ScalingStacks

We write ${\mathcal{T}}$ for the symmetric monoidal simplicial model category of simplicial sets and $\mathcal{S}$ for the symmetric monoidal simplicial model category of symmetric spectra [45]. Recall that a spectral category ${\mathcal{A}}$ is a category enriched in the category of symmetric spectra. Specifically, a spectral category is given by:

• •

  A class of objects $\mbox{obj}({\mathcal{A}})$,

• •

  for each pair of objects $(x,y)$ of ${\mathcal{A}}$, a symmetric spectrum ${\mathcal{A}}(x,y)$,

• •

  for each triple of objects $(x,y,z)$ of ${\mathcal{A}}$, a composition morphism in $\mathcal{S}$

  $${\mathcal{A}}(y,z)\wedge{\mathcal{A}}(x,y)\longrightarrow{\mathcal{A}}(x,z)\,,$$

  satisfying the usual associativity condition, and

• •

  for any object $x$ of ${\mathcal{A}}$, a morphism $\mathbb{S}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{A}}(x,x)$ in $\mathcal{S}$, satisfying the usual unit condition with respect to the above composition.

A spectral category is said to be small if its class of objects forms a set. We write $\Cat_{\mathcal{S}}$ the category of small spectral categories and spectral (enriched) functors. References on spectral categories are [10, §2], [69, Appendix A] and [74, §2].

We now briefly recall the Quillen model structure on spectral categories we work with in this paper. Given a spectral category ${\mathcal{A}}$, we can form a genuine category $[{\mathcal{A}}]$ by keeping the same set of objects and defining the set of morphisms between $x$ and $y$ in $[{\mathcal{A}}]$ to be the set of morphisms in the homotopy category $\Ho(\mathcal{S})$ from the sphere spectrum $\mathbb{S}$ to ${\mathcal{A}}(x,y)$. We obtain in this way a functor

with values in the category of small categories. Equivalently, we can think of $[-]$ as computed by passing to $\pi_{0}$ on the morphism spectra, and so we will also refer to $[{\mathcal{A}}]$ as the homotopy category $\Ho({\mathcal{A}})$.

Recall from [74, § 2] the natural adjunction

between spectral and simplicial categories, where $\Omega^{\infty}$ (also denoted $(-)_{0}$) is the space of maps from the unit (equivalently, restriction to the $0$-th space of the spectrum). We will use this adjunction to pass between spectral categories and $\infty$-categories. Using the model structure on simplicial categories of [6], the pair $(\Sigma^{\infty}_{+},\Omega^{\infty})$ is a Quillen adjunction.

For technical control, we require the following corollary which sharpens the description of the model structure, providing a combinatorial model category. (For references for Jeff Smith’s theory of combinatorial model categories, see [2] or [24].)

Let ${\mathcal{A}}$ be a (fixed) small spectral category and let ${\mathcal{A}}^{\op}$ denote the opposite spectral category, defined by ${\mathcal{A}}^{\op}(x,y)={\mathcal{A}}(y,x)$.

By [69, A.1.1], $\widehat{{\mathcal{A}}}$ can be given a combinatorial spectral model structure in which the weak equivalences are the pointwise stable equivalences and the fibrations are pointwise fibrations (referred to as the projective model structure). We will denote by $\widehat{{\mathcal{A}}}^{\cf}$ the full spectral subcategory of $\widehat{{\mathcal{A}}}$ on the cofibrant and fibrant ${\mathcal{A}}$-modules, and by ${\mathcal{D}}({\mathcal{A}})$ the derived category of ${\mathcal{A}}$, i.e., the homotopy category $\Ho(\widehat{{\mathcal{A}}})$ associated to the model structure. As usual, there is an equivalence $[\widehat{{\mathcal{A}}}^{\cf}]\simeq{\mathcal{D}}({\mathcal{A}})$.

