{𝔜} … ×_{𝔜} U}^{n+1} .

Resolutions of this sort were used by Olsson in [Ols] to study quasi-coherent sheaves on Artin stacks.

Questions.

(1) Which simplicial affine schemes correspond to schemes, Artin stacks or Deligne– Mumford stacks in this way?

(2) What about higher stacks?

[For an example of a higher stack, moduli of perfect complexes ℱ on X will give an n-stack provided we restrict to complexes with Ext_{X}^{i}(ℱ , ℱ) = 0 for all i ≤ −n. Similarly, a 2-stack governs moduli of stacky curves.]

(3) What about derived schemes and stacks?

(4) What do quasi-coherent sheaves then look like?

The first two questions will be addressed in Theorem 3.3, and the third in Theorem 5.11. The fourth will be addressed in §§4 and 5.4.

I would like to thank João Pedro dos Santos for making many helpful suggestions and correcting several errors.

1. Hypergroupoids

1.1. Simplicial sets.

Definition 1.1. Define |Δ^{n}| to be the geometric n-simplex {x ∈ ℝ_{+}^{n+1} : ∑_{i=0}^{n} x_{i} = 1}. Write ∂^{i}: |Δ^{n−1}| → |Δ^{n}| for the inclusion of the ith face, and σ^{i}: |Δ^{n+1}| → |Δ^{n}| for the map given by collapsing the edge (i, i + 1).

Definition 1.2. Given a topological space X, define Sing(X)_{n} to be the set of continuous maps from |Δ^{n}| to X. These fit into a diagram

[figure: Sing(X)_{0}; σ_{0}; Sing(X)_{1}; ∂_{0}; ∂_{1}; Sing(X)_{2}; Sing(X)_{3}; ·; ·; ·; …; …,]

where the arrows satisfy various relations such as ∂_{i}σ_{i} = id. (Note that contravariance has turned superscripts into subscripts).

Any diagram of this form is called a simplicial set. (For a rigorous definition, see [Wei, §8.1]) We will denote the category of simplicial sets by sSet. We can define simplicial diagrams in any category similarly, while cosimplicial diagrams are given by reversing all the arrows.

If A_{•} is a simplicial abelian group, then note that setting d := ∑_{i=0}^{n}(−1)^{i}∂_{i}: A_{n} → A_{n−1} gives maps satisfying d^{2} = 0, so A_{•} becomes a chain complex.

Definition 1.3. For n ≥ 0, the combinatorial n-simplex Δ^{n} ∈ sSet is characterised by the property that Hom_{sSet}(Δ^{n}, K) ≅ K_{n} for all simplicial sets K. Its boundary ∂Δ^{n} ⊂ Δ^{n} is given by ⋃_{i=0}^{n} ∂^{i}(Δ^{n−1}) (taken to include the case ∂Δ^{0} = ∅), and for n ≥ 1 the kth horn

Λ^{n,k} is given by ⋃\begin{matrix}0≤i≤n \\ i≠k \end{matrix} ∂^{i}(Δ^{n−1}).

Definition 1.4. There is a geometric realisation functor | − |: sSet → Top, left adjoint to Sing. This is characterised by the properties that it preserves colimits and that |Δ^{n}| = |Δ^{n}|.

Draw a picture of |Λ^{2,k}| or |Λ^{3,k}| and you will see the reasoning for both the term horn and the notation Λ.

Definition 1.5. A map f: K → L in sSet is said to be a weak equivalence if |f|: |K| → |L| is a weak equivalence (i.e. induces isomorphisms on all homotopy groups).

Note that the canonical maps |Sing(X)| → X and hence K → Sing(|K|) are always weak equivalences, so sSet and Top have the same homotopy theory ([GJ, Theorem 11.4]).

