ScalingStacks

A.2 Some basic notions in \(\infty\)-category theory[00IA]

Here we highlight a few \(\infty\)-categorical notions that we use repeatedly throughout this paper. We make no effort to give a comprehensive account, and instead refer the interested reader to [Lur09] for a more thorough treatment. Indeed, a remarkable number of notions in ordinary category theory port over to \(\infty\)-category theory with minimal modification (though see Subsection A.3 for a prominent non-example, and see Subsection A.2.6 for another non-example).

A.2.1 Basic notions[00IB]

Broadly speaking, the fundamental role played by sets in ordinary category theory is played by spaces in \(\infty\)-category theory. In particular, as noted in Subsection A.1, an \(\infty\)-category \(\mathcal C\) is enriched in spaces (i.e. \(\infty\)-groupoids): for any pair of objects \(c,d \in \mathcal C\) we obtain a space \(\mathrm{Hom}_\mathcal C(c,d) \in \mathcal S\). These hom-spaces admit a composition law, which is associative and unital up to coherent homotopy.

Any \(\infty\)-category has an associated ordinary \(1\)-category \(h_1\mathcal C\), called its homotopy category, with the same objects as \(\mathcal C\) and hom-sets \(\mathrm{Hom}_{h_1 \mathcal C}(c,d) \coloneqq \pi_0 \mathrm{Hom}_{\mathcal C}(c,d)\), i.e. identifying \(1\)-morphisms in \(\mathcal C\) if there is an invertible \(2\)-morphism between them.

A presheaf on an \(\infty\)-category \(\mathcal C\) is a functor \(\mathcal C^\mathrm{op}\rightarrow\mathcal S\). These assemble into the \(\infty\)-category \(\mathcal P(\mathcal C) \coloneqq \mathrm{Fun}(\mathcal C^\mathrm{op},\mathcal S)\), which receives a fully faithful Yoneda embedding \(\mathcal C\xrightarrow{\mathrm{Hom}_\mathcal C(=,-)} \mathcal P(\mathcal C)\) [Lur09, Prop. 5.1.3.1].

As a matter of terminology, we interchangeably use the terms “isomorphism”, as in ordinary category theory and “equivalence” (in order to emphasize that one is working in a higher-categorical context).

In ordinary categories, objects characterized by universal properties (e.g. limits and colimits) are unique up to unique isomorphism when they exist: said differently, the collection of objects satisfying the characterization assemble into an empty or contractible groupoid. In \(\infty\)-categories, objects characterized by a universal property instead assemble into an empty or contractible \(\infty\)-groupoid. For instance, an object \(c \in \mathcal C\) is called initial if for every \(d \in \mathcal C\) the space \(\mathrm{Hom}_\mathcal C(c,d)\) is contractible, and the initial objects of \(\mathcal C\) assemble into an empty or contractible \(\infty\)-groupoid.

In classical category theory, the term “unique up to unique isomorphism” is sometimes replaced by the shorter term “essentially unique”. The word “essentially” here is meant to indicate that object is not literally unique (e.g. there exist many terminal objects in the category \(\mathrm{Set}\) of sets (namely the singletons)), but rather that it is unique in the appropriate category-theoretic sense. However, in \(\infty\)-category theory one is emphatically never interested in uniqueness beyond that in the \(\infty\)-categorical sense (i.e. parametrized by a contractible \(\infty\)-groupoid), and so we generally omit all technical uses of the word “essentially”. Relatedly, we will refer to a functor \(\mathcal C\xrightarrow{F} \mathcal D\) simply as surjective (rather than “essentially surjective”) if for every object \(d \in \mathcal D\) there exists an object \(c \in \mathcal C\) and an equivalence \(F(c) \simeq d\).

A.2.2 Monomorphisms and subcategories[00IC]

A general pattern in higher category theory is that one must keep track of “higher coherence data” (see e.g. Subsection A.3). Thus, it is notable when a given construction does not require this. Given a construction that a priori might involve coherence data, we say that the data is in fact (merely) a condition in order to indicate that such data is unique if it exists (i.e. that the \(\infty\)-category of such assembles into an empty or contractible \(\infty\)-groupoid).

