ScalingStacks

The definitions and results of this paper depend in an essential way on noncommutative and commutative algebra for $\infty$-categories, as developed by Lurie in [L4] and [L5]. We briefly summarize this theory in this section, giving detailed references for the benefit of the reader.

The definition of a monoidal $\infty$-category is given in [L4, 1.1]. The homotopy category of a monoidal $\infty$-category is an ordinary monoidal category. The $\infty$-categorical notion incorporates not only the naive notion of multiplication and unit on an $\infty$-category $\mathcal{C}$ but also all of the higher coherences for associativity, which are packaged in the data of a fibration over $\Delta^{\rm op}$ whose fiber over $[n]$ is $\mathcal{C}^{\times n}$. (Alternatively, it can be captured concretely by a bisimplicial set with compatibilities, or as a monoid object in $\mathcal{C}$ [L4, Remark 1.2.15], that is, a simplicial object in $\mathcal{C}$ mimicking the classifying space of a monoid.) An algebra object in $\mathcal{C}$ can then be defined [L4, 1.1.14] as an appropriate section of this fibration. The $\infty$-categorical notion of algebra reduces to the more familiar notions of $A_{\infty}$ $k$-algebra or $A_{\infty}$-ring spectrum when the ambient monoidal $\infty$-category is that of differential graded $k$-modules or that of spectra [L4, 4.3]. In other words, it encodes a multiplication associative up to coherent homotopies. Likewise, (left) modules over an algebra are defined by the same simplicial diagrams as algebras, except with an additional marked vertex at which we place the module [L4, 2.1]. There is also a pairing to $\mathcal{C}$ between left and right modules over an algebra object $A$ in $\mathcal{C}$, namely, the relative tensor product $\cdot\otimes_{A}\cdot$ defined by the two-sided bar construction [L4, 4.5]. Monoidal $\infty$-categories, algebra objects in a monoidal category, and module objects over an algebra object themselves form $\infty$-categories, some of whose properties (in particular behavior of limits and colimits) are worked out in [L4, 1,2]. In particular, limits of algebra objects are calculated on the underlying objects [L4, 1.5] and module categories are stable [L4, Proposition 4.4.3].

The definition of a symmetric monoidal $\infty$-category is given in [L5, 1], modeled on the Segal machine for infinite loop spaces. Namely, we replace $\Delta^{\rm op}$ in the definition of monoidal $\infty$-categories by the category of pointed finite sets, thus encoding all the higher compatibilities of commutativity. Likewise, commutative algebra objects are defined as suitable sections of the defining fibration. On the level of homotopy categories, we recover the notion of commutative algebra object in a symmetric monoidal category, but on the level of chain complexes or spectra this notion generalizes the notion of $\mathcal{E}_{\infty}$-algebra or $\mathcal{E}_{\infty}$-ring spectrum (as developed in [EKMM, HSS]). An important feature of the $\infty$-category of commutative algebra objects is that coproducts of commutative algebra objects are calculated by the underlying monoidal structure [L5, Proposition 4.7]. Section [L5, 5] introduces commutative modules over commutative algebra objects $A$, which are identified with both left and right modules over the underlying algebras. The key feature of the $\infty$-category of $A$-modules is that it has a canonical symmetric monoidal structure [L5, Proposition 5.7], extending the relative tensor product of modules. Moreover, commutative algebras for this structure are simply commutative algebras over $A$ [L5, Proposition 5.9].

One of the key developments of [L4] is the $\infty$-categorical version of tensor products of abelian categories [De]. Namely, in [L4, 4.1] it is shown that the $\infty$-category ${\mathcal{P}r}^{\rm L}$ of presentable $\infty$-categories has a natural monoidal structure. In this structure, the tensor product $\mathcal{C}\otimes\mathcal{D}$ of presentable $\mathcal{C},\mathcal{D}$ is a recipient of a universal functor from the Cartesian product $\mathcal{C}\times\mathcal{D}$ which is “bilinear” (commutes with colimits in each variable separately). Moreover, [L5, Proposition 6.18] lifts this to a symmetric monoidal structure in which the unit object is the $\infty$-category of spaces. This structure is in fact closed, in the sense that ${\mathcal{P}r}^{\rm L}$ has an internal hom functor compatible with the tensor structure, see [L2, Remark 5.5.3.9] and [L4, Remark 4.1.6]. The internal hom assigns to presentable $\infty$-categories $\mathcal{C}$ and $\mathcal{D}$ the $\infty$-category of colimit-preserving functors $\Fun^{\rm L}(\mathcal{C},\mathcal{D})$, which is presentable by [L2, Proposition 5.5.3.8]. In Section 5.1 below, we use the monoidal structure on ${\mathcal{P}r}^{\rm L}$ to define an analogue for $\infty$-categories of the Hochschild cohomology of algebras or topological Hochschild cohomology of ring spectra.

The symmetric monoidal structure on the $\infty$-category $\mathcal{P}r^{\rm L}$ of presentable $\infty$-categories restricts to one on the full $\infty$-subcategory of stable presentable $\infty$-categories ([L4, 4.2] and [L5, 6.22]). The unit of the restricted monoidal structure is the stable category of spectra. In particular, this induces a symmetric monoidal structure on spectra and exhibits presentable stable categories as tensored over spectra. Thus if $\mathcal{C}$ is a symmetric monoidal stable $\infty$-category (that is, a stable commutative ring object in stable $\infty$-categories), we may consider module categories over $\mathcal{C}$. These modules themselves will form a symmetric monoidal $\infty$-category under the operation $\otimes_{\mathcal{C}}$, which is characterized by the two-sided bar construction with respect to $\mathcal{C}$. In our applications, $\mathcal{C}$ will be the $\infty$-category $\qc(Y)$ of quasi-coherent sheaves on a derived stack $Y$, and we will consider the tensor products of module categories of the form $\qc(X)$ for derived stacks $X\to Y$.

In Section 5.1, we use this general formalism to define the center (or Hochschild cohomology) and universal trace (or Hochschild homology) for algebra objects in any symmetric monoidal $\infty$-category. The case of spectra recovers topological Hochschild (co)homology, while the case of presentable $\infty$-categories ${\mathcal{P}r}^{\rm L}$ provides a derived generalization of the Drinfeld center and will be the focus of our applications in Section 5.

In Section 4.1, we discuss basic properties of $\infty$-categories of modules, and the tensor product of small stable $\infty$-categories.