1.2. Hypergroupoids.

Definition 1.6. [Dus, Gle]: A (Duskin–Glenn) n-hypergroupoid (often also called a weak n-groupoid) is an object X ∈ sSet for which the partial matching maps

X_{m} = Hom_{sSet}(Δ^{m}, X) → Hom_{sSet}(Λ^{m,k}, X)

are surjective for all m, k, and isomorphisms for all m > n and all k.

The first condition is equivalent to saying that X is a Kan complex, or fibrant.

Examples 1.7. (1) A 0-hypergroupoid is just a set X = X_{0}.

(2) [Gle, §2.1] (see also [GJ, Lemma I.3.5]): Every 1-hypergroupoid X arises as the nerve BΓ of some groupoid Γ, so is given by

(BΓ)_{n} = ∐_{x_{0},...,x_{n}} Γ(x_{0}, x_{1}) × Γ(x_{1}, x_{2}) × … × Γ(x_{n−1}, x_{n}).

Moreover, Γ can be recovered from X by taking objects X_{0}, morphisms X_{1}, source and target ∂_{0}, ∂_{1} : X_{1} → X_{0}, identity σ_{0} : X_{0} → X_{1} and multiplication

X_{1} ×_{∂_{0},X_{0},∂_{1}} X_{1} ⟶^{(∂_{2},∂_{0})^{−1}} X_{2} ⟶^{∂_{1}} X_{1}.

Equivalently, Γ is the fundamental groupoid π_{f}X of X (as in [GJ, §I.8]).

(3) Under the Dold–Kan correspondence between non-negatively graded chain complexes and abelian groups ([Wei, §8.4]), n-hypergroupoids in abelian groups correspond to chain complexes concentrated in degrees [0, n].

Properties 1.8. (1) For an n-dimensional hypergroupoid X, π_{m}X = 0 for all m > n.

(2) [Pri, Lemma 2.12]: An n-hypergroupoid X is completely determined by its truncation X_{≤n+1}. Explicitly, X = cosk_{n+1}X, where the m-coskeleton cosk_{m}X is given by (cosk_{m}X)_{i} = Hom((Δ^{i})_{≤m}, X_{≤m}). Moreover, a simplicial set of the form cosk_{n+1}X is an n-hypergroupoid if and only if it satisfies the conditions of Definition 1.6 up to level n + 2.

When n = 1, these statements amount to saying that a groupoid is uniquely determined by its objects (level 0), morphisms and identities (level 1) and multiplication (level 2). However, we do not know we have a groupoid until we check associativity (level 3).

There is also a notion of relative n-hypergroupoids X → Y, expressed in terms of (unique) liftings of

\xymatrix{Λ^{m,k} \ar[d] \ar[r] & X \ar[d] \\ Δ^{m} \ar@{.>}[ur] \ar[r] & Y.}

For example, a relative 0-dimensional hypergroupoid f: X → Y is a Cartesian morphism, in the sense that the maps

X_{n} ⟶^{(∂_{i},f)} X_{n−1} ×_{Y_{n−1},∂_{i}} Y_{n}

are all isomorphisms. Given y ∈ Y_{0}, we can write F (y) := f_{0}^{−1}{y}, and observe that f is equivalent to a local system on Y with fibres F. The analogue of Property 1.8.2 above also holds in the relative case. Level 0 gives us the fibres F (y), level 1 gives us the descent data θ(z): F (∂_{0}z) ≅ F (∂_{1}z) for z ∈ Y_{1} (thereby determining the local system uniquely), but we do not know we have a groupoid until we check the cocycle condition (level 2): θ(∂_{2}w) ∘ θ(∂_{0}w) = θ(∂_{1}w) for all w ∈ Y_{2}.

2. Higher stacks

We will now show how to develop the theory of higher Artin stacks. For other types of stack, just modify the notion of covering. In particular, for Deligne–Mumford stacks, replace “smooth” with “étale” throughout. For simplicity of exposition, we will assume that everything is quasi-compact, quasi-separated etc. (strongly quasi-compact in the terminology of [TV]) — to allow more general objects, replace affine schemes with arbitrary disjoint unions of affine schemes.