Notice that we have a (fully faithful) spectral Yoneda embedding ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widehat{{\mathcal{A}}}$ which sends the object $z$ to the functor ${\mathcal{A}}(-,z)\colon{\mathcal{A}}^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{S}$ represented by $z$. Note that when ${\mathcal{A}}$ is fibrant, the Yoneda embedding lands in $\widehat{{\mathcal{A}}}^{\cf}$. By [69, §A.1], a spectral functor $F\colon{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ gives rise to a restriction/extension Quillen adjunction

and therefore a total left-derived functor $\mathbb{L}F_{!}\colon{\mathcal{D}}({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{D}}({\mathcal{B}})$.

We will be most interested in spectral categories ${\mathcal{A}}$ for which the homotopy category $\Ho({\mathcal{A}})$ has a triangulated structure compatible with the mapping spectra; we refer to [10, 4.4] for the definition of a pretriangulated spectral category, and highlight the essential consequence [10, 4.6] that the homotopy category of a pretriangulated spectral category is triangulated (with distinguished triangles given by the Puppe sequences). This is the stable homotopy theory analogue of the notion of a pretriangulated dg-category. A spectral functor between pretriangulated spectral categories is a DK-equivalence if and only if it induces an equivalent on homotopy categories [11, 5.7]. Using the Yoneda embedding, we can construct minimal pretriangulated categories containing the spectral category ${\mathcal{A}}$.

Given a spectral category ${\mathcal{A}}$, the proof of [10, 4.5] constructs a functorial “triangulated closure” $\widehat{{\mathcal{A}}}_{\tri}$ which is a pretriangulated spectral category. Briefly, $\widehat{{\mathcal{A}}}_{\tri}$ consists of the subcategory of cofibrant-fibrant objects in $\widehat{{\mathcal{A}}}$ which have the homotopy type of finite cell objects (in the projective model structure). Using retracts of finite cell objects instead [10, 4.5] produces a functorial “thick closure” $\widehat{{\mathcal{A}}}_{\perf}$, which is an idempotent-complete pretriangulated spectral category.

We start with a spectral functor $F\colon{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$, and tacitly assume we have performed a functorial fibrant replacement. We denote by $F_{!}^{\cf}\colon\widehat{{\mathcal{A}}}^{\cf}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widehat{{\mathcal{B}}}^{\cf}$ the composite of $F_{!}$ with a fibrant replacement (note that we do not need a cofibrant replacement here since $F_{!}$ preserves cofibrant objects) to obtain

Since this is a model of the derived functor of $F_{!}$ as a left Quillen functor, it preserves homotopy colimits and thus sends modules of the homotopy type of finite cell ${\mathcal{A}}$-modules to modules of the homotopy type of finite cell ${\mathcal{B}}$-modules and hence perfect ${\mathcal{A}}$-modules to perfect ${\mathcal{B}}$-modules. Therefore, the following definitions make sense.

We can relate these notions to definitions purely on the level of triangulated categories (the relationship between triangulated constructions and enriched constructions is discussed further in Section 5). For a spectral category ${\mathcal{A}}$, let ${\mathcal{D}}_{\tri}({\mathcal{A}})$ denote the smallest triangulated subcategory of ${\mathcal{D}}({\mathcal{A}})$ containing the image of the ${\mathcal{A}}$ under the Yoneda embedding, and ${\mathcal{D}}_{\perf}({\mathcal{A}})$ denote the smallest thick subcategory of ${\mathcal{D}}({\mathcal{A}})$ containing the image of ${\mathcal{A}}$ under the Yoneda embedding. Observe that ${\mathcal{D}}_{\tri}({\mathcal{A}})\simeq\Ho(\widehat{{\mathcal{A}}}_{\tri})$ and ${\mathcal{D}}_{\perf}({\mathcal{A}})\simeq\Ho(\widehat{{\mathcal{A}}}_{\perf})$. As a consequence, we obtain the following proposition.

