ScalingStacks

The basic setting for our work is the theory of $\infty$-categories (and particularly stable $\infty$-categories), which provide a tractable way to handle a “homotopical category of homotopical categories” as well as homotopically meaningful categories of homotopical functors. There are now many competing models of $\infty$-categories, including Rezk’s complete Segal spaces [77], the Segal categories [43, 78] of Simpson and Tamsamani, the quasicategories (weak Kan complexes) of Boardman and Vogt, the homotopy theory of simplicial categories as studied by Dwyer-Kan and Bergner  [28, 6], and others, all of which are known to be equivalent (see [7] for a nice discussion of the situation). In a sense the situation is analogous to the situation with the varied modern categories of spectra (e.g., symmetric spectra, orthogonal spectra, EKMM $S$-modules). None of the work of this paper depends in any way on particular properties of the model of $\infty$-categories chosen; given certain basic structural properties, one could carry out our arguments in any of them.

We have chosen to work in this paper with the theory of quasicategories. These first appeared in the work of Boardman and Vogt, where they were referred to as weak Kan complexes [15]. The theory was subsequently developed by Joyal [46] and then extensively studied by Lurie. In this section we give a rapid review of the relevant background on the theory of quasicategories as a model of $\infty$-categories. Our basic references for this material are Lurie’s books [52, 53].

We will write $\Cat_{\infty}$ to denote the $\infty$-category of small $\infty$-categories and functors, which we explicitly model as the category of simplicial sets with the Joyal model structure [46]. There is a simplicial nerve functor $\mathrm{N}$ from simplicial categories to simplicial sets which is the right Quillen functor of a Quillen equivalence [52, §1.1.5.5, 1.1.5.13, 2.2.5.1]

Here the model structure on the top is Bergner-Dwyer-Kan’s model structure on simplicial categories [6] and the model structure on the bottom is Joyal’s model structure on simplicial sets.

There are a number of options for producing the “underlying” $\infty$-category of a category equipped with a notion of “weak equivalence”. The most structured setting is that of a simplicial model category ${\mathcal{C}}$, where the $\infty$-category can be obtained by restricting to the full simplicial subcategory ${\mathcal{C}}^{\cf}$ of cofibrant-fibrant objects and then applying the simplicial nerve functor $\mathrm{N}$. More generally, if ${\mathcal{C}}$ is a category equipped with a subcategory of weak equivalences $w{\mathcal{C}}$, the Dwyer-Kan simplicial localization $L{\mathcal{C}}$ [28] provides a corresponding simplicial category, and then $\mathrm{N}((L{\mathcal{C}})^{\textrm{fib}})$, where $(-)^{\textrm{fib}}$ denotes fibrant replacement in simplicial categories, yields an associated $\infty$-category. Barwick and Kan [4] have studied this procedure in the context of Segal spaces and Lurie has given a version of this approach in [53, §1.3.3]: we associate to a (not necessarily simplicial) category ${\mathcal{C}}$ with weak equivalences $W$ an $\infty$-category $\mathrm{N}({\mathcal{C}})[W^{-1}]$; when ${\mathcal{C}}$ is a model category, for functoriality reasons it is usually convenient to restrict to the cofibrant objects ${\mathcal{C}}^{\mathrm{c}}$ and consider $\mathrm{N}({\mathcal{C}}^{\mathrm{c}})[W^{-1}]$.

All of these constructions produce equivalent $\infty$-categories [53, 1.3.7]. Furthermore, all of them are functorial. Although a simplicial left Quillen functor ${\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{D}}$ does not typically induce a functor between the subcategories of cofibrant-fibrant objects in ${\mathcal{C}}$ and ${\mathcal{D}}$ respectively, composing with a fibrant replacement functor as in definition 2.7, does yield an induced functor on simplicial nerves. Furthermore, given a simplicial Quillen adjunction $(F,G)$, there is an induced adjunction of functors on the level of $\infty$-categories by [52, 5.2.4.6]. (Alternatively, it can be seen directly that a functor ${\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{D}}$ which preserves weak equivalences between cofibrant objects induces a functor $\mathrm{N}({\mathcal{C}})[W^{-1}]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{N}({\mathcal{D}})[W^{-1}]$.)

Given an $\infty$-category ${\mathcal{C}}$, we can form its homotopy category $\Ho({\mathcal{C}})$, which is an ordinary category [52, §1.2.3]. In addition, given an $\infty$-category ${\mathcal{C}}$, there is a maximal $\infty$-groupoid (Kan complex) ${\mathcal{C}}_{\mathrm{iso}}$ inside of ${\mathcal{C}}$, obtained by restricting to the subcategory of ${\mathcal{C}}$ consisting of those arrows which become isomorphisms in the homotopy category $\Ho({\mathcal{C}})$. The functor which associates to the $\infty$-category ${\mathcal{C}}$ its maximal subgroupoid ${\mathcal{C}}_{\mathrm{iso}}$ is right adjoint to the inclusion of $\infty$-groupoids into $\infty$-categories. We have the following proposition relating this to other, possibly more familiar, notions (see also [81, 2.3]).

The $\infty$-category of functors between two $\infty$-categories ${\mathcal{C}}$ and ${\mathcal{D}}$ is denoted $\mathrm{Fun}({\mathcal{C}},{\mathcal{D}})$. As a point-set object, in this setting $\mathrm{Fun}({\mathcal{C}},{\mathcal{D}})$ is the simplicial set of maps between the quasicategories ${\mathcal{C}}$ and ${\mathcal{D}}$, which is itself a quasicategory. Note that the space of functors from ${\mathcal{C}}$ to ${\mathcal{D}}$ is precisely the maximal subgroupoid $\mathrm{Fun}({\mathcal{C}},{\mathcal{D}})_{\mathrm{iso}}$ of $\mathrm{Fun}({\mathcal{C}},{\mathcal{D}})$ [52, 1.2.5.3, 3.0.0.1].

For a small stable $\infty$-category ${\mathcal{C}}$, the homotopy category $\Ho({\mathcal{C}})$ is triangulated, with the exact triangles determined by the cofiber sequences in ${\mathcal{C}}$ [53, 1.1.2.13].

Recall that an $\infty$-category ${\mathcal{C}}$ is idempotent-complete if the image of ${\mathcal{C}}$ under the Yoneda embedding ${\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Pre}({\mathcal{C}})$ is closed under retracts (see also [52, §4.4.5]); here $\mathrm{Pre}({\mathcal{C}})$ denotes the $\infty$-category $\mathrm{Fun}({\mathcal{C}}^{\op},\mathrm{N}({\mathcal{T}}^{\cf}))$ of presheaves of spaces on ${\mathcal{C}}$. Let $\Cat_{\infty}^{\perf}$ denote the $\infty$-category of small idempotent-complete stable $\infty$-categories. There is an idempotent completion functor given as the left adjoint to the inclusion $\Cat_{\infty}^{\perf}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\infty}^{\ex}$ [52, 5.1.4.2], which we denote by $\Idem$.

We will verify shortly that this notion of Morita equivalence is compatible with the definition given in terms of spectral categories in definition 2.7.