Given a simplicial set K and a simplicial affine scheme X, there is an affine scheme Hom_{sSet}(K, X) with the property that for all rings A, Hom_{sSet}(K, X)(A) = Hom_{sSet}(K, X(A)). Explicitly, when K = Λ^{m,k} this is given by the equaliser of a diagram

∏_{\begin{matrix}0≤i≤m \\ i≠k \end{matrix}} X_{m−1} ⟹ ∏_{\begin{matrix}0≤i<j≤m \\ i,j≠k \end{matrix}} X_{m−2}.

Definition 2.1. Define an Artin n-hypergroupoid to be a simplicial affine scheme X_{•}, such that the partial matching maps

X_{m} = Hom_{sSet}(Δ^{m}, X) → Hom_{sSet}(Λ^{m,k}, X)

are smooth surjections for all k, m, and isomorphisms for all m > n and all k.

The idea of using such objects to model higher stacks is apparently originally due to Grothendieck, buried somewhere in [Gro].

Remark 2.2. Note that hypergroupoids can be defined in any category containing pullbacks along covering morphisms. In [Zhu], this is used to define Lie n-groupoids (taking the category of manifolds, with coverings given by submersions). A similar approach could be used to define higher topological stacks (generalising [Noo]), taking surjective local fibrations as the coverings in the category of topological spaces. Similar constructions could be made in non-commutative geometry and in synthetic differential geometry.

Fix a base ring R. Given any Artin n-hypergroupoid X over R, there is an associated functor X: Alg_{R} → sSet, given by X(A)_{n} := X_{n}(A). The following is [Pri, Theorem 4.7]:

Theorem 2.3. If X is an Artin n-hypergroupoid X over R, then its hypersheafification X^{♯}: Alg_{R} → sSet is an n-geometric Artin stack in the sense of [TV, Definition 1.3.3.1]. Every n-geometric Artin stack arises in this way.

For set-valued functors, hypersheafification is just ordinary sheafification. For groupoidvalued functors, hypersheafification is stackification. An analogous definition can be made for sSet-valued functors, but in the next section we will give an explicit description.

Remark 2.4. Beware that there are slight differences in terminology between [TV] and [Lur1]. In the former, only affine schemes are 0-representable, so arbitrary schemes might only be 2-geometric, while Artin stacks are 1-geometric stacks if and only if they have affine diagonal. In the latter, algebraic spaces are 0-stacks.

An n-stack 𝔛 in the sense of [Lur1] is called n-truncated in [TV], and can be characterised by the property that that π_{i}(𝔛(A)) = 0 for all i > n and A ∈ Alg_{R}. For a hypergroupoid X, this amounts to saying that the associated stack X^{♯} is n-truncated if and only if the maps

X_{i} = Hom_{sSet}(Δ^{i}, X) → Hom_{sSet}(∂Δ^{i}, X)

are monomorphisms (i.e. immersions) of affine schemes for all i > n.

It follows easily from Property 1.8.2 that every n-geometric stack in [TV] is n-truncated; conversely, any n-truncated stack 𝔛 is (n + 2)-geometric. Any Artin stack with affine diagonal (in particular any separated algebraic space) is 1-geometric.

If we used algebraic spaces instead of affine schemes in Definition 2.1, then Theorem 2.3 would adapt to give a characterisation of n-truncated Artin stacks. Our main motivation for using affine schemes as the basic objects is that they will be easier to translate to a derived setting.

3. Morphisms and equivalences

Theorem 2.3 is all very well, but is clearly not the whole story. For a start, it gives us no idea of how to construct the hypersheafification. Thus we have no way of understanding morphisms between n-geometric stacks (as the hypersheafification is clearly not full), or even of knowing when two hypergroupoids will give us equivalent n-geometric stacks. If we think of the hypergroupoid as analogous to the atlas of a manifold, then we need a notion similar to refinement of an open cover.