Finally, note that we can use $\Omega^{\infty}$ to obtain simplicial models of the triangulated and thick closures. Define $\Mod({\mathcal{A}})$ to be the simplicial category $\Omega^{\infty}\widehat{{\mathcal{A}}}^{\cf}$, $\Mod({\mathcal{A}})_{\tri}$ to be the simplicial category $\Omega^{\infty}\widehat{{\mathcal{A}}}_{\tri}$, and $\Mod({\mathcal{A}})_{\perf}$ to be the simplicial category $\Omega^{\infty}\widehat{{\mathcal{A}}}_{\perf}$. Of course, it is also possible to give intrinsic definitions of the latter two categories in terms of $\Mod({\mathcal{A}})$.

Summarizing the relationships between the various categories, we have the following commutative diagram (with horizontal arrows induced by the Yoneda embedding and subsequent inclusions):

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.

Given any $\infty$-category ${\mathcal{C}}$ with finite limits, we can form the stabilization $\Stab({\mathcal{C}})$ [53, §1.4]. The $\infty$-category $\Stab({\mathcal{C}})$ is stable and comes equipped with a limit-preserving functor

If in addition ${\mathcal{C}}$ is presentable, then $\Omega^{\infty}$ admits a left adjoint

by [53, 1.4.4.4].

We now recall an explicit model of the stabilization of an $\infty$-category in terms of spectrum objects [53, §1.4.2]. Recall that a spectrum object of a pointed $\infty$-category $\mathcal{C}$ consists of a functor $\mathrm{N}({\mathbb{Z}}\times{\mathbb{Z}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{C}$. In particular, there are families of objects $A(i,j)$ of $\mathcal{C}$ and maps $A(i,j)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A(i+1,j)$, $A(i,j)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A(i,j+1)$ such that $A(i,j)$ is zero object whenever $i\neq j$ and the square

is cartesian for all $i$; consult [53, 1.4.2.4] for further details. Since the restriction of $A$ to the diagonal carries the nontrivial objects in $A$, we set $A_{i}=A(i,i)$ and often refer to $A$ simply by the collection of pointed objects $\{A_{i}\}$. We write $\mathrm{Sp}(\mathcal{C})$ for the $\infty$-category of spectrum objects in $\mathcal{C}$; $\mathrm{Sp}(\mathcal{C})$ comes equipped with a functor $\Omega^{\infty}\colon\mathrm{Sp}(\mathcal{C})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{C}$ which associates to the spectrum object $A$ its zero space $A_{0}=A(0,0)$. This is an explicit model for the stabilization $\Stab({\mathcal{C}})$ discussed previously. To ease notation, we will usually just write ${\mathcal{T}}_{\infty}\simeq\mathrm{N}(\mathcal{T}^{\cf})$ for the $\infty$-category of spaces and ${\mathcal{S}}_{\infty}\simeq\mathrm{Sp}({\mathcal{T}}_{\infty})\simeq\mathrm{N}({\mathcal{S}}^{\cf})$ for the $\infty$-category of spectra.

Now suppose that $\mathcal{C}$ is an arbitrary $\infty$-category. The Yoneda embedding $\mathcal{C}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}(\mathcal{C}^{\op},{\mathcal{T}}_{\infty})$ preserves finite limits (when they exist), so it induces a functor

on the level of spectrum objects, where $\mathcal{C}_{*}$ denotes the category of pointed objects in $\mathcal{C}$. Here the last equivalence follows from the fact that limits in functor categories are computed pointwise, and observe also that $\mathcal{C}_{*}$ will be empty unless $\mathcal{C}$ has a final object. On the other hand, if $\mathcal{C}$ is a stable $\infty$-category, then $\mathcal{C}\simeq\mathcal{C}_{*}$ and $\Omega^{\infty}\colon\mathrm{Sp}(\mathcal{C})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{C}$ is an equivalence with inverse $\Sigma^{\infty}\colon\mathcal{C}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Sp}(\mathcal{C})$ given by $(\Sigma^{\infty}a)_{i}=\Sigma^{i}a$ [53, 1.4.2.20]. This motivates the following definition:

Informally, the mapping spectrum is described by the formula

Note that this is a functor to the $\infty$-category of spectra; this is in contrast to the (point-set) mapping space functors from the category of quasicategories to the category of simplicial sets described in [52, 1.2.2] or [25].

