ScalingStacks

A.10 Enriched \(\infty\)-categories[00J8]

Our work makes crucial use of the theory of enriched \(\infty\)-categories of [GH15], which we briefly review here. Given a monoidal \(\infty\)-category \(\mathbb V\), we write \(\mathrm{Cat}[\mathbb V]\) for the (large) \(\infty\)-category of (small) \(\mathbb V\)-enriched \(\infty\)-categories. Similarly, let \(\widehat{\mathrm{Cat}}[\mathbb V]\) be the (huge) \(\infty\)-category of \(\mathbb V\)-enriched \(\infty\)-categories with large spaces of objects.

We note from the outset that this formalism enjoys a convenient univalence property: the equivalences in \(\mathrm{Cat}[\mathbb V]\) are precisely the (enrichedly) fully faithful and surjective functors. This may be contrasted with the classical notion of an “equivalence of categories”, which is not generally an isomorphism in the ordinary category of ordinary categories since it is not generally an isomorphism on objects. Of course, achieving this univalence requires an additional step, which is itself the imposition of a univalence condition.⁶²

Given a monoidal \(\infty\)-category \(\mathbb V\), a categorical \(\mathbb V\)-algebra \(\mathcal C\) with space of objects \(X \in \mathcal S\) heuristically consists of a functor \(X^{\times 2} \xrightarrow{\underline{\mathrm{Hom}}_\mathcal C(-,-)} \mathbb V\) specifying hom-objects as well as an associative and unital composition operation. These assemble into an \(\infty\)-category \(\mathrm{Alg}_{\mathrm{Cat}}[\mathbb V]\). Given a categorical \(\mathbb V\)-algebra \(\mathcal C\in \mathrm{Alg}_{\mathrm{Cat}}[\mathbb V]\) we generally write \(\iota_0 \mathcal C\in \mathcal S\) for its space of objects, and the functor \(\mathrm{Alg}_{\mathrm{Cat}}[\mathbb V] \xrightarrow{\iota_0} \mathcal S\) is a Cartesian fibration (with Cartesian monodromy functors given by pulling back the hom-objects along a map of spaces). If \(\mathbb V\) is in fact symmetric monoidal, then \(\mathrm{Alg}_{\mathrm{Cat}}[\mathbb V]\) admits a symmetric monoidal structure as well [GH15, Cor. 5.7.12], with \(\iota_0 (\mathcal C\otimes \mathcal D) \simeq (\iota_0 \mathcal C) \times (\iota_0 \mathcal D)\) and \(\underline{\mathrm{Hom}}_{\mathcal C\otimes \mathcal D}((c,d) , (c',d')) \simeq \underline{\mathrm{Hom}}_\mathcal C(c,c') \otimes \underline{\mathrm{Hom}}_\mathcal D(d,d')\).

Now, given a categorical \(\mathbb V\)-algebra \(\mathcal C\in \mathrm{Alg}_{\mathrm{Cat}}[\mathbb V]\) we can extract a space \(\mathcal C^\simeq \in \mathcal S\) of equivalences (with respect to its internal category theory), and this comes equipped with a morphism \(\iota_0 \mathcal C\rightarrow\mathcal C^\simeq\) from its space of objects (which heuristically sends each object to its identity morphism). A morphism \(\mathcal C\rightarrow\mathcal D\) in \(\mathrm{Alg}_{\mathrm{Cat}}[\mathbb V]\) is surjective on objects if the induced map \(\mathcal C^{\simeq} \rightarrow\mathcal D^{\simeq}\) is surjective (i.e. surjective on \(\pi_0\)). A morphism \(F \colon \mathcal C\rightarrow\mathcal D\) in \(\mathrm{Alg}_{\mathrm{Cat}}[\mathbb V]\) is fully faithful if for any two objects \(c, d \in \mathcal C\), the induced map \(\underline{\mathrm{Hom}}_{\mathcal C}(c,d) \rightarrow\underline{\mathrm{Hom}}_{\mathcal D}(Fc, Fd)\) in \(\mathbb V\) is an isomorphism in \(\mathcal D\).

We say that \(\mathcal C\) is univalent if the morphism \(\iota_0 \mathcal C\rightarrow\mathcal C^\simeq\) is an equivalence; in essence, this is the condition that its internally- and externally-defined spaces of objects coincide. Finally, a \(\mathbb V\)-enriched \(\infty\)-category is a univalent categorical \(\mathbb V\)-algebra. These define a full subcategory \(\mathrm{Cat}[\mathbb V] \subseteq \mathrm{Alg}_{\mathrm{Cat}}[\mathbb V]\). Furthermore, the inclusion has a left adjoint (i.e. a reflective localization), which we may refer to as univalent completion, which exhibits \(\mathrm{Cat}[\mathbb V]\) as the localization of \(\mathrm{Alg}_{\mathrm{Cat}}[\mathbb V]\) with respect to fully faithful and essentially surjective functors [GH15, Thm. 2.4.11].

If we assume that \(\mathbb V\) is symmetric monoidal, then the reflective localization is compatible with the symmetric monoidal structure on \(\mathrm{Alg}_{\mathrm{Cat}}[\mathbb V]\) in the sense of Subsection A.8.9. It follows that \(\mathrm{Cat}[\mathbb V]\) also inherits a symmetric monoidal structure, given by taking the tensor product in categorical \(\mathbb V\)-algebras and then univalently completing the result.⁶³ By the same argument, \(\widehat{\mathrm{Cat}}_{\infty}[\mathbb V]\) also inherits a symmetric monoidal structure.

If \(\mathbb V\) is presentably monoidal, then both \(\infty\)-categories \(\mathrm{Cat}[\mathbb V]\) and \(\mathrm{Alg}_{\mathrm{Cat}}[\mathbb V]\) are presentable [GH15, Prop. 5.7.8]. If \(\mathbb V\) is furthermore presentably symmetric monoidal, then so is \(\mathrm{Cat}[\mathbb V]\) [GH15, Prop. 5.7.16]. This assembles into a functor \(\mathrm{Cat}[-] \colon \mathrm{CAlg}(\mathrm{Pr}^\mathrm{L}) \rightarrow\mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\).

If \(\mathbb V\) is presentably monoidal, there also exists a categorical suspension functor \(\mathbb V\xrightarrow{\Sigma[-]} \mathrm{Alg}_{\mathrm{Cat}}[\mathbb V]\) [GH15, Def. 4.3.21], which is characterized by the universal property that morphisms \(\Sigma[V] \rightarrow\mathcal C\) are equivalent to a pair of objects \(c,d \in \mathcal C\) and a morphism \(V \rightarrow\underline{\mathrm{Hom}}_\mathcal C(c,d)\) in \(\mathbb V\).⁶⁴ We also simply write \(\Sigma[-]\) for the composite \(\mathbb V\rightarrow\mathrm{Alg}_{\mathrm{Cat}}[\mathbb V] \rightarrow\mathrm{Cat}[\mathbb V]\), which has the same universal property in \(\mathrm{Cat}[\mathbb V]\).