3.1. Trivial relative hypergroupoids.

Definition 3.1. Say that a morphism f: X → Y of simplicial affine schemes is a trivial relative Artin (resp. Deligne–Mumford) n-hypergroupoid if the relative matching maps

X_{m} → Hom_{sSet}(∂Δ^{m}, X) ×_{Hom_{sSet}(∂Δ^{m},Y)} Y_{m}

are smooth (resp. étale) surjections for all m, and isomorphisms for all m ≥ n.

An example of a trivial relative Artin 1-hypergroupoid in stacks is the Čech nerve Y̌ → 𝔜 constructed in the introduction.

Property 3.2. [Pri, Lemma 2.9]: Trivial relative n-hypergroupoids are completely determined by their truncations in levels < n. Explicitly, a morphism f : X → Y is a trivial relative Artin (resp. Deligne–Mumford) n-hypergroupoid if and only if X = Y ×_{cosk_{n−1}Y} cosk_{n−1}X, and the (n − 1)-truncated morphism X_{<n} → Y_{<n} satisfies the conditions of Definition 3.1 (up to level n − 1).

3.2. Sheafification and morphisms.

Theorem 3.3. The homotopy category of strongly quasi-compact n-geometric Artin stacks is given by taking the full subcategory of sAff consisting of Artin n-hypergroupoids, and formally inverting the trivial relative Artin (resp. Deligne–Mumford) n-hypergroupoids.

In fact, a model for the ∞-category of strongly quasi-compact n-geometric Artin stacks is given by the relative category consisting of Artin n-hypergroupoids and the class of trivial relative Artin n-hypergroupoids.

The same results hold true if we substitute “Artin” with “Deligne–Mumford” throughout.

The relative categories of [BK] give the cleanest description of the ∞-category (as suggested to the author by the referee of [Pri]). Before proving Theorem 5.11, we will give a more explicit description of the ∞-category.

Lemma 3.4. If f: X → Y is a trivial relative Artin n-hypergroupoid, then for all rings A, the map X^{♯}(A) → Y^{♯}(A) is a weak equivalence in sSet.

Definition 3.5. Define the simplicial Hom functor on simplicial affine schemes by letting \underline{Hom}_{sAff}(X, Y) be the simplicial set given by

\underline{Hom}_{sAff}(X, Y)_{n} := Hom_{sAff}(Δ^{n} × X, Y).

Definition 3.6. Given an Artin n-hypergroupoid Y and X ∈ sAff, define

\underline{Hom}_{sAff}^{♯}(X, Y) := \underrightarrow{lim} \underline{Hom}_{sAff}(X̃, Y),

where X̃ → X runs over any weakly initial filtered inverse system of trivial relative Artin n-hypergroupoids. [In fact, [Pri, Corollary 3.32] shows that there is a suitable inverse system X̃ → X of trivial relative Deligne–Mumford n-hypergroupoids.]

The following is [Pri, Corollary 4.10]:

Theorem 3.7. If X ∈ sAff and Y is an Artin n-hypergroupoid, then the derived Hom functor on hypersheaves is given (up to weak equivalence) by

R\underline{Hom}(X^{♯}, Y^{♯}) ≃ \underline{Hom}_{sAff}^{♯}(X, Y).

Remarks 3.8. Given a ring A, set X = Spec A, and note that \underline{Hom}_{sAff}^{♯}(X, Y) = Y^{♯}(A), the hypersheafification of the functor Y : Alg_{R} → sSet, so we can take Definition 3.6 as a definition of sheafification for Artin hypergroupoids, giving an explicit description of Y^{♯}. The n = 1 case should be familiar as the standard definition of sheafification.

Between them, Theorems 2.3 and 3.7 recover the simplicial category A_{n} of strongly quasi-compact n-geometric Artin stacks, with Theorem 2.3 giving the objects and Theorem 3.7 the morphisms. We could thus take those theorems to be a definition of that simplicial category.