We wish to characterize the image of ${\mathcal{C}}$ under the spectral Yoneda embedding:

When $\mathcal{C}$ is stable already, the following proposition gives an easy characterization of stably representable functors.

The categorical data which serves as the input to algebraic $K$-theory is typically obtained as the objects in a larger ambient category (with weak equivalences and extension sequences) that satisfy some sort of “smallness” condition; e.g., the perfect complexes as a subcategory of all complexes. A key insight initially codified by Thomason-Trobaugh [79] and subsequently elaborated upon by Neeman [61] is that this example is generic in algebraic $K$-theory, and the typical situation involves working with the compact objects in some model of a triangulated category, which is generated under homotopy colimits by those compact objects. Thus, we will systematically regard the small stable idempotent-complete $\infty$-categories that are the domain of the algebraic $K$-theory functor as arising as the compact objects in a larger category.

This notion of looking at large categories which are in some sense determined by the compact objects is axiomatized in category theory with the formalism of accessible and locally presentable categories, introduced by Makkai and Paré [54] and further developed by Adámek and Rosický [1]. This theory was integrated into homotopy theory in Jeff Smith’s theory of combinatorial model categories and developed further in this context in the seminal work of Dugger [24].

A version of this theory forms the basis for Lurie’s theory of presentable $\infty$-categories, which is the analogue in the $\infty$-category setting of the homotopy theories encoded by presentable combinatorial model category structures (see also Simpson’s related work in the context of Segal spaces [71]). We use this approach to handle the set-theoretic issues that arise in our work, along the lines described in [52, 1.2.15, 5.4.1]. As indicated in remark 2.6, it is also possible to handle some of the set-theoretic technicalities that arise (i.e., in the context of the Yoneda lemma) by explicit size bounds.

This framework is related to Grothendieck’s universe formalization, allowing us to handle small and large $\infty$-categories on similar grounds. In particular, [52, §5] has extensive discussion of the interaction of the Yoneda embeddings (which arise pervasively in this context) with set-theoretic concerns. In addition to Lurie’s work, the paper of Ben-Zvi, Francis, and Nadler [8] provides a nice exposé of this theory in the context of the study of geometric function theory from a perspective with its origin in Thomason-Trobaugh, and we refer the interested reader to sections 2 and 4.1 of that paper.

Roughly speaking, presentable $\infty$-categories are large $\infty$-categories that are generated under sufficiently large filtered colimits by some small $\infty$-category. To make this precise, we need to discuss the notion of the $\Ind$-category. Given any small $\infty$-category ${\mathcal{C}}$, we can form the $\infty$-category $\mathrm{Pre}({\mathcal{C}})$ of presheaves of simplicial sets on ${\mathcal{C}}$, which is the formal closure of ${\mathcal{C}}$ under colimits; that is, there is a fully faithful Yoneda embedding ${\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Pre}({\mathcal{C}})$, and $\mathrm{Pre}({\mathcal{C}})$ is generated by the image of ${\mathcal{C}}$ under small colimits [52, 5.1.5.8]. For any $\infty$-category ${\mathcal{C}}$ and infinite regular cardinal $\kappa$, we can form the $\Ind$-category $\Ind_{\kappa}({\mathcal{C}})$, which is the formal closure under $\kappa$-filtered colimits of ${\mathcal{C}}$ [52, §5.3.5]. The $\infty$-category $\Ind_{\kappa}({\mathcal{C}})$ is a full subcategory of $\mathrm{Pre}({\mathcal{C}})$, and the Yoneda embedding ${\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Pre}({\mathcal{C}})$ factors as ${\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind_{\kappa}({\mathcal{C}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Pre}({\mathcal{C}})$. We record here the following useful properties of the construction of the $\Ind$-category.

