ScalingStacks

A.1 From \(n\)-categories to \((\infty,n)\)-categories[00I9]

As a starting point, let us recall the distinction between strict and weak 2-categories [Bén+67]. In a strict 2-category, one requires associativity and unitality of composition to hold up to equality. By contrast, in a weak 2-category, one only requires these to hold up to natural isomorphism. Moreover, these natural isomorphisms are then required to satisfy further coherence conditions.⁴⁰ We can summarize the situation with the slogan that a weak 2-category is “a category that is enriched in 1-categories up to coherent natural isomorphism”. More generally, one would like to define a weak \(n\)-category as “a category that is enriched in weak \((n-1)\)-categories up to coherent natural isomorphism”. However, making this notion rigorous for higher values of \(n\) – with all of the desired coherence conditions – becomes increasingly infeasible as \(n\) grows [GPS95].

Homotopy theory provides a remarkable alternative perspective on this problem, which leads to a uniform and robust solution. To explain it, let us recall Grothendieck’s homotopy hypothesis: any appropriate definition of “weak \(n\)-category” should have that its weak \(n\)-groupoids are equivalent (in a suitably homotopical sense) to homotopy \(n\)-types (i.e. topological spaces with homotopy groups above dimension \(n\) all vanishing, taken up to weak homotopy equivalence).⁴¹ Note that under the homotopy hypothesis, natural isomorphisms on the categorical side correspond to homotopies on the topological side. Hence, we arrive at an alternative proposed definition for “weak \((n,1)\)-categories”,⁴² namely as categories that are enriched either in weak \((n-1)\)-groupoids up to coherent natural isomorphism or in homotopy \(n\)-types up to coherent homotopy. In the limit, we find that “weak \((\infty,1)\)-categories” should be categories that are enriched in (arbitrary) homotopy types up to coherent homotopy.

Before continuing our discussion, we pause to note a few conventions. First of all, just as we may refer to 1-categories simply as “categories”, we will also refer to \((\infty,1)\)-categories simply as “\(\infty\)-categories”. Moreover, given that we will only be interested in “weak” notions, we usually leave this term implicit henceforth.

Now, there exist a number of robust models for \(\infty\)-categories (i.e. categories enriched in homotopy types up to coherent homotopy), although all are known to be equivalent (in a suitably homotopical sense) [Toë05]. The most developed is that of quasicategories, thanks to Lurie’s foundational work [Lur09], which we take as a primary reference. However, we stress that throughout this paper we work in an entirely model-independent fashion: we only manipulate \(\infty\)-categories in a manner that makes no reference to a specific model (so e.g. we never make reference to the individual simplices of a quasicategory).

The theory of \(\infty\)-categories reifies the limiting case of Grothendieck’s homotopy hypothesis: among \(\infty\)-categories, the \(\infty\)-groupoids are equivalent to homotopy types. We refer to such objects alternately as \(\infty\)-groupoids or as spaces, depending on the context. We write \(\mathrm{Cat}_\infty\) for the \(\infty\)-category of (small) \(\infty\)-categories (see Subsection A.6 for a brief discussion of set-theoretic matters), and we write \(\mathcal S\subset \mathrm{Cat}_\infty\) for the full subcategory of spaces.

We can now return to the problem of defining weak \(n\)-categories. The essential observation is as follows: all of the desired coherence conditions articulate equivalences between various composite operations. Thus, in order to obtain a robust theory of \((\infty,n)\)-categories, it suffices to have a robust theory of \(\infty\)-categories enriched in a given one: then, we can recursively define \((\infty,n)\)-categories to be \(\infty\)-categories that are enriched in the \((\infty,1)\)-category of \((\infty,n-1)\)-categories. As we explain further in Subsection A.10, such a robust formalism is provided by [GH15]. Hence, writing \(\mathrm{Cat}[\mathbb V]\) for the \(\infty\)-category of \(\mathbb V\)-enriched \(\infty\)-categories, we may recursively define the \((\infty,1)\)-category of \((\infty,n)\)-categories as \(\mathrm{Cat}_{(\infty, {n})} \coloneqq \mathrm{Cat}[\mathrm{Cat}_{(\infty, {n-1})}]\), the \(\infty\)-category of \(\infty\)-categories enriched in (small) \((\infty,n-1)\)-categories; as a base case we define \(\mathrm{Cat}_{(\infty, {0})} \coloneqq \mathcal S\), and as a consistency check we have an equivalence \(\mathrm{Cat}_{(\infty, {1})} \coloneqq \mathrm{Cat}[\mathrm{Cat}_{(\infty, {0})}] \simeq \mathrm{Cat}_\infty\) [GH15, Thm. 5.4.6]. As explained in Subsection A.10, for \(k \geq 0\), \(\mathrm{Cat}_{(\infty, {k})}\) is Cartesian presentably symmetric monoidal. Among the \((\infty,n)\)-categories, weak \((n,n)\)-categories can then be defined simply as those satisfying certain discreteness conditions [GH15, § 6.1].

In fact, this definition ultimately affords an \((\infty,n+1)\)-category (as opposed to just an \((\infty,1)\)-category) of \((\infty,n)\)-categories, using the fact that \(\mathrm{Cat}_{(\infty, {n})}\) is Cartesian closed [Rez10]: for any \((\infty,n)\)-categories \(\mathcal C\) and \(\mathcal D\) we have an \((\infty,n)\)-category \(\mathrm{Fun}(\mathcal C,\mathcal D)\) of functors between them, which is uniquely characterized by the universal property that we have a natural equivalence \[\mathrm{Hom}_{\mathrm{Cat}_{(\infty, {n})}}(\mathcal E, \mathrm{Fun}(\mathcal C,\mathcal D)) \simeq \mathrm{Hom}_{\mathrm{Cat}_{(\infty, {n})}}(\mathcal E\times \mathcal C, \mathcal D)\] of hom-spaces for any \((\infty,n)\)-category \(\mathcal E\in \mathrm{Cat}_{(\infty, {n})}\).