Moreover, [Pri, Remark 3.27] shows that

π_{0}\underline{Hom}_{sAff}^{♯}(X, Y) ≃ \underrightarrow{lim}_{X′} π_{0}\underline{Hom}_{sAff}(X′, Y),
π_{n}(\underline{Hom}_{sAff}^{♯}(X, Y), f) ≃ \underrightarrow{lim}_{X′} π_{n}(\underline{Hom}_{sAff}(X′, Y), f),

where the limit is taken over the homotopy category of all trivial relative Deligne–Mumford n-hypergroupoids X′ → X.

Even for classical Artin or Deligne–Mumford stacks, this gives a shorter (and arguably more satisfactory) definition, since classical algebraic stacks are just 1-truncated (see Remark 2.4) geometric stacks. For semi-separated algebraic spaces or schemes (0-truncated étale and Zariski 1-hypergroupoids, respectively), this definition is at least comparable in complexity to the classical one. Note that making use of Properties 1.8.2 and 3.2 allows us characterise all of these concepts in terms of finite diagrams of affine schemes.

Proof of Theorem 3.3. We just need to show that the functor \underline{Hom}_{sAff}^{♯} on pairs of Artin (resp. Deligne–Mumford) n-hypergroupoids is the right-derived functor of Hom_{sAff} with respect to the class T_{n} of trivial relative Artin (resp. Deligne–Mumford) n-hypergroupoids. In other words, we need to show that \underline{Hom}_{sAff}^{♯} is the universal bifunctor under Hom_{sAff} mapping T_{n} to isomorphisms in Ho(sSet).

Lemma 3.4 ensures that \underline{Hom}_{sAff}^{♯} sends T_{n} to weak equivalences, so we need only prove universality. For this, we just observe that if Y is an Artin (resp. Deligne–Mumford) nhypergroupoid, then the map Y → Y^{Δ^{n}} is the section of a morphism in T_{n}. Since X̃ → X is a system in T_{n}, this means that \underline{Hom}_{sAff}^{♯} is constructed from Hom_{sAff} and T_{n}, so any bifunctor under Hom_{sAff} must lie under \underline{Hom}_{sAff}^{♯} if it sends T_{n} to weak equivalences. □

3.3. Eilenberg–Mac Lane spaces and cohomology. Given any abelian group n, the Dold–Kan correspondence allows us to form a simplicial abelian group K(A, n) given by denormalising the chain complex A[−n]. Explicitly, K(A, n) is freely generated under degeneracy operations σ_{i} by a copy of A in level n, so K(A, n)_{m} ≅ A^{\binom{m}{n}} (see [Wei, 8.4.4] for details).

Given a smooth commutative affine group scheme A, this construction gives us an Artin n-hypergroupoid K(A, n) (in fact, if n = 1, A need not be commutative), and then for any Artin stack 𝔛,

H_{ét}^{n−i}(𝔛, A) ≅ π_{i}R\underline{Hom}(X^{♯}, K(A, n)^{♯}).

In particular, taking A = 𝔾_{a} gives us H^{∗}(𝔛, 𝒪_{𝔛}), while taking A = ℤ/ℓ^{m} (regarded as a finite scheme) gives H_{ét}^{∗}(𝔛, ℤ/ℓ^{m}).

We could generalise this by allowing A to be a smooth commutative algebraic group space, in which case K(A, n) would be an Artin n-hypergroupoid in algebraic spaces, making K(A, n)^{♯} an n-truncated geometric stack (see Remark 2.4), and hence representable by an Artin (n + 2)-hypergroupoid (in affine schemes).

4. Quasi-coherent sheaves

The following is [Pri, Corollary 5.12]:

Proposition 4.1. For an Artin n-hypergroupoid X, giving a quasi-coherent module on the n-geometric stack X^{♯} is equivalent to giving

(1) a quasi-coherent sheaf ℱ^{n} on X_{n} for each n, and

(2) isomorphisms ∂^{i}: ∂_{i}^{∗}ℱ^{n−1} → ℱ^{n} for all i and n, satisfying the usual cosimplicial identities.