We now recall the following definitions [52, 5.4.2.1,5.5.1.1].

A morphism of presentable $\infty$-categories is a left adjoint functor; by the adjoint functor theorem [52, 5.5.2.9], a functor between presentable $\infty$-categories is a left adjoint if and only if it preserves colimits. We let ${\mathcal{P}\mathrm{r}}^{\mathrm{L}}$ denote the $\infty$-category of presentable $\infty$-categories and colimit-preserving functors; the $\infty$-category of colimit-preserving functors is denoted by $\mathrm{Fun}^{\mathrm{L}}(-,-)$. In fact, $\mathrm{Fun}^{\mathrm{L}}(-,-)$ is in fact itself a presentable $\infty$-category [52, 5.5.3.8], yielding an internal $\hom$ object for ${\mathcal{P}\mathrm{r}}^{\mathrm{L}}$.

We now restrict attention to the situation in which $\kappa=\omega$. Recall that an object $x$ of an $\infty$-category ${\mathcal{C}}$ is compact if the functor ${\mathcal{C}}^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{T}}$ represented by $x$ commutes with filtered colimits [52, §5.3.4]. Given an an $\infty$-category ${\mathcal{C}}$, let ${\mathcal{C}}^{\omega}$ denote the full subcategory of ${\mathcal{C}}$ consisting of the compact objects of ${\mathcal{C}}$. A presentable $\infty$-category ${\mathcal{C}}$ is compactly generated if the natural functor

which sends a filtered diagram in ${\mathcal{C}}^{\omega}$ to its colimit in ${\mathcal{C}}$, is an equivalence. There is a correspondence between small idempotent-complete $\infty$-categories and compactly generated $\infty$-categories given by the construction of the $\Ind$-category [52, §5.5.7]. More generally, the construction of the $\Ind$-category sets up a correspondence between the $\infty$-category of compactly-generated presentable $\infty$-categories with morphisms colimit-preserving functors that preserve compact objects and $\Cat_{\infty}$; the other direction is given by passage to compact objects [52, 5.5.7.10].

The preceding discussion carries over when we restrict attention to stable categories. In this setting, the stabilization $\Stab({\mathcal{C}})$ is initial amongst presentable stable $\infty$-categories admitting a functor from ${\mathcal{C}}$ [53, 1.4.5.5], in the sense that if ${\mathcal{D}}$ is a presentable stable $\infty$-category then $\Sigma^{\infty}_{+}$ induces an equivalence

The $\infty$-category of stable presentable $\infty$-categories ${{\mathcal{P}\mathrm{r}}^{\mathrm{L}}_{\mathrm{St}}}$ is a full subcategory of ${\mathcal{P}\mathrm{r}}^{\mathrm{L}}$, and the $\Ind$-category sets up a correspondence between $\Cat_{\infty}^{\perf}$ and compactly generated stable $\infty$-categories. We may also apply $\Ind$ to non-idempotent-complete stable $\infty$-categories to obtain a correspondence between $\Cat_{\infty}^{\ex}$ and compactly generated stable $\infty$-categories; however, these two $\infty$-categories are rather less closely related, as the full subcategory of compact objects is always idempotent-complete.

Given an $\infty$-category ${\mathcal{C}}$ and a suitable collection of morphisms $S$, one might hope to form the localization ${\mathcal{C}}[S^{-1}]$. This is by definition an $\infty$-category equipped with a functor $f\colon{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}[S^{-1}]$ which satisfies the following universal property: for any other $\infty$-category ${\mathcal{D}}$, restriction along $f$ identities

as the full subcategory of $\mathrm{Fun}({\mathcal{C}},{\mathcal{D}})$ spanned by those functors which send the morphisms in $S$ to equivalences in ${\mathcal{D}}$. If $S$ is a proper set, then ${\mathcal{C}}[S^{-1}]$ exists in the same universe as ${\mathcal{C}}$; indeed, without loss of generality we may assume that $S$ contains all degenerate edges of the simplicial set ${\mathcal{C}}$, in which case ${\mathcal{C}}[S^{-1}]$ may be constructed as a fibrant replacement of $({\mathcal{C}},S)$ in the model category of marked simplicial sets.

