ScalingStacks

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.