Given a morphism f: X → Y of Artin n-hypergroupoids, inverse images are easy to compute: we just have (f^{∗}ℱ)^{n} := f_{n}^{∗}ℱ^{n}. Direct images are much harder to define, as taking f_{∗} levelwise destroys the Cartesian property. See [Pri, §5.4.3] for an explicit description of the derived direct image functor Rf_{∗}^{cart}.

5. Derived stacks

Motivated by the need for good obstruction theory and cotangent complexes, derived algebraic geometry replaces rings with simplicial rings. There is a normalisation functor N: sAlg_{R} → dg_{+}Alg_{R} from simplicial R-algebras to commutative chain R-algebras in nonnegative degrees. If R is a ℚ-algebra, then N induces an equivalence on the homotopy categories (and also on the derived Hom-spaces). [When R is not a ℚ-algebra, dg_{+}Alg_{R} is not even a model category.]

We will write dAlg for either of the categories sAlg_{R}, dg_{+}Alg_{R}, and write dAff for the opposite category (derived affine schemes over R), denoting objects as Spec A. Any derived ring A ∈ dAlg can be thought of as essentially an exotic nilpotent thickening of H_{0}A, so there are equivalent variants of this theory replacing A with its localisation ([Pri, §6.2]), its henselisation ([Pri, §6.3]), or even (in Noetherian cases) its completion ([Pri, Propositions 6.6 and 6.37]) over H_{0}A.

Remark 5.1. The constructions in this section will work for any model category with a suitable notion of coverings. In particular, they work for symmetric spectra, the basis of a theory known as topological, spectral, brave new or (unfortunately) derived algebraic geometry. Roughly speaking (as explained in the introduction to [Lur2]), simplicial rings serve to apply homotopy theory to algebraic geometry, while symmetric spectra are used to do the opposite. For a detailed discussion, see [Lur1, §2.6].

5.1. Derived hypergroupoids.

Definition 5.2. Say that a morphism in sAlg_{R} is quasi-free if it is freely generated in each level, with the generators closed under the degeneracy operations σ_{i}. Say that a morphism in dg_{+}Alg_{R} is quasi-free if the underlying morphism of skew-commutative graded algebras is freely generated.

Say that a morphism in dAlg is a cofibration if it is a retract of a quasi-free map, and that a morphism in dAff is a fibration if it is Spec of a cofibration.

Write sdAff for the category of simplicial derived affine schemes, i.e. simplicial diagrams in dAff. Weak equivalences in this category are maps Spec B → Spec A inducing isomorphisms H_{∗}(B) ≅ H_{∗}(A).

Definition 5.3. An object X_{•} ∈ sdAff is said to be Reedy fibrant if the matching maps

X_{n} = Hom_{sSet}(Δ^{n}, X) → Hom_{sSet}(∂Δ^{n}, X)

are fibrations in dAff for all n.

Example 5.4. Given a cofibration R → A in sAlg_{R}, write X = Spec A. We may form an object A ⊗ Δ^{n} ∈ sAlg_{R} by

(A ⊗ Δ^{n})_{i} := \overbrace{A_{i} ⊗_{R} A_{i} ⊗_{R} … ⊗_{R} A_{i}}^{Δ_{i}^{n}} .

Then the simplicial derived affine scheme \underline{X} given by \underline{X}_{n} := Spec (A ⊗ Δ^{n}) is Reedy fibrant, and X → \underline{X} is a weak equivalence levelwise.

\underline{X} will be familiar to some readers from the construction of simplicial Hom, since by definition

\underline{Hom}_{sAlg_{R}}(A, B) = \underline{X}(B) ∈ sSet

for A ∈ sAlg_{R}. More generally, for any A ∈ sAlg_{R}, we can always take a quasi-isomorphism A′ → A for A′ cofibrant, and then set

R\underline{Hom}_{sAlg_{R}}(A, B) := \underline{Hom}(A′, B).