Often in practice, however, $S$ is not small, and the existence of ${\mathcal{C}}[S^{-1}]$ (without passing to a higher universe) requires more delicate analysis. One standard method is to show that ${\mathcal{C}}$ is presentable and $S$ is (generated by) a small set of arrows in a certain sense: this is the theory of Bousfield localization, following Bousfield’s seminal work on the subject [18]. In this case we may identify the localization as the full subcategory of ${\mathcal{C}}$ spanned by the $S$-local objects.

In model categories, there is a well-developed theory of Bousfield localization (e.g., Hirschhorn’s comprehensive discussion in  [42], Goerss and Jardine’s treatment in the simplicial setting [37], or the exposition of Smith’s theory for combinatorial model categories in  [2]). Because localization is a central technical device in our work, in this section we provide a brief review of Lurie’s version of localization in the setting of presentable $\infty$-categories from [52, §5.2.7] and [52, §5.5.4].

Specifically, we say that a colimit preserving functor $f\colon{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{D}}$ of presentable $\infty$-categories ${\mathcal{C}}$ and ${\mathcal{D}}$ is a Bousfield localization if the right adjoint of $f$ (which exists by the adjoint functor theorem) is fully faithful [52, 5.2.7.2]. When the context is clear, we tend to abuse notation and simply refer to this as a localization. A useful observation is that this data induces an equivalence between ${\mathcal{D}}$ and a full subcategory of ${\mathcal{C}}$, called the subcategory of local objects. In fact, [52, 5.2.7.4] gives a useful criterion for an endofunctor $L\colon{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ to be a localization. Specifically, the following are equivalent:

1. (i)

   There exists a functor $f\colon{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{D}}$ with a fully faithful right adjoint $g$ and an equivalence $g\circ f\simeq L$.

2. (ii)

   When regarded as a functor ${\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}L{\mathcal{C}}$, $L$ is the left adjoint of the inclusion $L{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$.

3. (iii)

   There exists a natural transformation $\alpha\colon\id_{{\mathcal{C}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}L$ such that for objects $X$ in ${\mathcal{C}}$, the morphisms $L(\alpha(X))$ and $\alpha(LX)$ are both equivalences.

Recall that a functor is accessible if it is $\kappa$-continuous (preserves $\kappa$-filtered colimits) for some sufficiently large regular cardinal $\kappa$ [52, 5.4.2.5]. A localization is accessible if $g$ or $L$ are accessible functors (equivalently, see [52, 5.5.1.2]) or the the essential image $L{\mathcal{C}}$ is an accessible subcategory.

Accessible localizations of presentable categories can be completely classified as follows. Recall from [52, 5.5.4] that associated to any set of arrows $S$ in a presentable $\infty$-category ${\mathcal{C}}$, the Bousfield localization $S^{-1}{\mathcal{C}}$ is equivalent to the ordinary localization ${\mathcal{C}}[T^{-1}]$ of ${\mathcal{C}}$ at the strongly saturated class $T$ generated by $S$ [52, 5.5.4.5]. In particular, many different sets $S$ can generate the same strongly saturated class $T$; they all define the same full subcategory $S^{-1}{\mathcal{C}}$ of ${\mathcal{C}}$ of $S$-local objects [52, 5.5.4.15], where $S$-local is defined in the standard fashion [52, 5.5.4.1]. An accessible localization of a presentable $\infty$-category is presentable. As the notation suggests, Bousfield localization is characterized by the following universal property [52, 5.5.4.20]: for any other presentable $\infty$-category ${\mathcal{D}}$, composition with $L$ induces a functor

which is fully faithful and whose essential image consists of those colimit-preserving functors which take elements of $S$ to equivalences.