ScalingStacks

1.4.2 Higher-categorical notions of faithfulness and homotopy categories[000B]

To prove our main theorems, we develop machinery to reduce the construction of algebraic structures on \((\infty,2)\)-categories to corresponding structures on their underlying ordinary homotopy categories, as well as higher-dimensional variants. We outline some of these results, which might be of independent interest.

As a toy case, observe that given an \(\infty\)-category \(\mathcal C\), a full subcategory \(\mathcal C'\) is determined entirely by the subset \(h_0\mathcal C'\) of the set \(h_0\mathcal C\) of equivalences classes of objects in \(\mathcal C\) that are contained in \(\mathcal C'\). Furthermore, if \(\mathcal C\) is endowed with some multiplicative structure (e.g. monoidal, braided monoidal, or symmetric monoidal), then \(\mathcal C'\) inherits such a structure if and only if \(h_0\mathcal C'\) inherits the resulting structure from \(h_0\mathcal C\).

Given an \((\infty,2)\)-category \(\mathcal C\), we write \(h_1\mathcal C\) for the ordinary category obtained by first discarding its noninvertible 2-morphisms and then passing to the homotopy category, i.e. taking \(\pi_0\) of its hom-spaces. Moreover, we say that a functor between \((\infty,2)\)-categories is faithful if it is fully faithful on hom-\((\infty,1)\)-categories.⁷ Then, analogously to the above situation, we show that a faithful functor \(\mathcal C' \rightarrow\mathcal C\) is determined entirely by their corresponding faithful functor \(h_1\mathcal C' \rightarrow h_1\mathcal C\). Moreover, we show that if \(\mathcal C\) is braided monoidal equipped with the faithful functor \(\mathcal C' \rightarrow\mathcal C\), then endowing \(\mathcal C'\) together with a (compatible) braided monoidal structure is equivalent to endowing \(h_1\mathcal C'\) and \(h_1\mathcal C' \rightarrow h_1\mathcal C\) with such a structure, see corollary 5.5.5. The task of defining an \(\mathbb E_2\)-algebra structure on \(\mathcal C'\) therefore reduces in such a situation to the task of defining a braiding (in the classical sense!) on its homotopy 1-category \(h_1\mathcal C'\).

In fact, we prove the above results in much broader generality: In section 5, we study the notion of \(n\)-faithfulness for functors between \((\infty,k)\)-categories, and show in subsection 5.3 that they define the right classes in factorization systems on \(\mathrm{Cat}_{(\infty,k)}\), whose corresponding left classes are given by the \(n\)-surjective functors, which are surjective on objects, and on parallel morphisms up to level \((n+1)\), see definition 5.3.1. Using the iterative definition of higher categories, \(\mathrm{Cat}_{(\infty,k)} \coloneqq \mathrm{Cat}[\mathrm{Cat}_{(\infty,k-1)}]\), we deduce this from general results that we prove regarding factorization systems on enriched \(\infty\)-categories in subsection B.4, also see [Hau23] for related recent result. Generalizing the above, we establish in corollary 5.5.3 that for any \(n\) and any \(k\), \((n-1)\)-faithful functors to an \((\infty,k)\)-category \(\mathcal C\) are equivalently determined by \((n-1)\)-faithful functors to its homotopy \(n\)-category.