There are analogous constructions in dg_{+}Alg_{R}, but they are not so easily described.

Definition 5.5. We say that a morphism Spec B → Spec A in dAff is a smooth (resp. étale) surjection if Spec H_{0}B → Spec H_{0}A is so, and the maps

H_{i}(A) ⊗_{H_{0}(A)} H_{0}(B) → H_{i}(B)

are all isomorphisms.

Definition 5.6. A derived Artin n-hypergroupoid is a Reedy fibrant object X_{•} ∈ sdAff for which the partial matching maps

X_{m} = Hom_{sSet}(Δ^{m}, X) → Hom_{sSet}(Λ^{m,k}X)

are smooth surjections for all k, m, and weak equivalences for all m > n and all k.

Remark 5.7. Given any derived Artin n-hypergroupoid X, we may form a simplicial scheme π^{0}X by setting

π^{0}X_{n} := Spec (H_{0}O(X_{n})) ∈ Aff.

Then observe that π^{0}X is an Artin n-hypergroupoid, equipped with a map π^{0}X → X. We call this the underived part of X.

5.2. Derived Artin stacks. The following is [Pri, Theorem 4.7]:

Theorem 5.8. If X is a derived Artin n-hypergroupoid X over R, then its hypersheafification X^{♯}: dAlg_{R} → sSet is an n-geometric derived Artin stack in the sense of [TV, Definition 1.3.3.1]. Every strongly quasi-compact n-geometric derived Artin stack arises in this way.

Remarks 5.9. As with Remarks 2.4, there is a difference in terminology between [Lur1] and [TV]. A geometric derived Artin ∞-stack 𝔛 is called an n-stack (Lurie) or n-truncated (Toën–Vezzosi) if the associated underived Artin ∞-stack π^{0}𝔛 is n-truncated. This implies that for A ∈ dAlg with H_{i}(A) = 0 for all i > m, we have π_{i}(𝔛(A)) = 0 for all i > m + n.

5.3. Morphisms and equivalences.

Definition 5.10. A trivial relative derived Artin n-hypergroupoid is a morphism X_{•} → Y_{•} in sdAff for which the matching maps

X_{m} → Hom_{sSet}(∂Δ^{m}, X) ×_{Hom_{sSet}(∂Δ^{m},Y)} Y_{m}

are smooth surjective fibrations for all m > 1 and weak equivalences for m ≥ n.

The results of §3.2 all now carry over:

Theorem 5.11. The homotopy category of strongly quasi-compact n-geometric derived Artin stacks is given by taking the full subcategory of sdAff consisting of derived Artin nhypergroupoids, and formally inverting the trivial relative derived Artin n-hypergroupoids.

In fact, a model for the ∞-category of strongly quasi-compact n-geometric derived Artin stacks is given by the relative category consisting of derived Artin n-hypergroupoid and the class of trivial relative derived Artin n-hypergroupoids.

Before proving this, we give the preliminary results necessary. We can first define Homspaces \underline{Hom}_{sdAff} by \underline{Hom}_{sdAff}(X, Y)_{n} = Hom_{sdAff}(X, Y^{Δ^{n}}), and then the following is [Pri, Theorem 4.10]:

Theorem 5.12. If X ∈ sdAff and Y is a derived Artin n-hypergroupoid, then the derived Hom functor on hypersheaves is given (up to weak equivalence) by

R\underline{Hom}(X^{♯}, Y^{♯}) ≃ \underrightarrow{lim} \underline{Hom}_{sdAff}(X̃, Y),

where X̃ → X runs over any weakly initial filtered inverse system of trivial relative derived Deligne–Mumford n-hypergroupoids.

Remark 5.13. Given a derived R-algebra A, set X = Spec A, and note that Y^{♯}(A) ≃ R\underline{Hom}(X^{♯}, Y^{♯}), so the theorem gives an explicit description of Y^{♯}. In fact, we can take the theorem to be a definition of hypersheafification, and even as a definition of the simplicial category DA_{n} of strongly quasi-compact n-geometric derived Artin stacks (with Theorem 5.8 giving the objects and Theorem 5.12 the morphisms).

