ScalingStacks

Algebraic geometry provides a wealth of examples of symmetric monoidal categories, as well as powerful tools to study such categories. To a scheme or stack, we may assign its category of quasi-coherent sheaves with its commutative multiplication given by tensor product. This construction generalizes the category of $R$-modules for $R$ a commutative ring (the case of $\operatorname{Spec}R$), and the category of representations of an algebraic group $G$ (the case of classifying stacks $BG$). Conversely, Tannakian formalism often allows us to reverse this process and assign a stack to a symmetric monoidal category.

Schemes and stacks are also natural sources of symmetric monoidal $\infty$-categories, which refine the familiar symmetric monoidal derived categories of quasi-coherent sheaves. For example, to any stack in characteristic zero, we can assign the differential graded enhancement of its derived category, constructed by taking the differential graded category of complexes with quasi-coherent cohomology and localizing the quasi-isomorphisms (following [Ke, D, To1]). In general, we can consider the stable symmetric monoidal $\infty$-category of quasi-coherent sheaves on any stack whose homotopy category is the familiar derived category.

Trying to geometrically describe algebraic operations on $\infty$-categories of quasi-coherent sheaves quickly takes us from ordinary algebraic geometry to an enhanced version which has been developed over the last few years known as derived algebraic geometry [L1, L3, L4, L5, ToVe1, ToVe2]. In fact, derived algebraic geometry also provides a far greater abundance of examples of stable symmetric monoidal $\infty$-categories. Thus it is most natural to both ask and answer algebraic questions about stable symmetric monoidal $\infty$-categories in this context.

Derived algebraic geometry generalizes the world of schemes simultaneously in two directions. Regarding schemes in terms of their functors of points, which are functors from rings to sets (satisfying a sheaf axiom with respect to a Grothendieck topology), we want to replace both the source and the target categories by suitable $\infty$-categories. First, we may consider functors from rings to spaces (or equivalently, simplicial sets) considered as an $\infty$-category (with weak homotopy equivalences inverted). If we consider only $1$-truncated spaces, or equivalently their fundamental groupoids, we recover the theory of stacks. This naturally leads to the introduction of higher stacks where we consider sheaves of (not necessarily $1$-truncated) spaces on the category of rings. For example, we can take any space and consider it as a higher stack by taking (the sheafification of) the corresponding constant functor on rings.

To pass from higher stacks to derived stacks, we replace the source category of commutative rings by an $\infty$-category of commutative ring objects in a symmetric monoidal $\infty$-category. There are at least three natural candidates that are commonly considered:

1. (1)

   connective (that is, homological or non-positively graded) commutative differential graded $k$-algebras over a commutative ring $k$ of characteristic zero;

2. (2)

   simplicial commutative rings, or simplicial commutative $k$-algebras over a commutative ring $k$;

3. (3)

   connective $\mathcal{E}_{\infty}$-ring spectra, or connective $\mathcal{E}_{\infty}$-algebras over the Eilenberg-MacLane spectrum $H(k)$ of a commutative ring $k$.¹¹ 1 Note that by the results of [L5] reviewed above, an $\mathcal{E}_{\infty}$-algebra over $H(k)$ is equivalently a commutative algebra object in the symmetric monoidal $\infty$-category of $H(k)$-module spectra.

When $k$ is a $\mathbb{Q}$-algebra, the $\infty$-categories of connective differential graded $k$-algebras, simplicial commutative $k$-algebras, and connective $\mathcal{E}_{\infty}$-algebras over $H(k)$ are all equivalent. For a general commutative ring $k$, simplicial commutative rings provide a setting for importing notions of homotopy theory into algebraic geometry over $k$ (for example, for the aim of describing centers of $\infty$-categories of sheaves on schemes or stacks). Connective $\mathcal{E}_{\infty}$-ring spectra are more subtle and provide the setting for importing algebro-geometric notions back into stable homotopy theory. General $\mathcal{E}_{\infty}$-ring spectra provide a radical generalization which is at the heart of stable homotopy theory. (See [Sh] and references therein for the relation of algebra over rings and over the corresponding Eilernberg-MacLane spectra.)

The techniques and results of this paper apply equally in any of the three settings, and we will refer to any of the three as commutative derived rings without further comment.

Roughly speaking, a derived stack is a functor from the $\infty$-category of commutative derived rings to the $\infty$-category of topological spaces. It should satisfy a sheaf property with respect to a Grothendieck topology on commutative derived rings. Examples of derived stacks include:

1. (1)

   spaces (constant functors),

2. (2)

   ordinary schemes and stacks,

3. (3)

   the spectrum of a derived commutative ring (the corresponding representable functor),

4. (4)

   objects obtained by various gluings or quotients (colimits) and intersections or fiber products (limits) of the above examples.

We now recall the precise definitions of derived stacks, following [ToVe1, ToVe2, To2].

The opposite of the $\infty$-category $\mathcal{A}lg_{k}$ of derived commutative $k$-algebras admits a Grothendieck topology with respect to étale morphisms. For $A,B\in\mathcal{A}lg_{k}$, a morphism $A\to B$ is étale if the induced morphism on connected component $\pi_{0}(A)\to\pi_{0}(B)$ is étale, and for $i>0$, the induced map on higher homotopy groups is an isomorphism

A finite family of morphisms $\{f_{i}:A\to B_{i}\}$ is a étale covering if each $f_{i}$ is étale and the induced morphism is surjective

This induces a Grothendieck topology on the (opposite of the) homotopy category of $\mathcal{A}lg_{k}$.

Now a derived stack $X$ is a covariant functor from $\mathcal{A}lg_{k}$ to the $\infty$-category $\mathcal{T}op$ of topological spaces which is a sheaf with respect to the étale topology. In particular, for any étale cover $A\to B$, the induced morphism

is an equivalence, where $B^{*}$ is the standard cosimplicial resolution of $A$ with $i$th simplex $B^{\otimes_{A}i+1}$.

We will work exclusively with derived stacks whose diagonal morphism $\Delta:X\to X\times X$ is representable and affine. Given a quasi-compact derived stack $X$ with affine diagonal, we can choose a cover $U\to X$ by an affine derived scheme, and obtain a C̆ech simplicial affine derived scheme $U_{*}\to X$ with $k$-simplices given by the $k$-fold fiber product $U\times_{X}\cdots\times_{X}U$ and whose geometric realization is equivalent to $X$.