Again, [Pri, Remark 3.27] shows that

π_{0}\underline{Hom}_{sAff}^{♯}(X, Y) ≃ \underrightarrow{lim}_{X′} π_{0}\underline{Hom}_{sAff}(X′, Y),
π_{n}(\underline{Hom}_{sAff}^{♯}(X, Y), f) ≃ \underrightarrow{lim}_{X′} π_{n}(\underline{Hom}_{sAff}(X′, Y), f),

where the limit is taken over the homotopy category of all trivial relative derived Deligne– Mumford n-hypergroupoids X′ → X.

Note that the proof of Theorem 5.11 is now exactly the same as that for Theorem 3.3, replacing Theorem 3.7 with Theorem 5.12.

5.4. Quasi-coherent complexes. Since the basic building blocks for derived algebraic geometry are simplicial rings or chain algebras, the correct analogue of quasi-coherent sheaves has to involve complexes.

Definition 5.14. Given a chain algebra A, an A-module M in complexes is a (possibly unbounded) chain complex M equipped with a distributive chain morphism A ⊗ M → M. Given a simplicial ring A, we just define an A-module in complexes to be an NA-module in complexes, where NA is the chain algebra given by Dold–Kan normalisation.

The following is [Pri, Proposition 5.12]:

Proposition 5.15. For a derived Artin n-hypergroupoid X, giving a quasi-coherent complex (in the sense of [Lur1, §5.2]) on the n-geometric derived stack X^{♯} is equivalent (up to quasi-isomorphism) to giving

(1) an O(X_{n})-module ℱ^{n} in complexes for each n, and

(2) quasi-isomorphisms ∂^{i}: ∂_{i}^{∗}ℱ^{n−1} → ℱ^{n} for all i and n, satisfying the usual cosimplicial identities.

In broad terms, quasi-coherent complexes correspond to complexes ℱ_{•} of presheaves of 𝒪_{X}-modules whose homology presheaves ℋ_{n}(ℱ_{•}) are quasi-coherent. To understand how these are related to complexes of quasi-coherent sheaves on schemes, see [Pri, Remarks 5.24].

As in §4, inverse images of quasi-coherent complexes are easy to compute, while derived direct images are more complicated — see [Pri, §5.4.3].

5.5. Derived schemes and algebraic spaces.

Definition 5.16. An n-geometric derived Deligne–Mumford stack X is called a derived algebraic space (resp. derived scheme) if the associated underived n-stack π^{0}X is represented by an algebraic space (resp. a scheme). For this to occur, we must have n ≤ 2 (or n ≤ 1 if π^{0}X is semi-separated).

The following is given by [Pri, Theorems 6.14, 6.23 and 6.42]:

Theorem 5.17. Fix a scheme Z over a ring R. Then the homotopy category of derived schemes X over R with π^{0}X ≃ Z is weakly equivalent to the homotopy category of presheaves 𝒜_{•} of derived R-algebras on the category of affine open subschemes of Z, satisfying the following:

(1) H_{0}(𝒜_{•}) = 𝒪_{Z};

(2) for all i, the presheaf H_{i}(𝒜_{•}) is a quasi-coherent 𝒪_{Z}-module.

The same result holds if we replace schemes with algebraic spaces, and open immersions with étale maps.

In this setting, Proposition 5.15 becomes:

Proposition 5.18. Take a derived scheme X, given by a pair (Z, 𝒜_{•}) as in Theorem 5.17. Then giving a quasi-coherent complex on X is equivalent (up to quasi-isomorphism) to giving a presheaf ℱ_{•} of 𝒜_{•}-modules in complexes for which the presheaves H_{i}(ℱ_{•}) are all quasi-coherent as 𝒪_{Z}-modules.

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