Most fundamentally, given a space \(X\) and a subset of its path components, it is merely a condition for a point \(x \in X\) to lie in one of these. In fact, the inclusions of path components are precisely the monomorphisms in the \(\infty\)-category of spaces: given an inclusion of path components \(Y \xhookrightarrow{i} X\), it is merely a condition for any map \(Z \rightarrow X\) to factor through it.

This notion generalizes: we say that a morphism \(c \rightarrow d\) in an \(\infty\)-category \(\mathcal C\) is a monomorphism if it is merely a condition for any morphism \(e \rightarrow d\) to factor through it.⁴³ This is equivalent to the condition that the resulting morphism \(\mathrm{Hom}_\mathcal C(-,c) \rightarrow\mathrm{Hom}_\mathcal C(-,d)\) in \(\mathcal P(\mathcal C)\) is a componentwise monomorphism.

As a notable example, the monomorphisms in \(\mathrm{Cat}_\infty\) are precisely the functors that are fully faithful on equivalences and monomorphisms on all hom-spaces.⁴⁴ We reserve the term subcategory for (the image of) a monomorphism (in \(\mathrm{Cat}_\infty\), or more generally in \(\mathrm{Cat}[\mathbb V]\) (again see Subsection A.10)).

As another notable example, it is merely a condition for a morphism in an \(\infty\)-category to be an equivalence. Said differently, the functor \([1] \rightarrow[1]^{\textup{gpd}}\simeq {\sf pt}\) is an epimorphism in \(\mathrm{Cat}_\infty\) (see §§A.2.4-A.2.5 for an explanation of the notation).

A.2.3 Adjunctions[00ID]

It is merely a condition for a functor \(\mathcal C\xrightarrow{F} \mathcal D\) to be a (say) left adjoint: its space of right adjoints is either empty or contractible. First of all, a pointwise right adjoint to \(F\) at an object \(d \in \mathcal D\) is a pair of an object \(c \in \mathcal C\) and a morphism \(F(c) \xrightarrow{\varepsilon_d} d\) such that for every \(c' \in \mathcal C\) the composite \(\mathrm{Hom}_\mathcal C(c',c) \xrightarrow{F} \mathrm{Hom}_\mathcal D(F(c'),F(c)) \xrightarrow{\varepsilon_d} \mathrm{Hom}_\mathcal D(F(c'),d)\) is an equivalence. Equivalently, this is the data of a representing object for the presheaf \(\mathcal C^\mathrm{op}\xrightarrow{\mathrm{Hom}_\mathcal D(F(-),d)} \mathcal S\) (which by definition comes equipped with the data of a universal element \(\varepsilon_d \in \mathrm{Hom}_\mathcal D(F(c),d)\) witnessing it as such). Then, a right adjoint exists if and only if a pointwise right adjoint exists at all objects of \(\mathcal D\); in this case, the right adjoint is the (necessarily unique) factorization of the functor \(\mathcal D\xrightarrow{\mathrm{Hom}_\mathcal D(F(=),-)} \mathcal P(\mathcal C)\) through the Yoneda embedding. (See Subsection A.5 for an alternative description of \(\infty\)-categorical adjunctions.)

As a basic example, there exists a right adjoint \(\mathcal S\xleftarrow{\iota_0} \mathrm{Cat}_\infty\) to the inclusion, which carries an \(\infty\)-category \(\mathcal C\) to its maximal subgroupoid \(\mathcal C^\simeq\) (which is obtained by discarding all of its noninvertible morphisms).

A.2.4 Simplicial objects[00IE]

We write \(\Delta\) for the simplicial indexing category (the full subcategory of \(\mathrm{Cat}_\infty\) on the finite nonempty totally ordered sets), and for any \(n \geq 0\) we write \([n] \coloneqq \{ 0 < 1 < \cdots < n \} \in \Delta\) for the indicated standard object. A simplicial object in an \(\infty\)-category \(\mathcal C\) is a functor \(\Delta^\mathrm{op}\xrightarrow{X} \mathcal C\); we use the term geometric realization to refer to its colimit, and denote this by \(|X| \coloneqq \mathrm{colim}_{\Delta^\mathrm{op}}(X) \in \mathcal C\).⁴⁵

A.2.5 Localizations[00IF]

Given an \(\infty\)-category \(\mathcal C\) and a collection \(\mathbf W\) of morphisms in \(\mathcal C\) (often assumed to be those defining a subcategory of \(\mathcal C\)), the localization of \(\mathcal C\) at \(\mathbf W\) is the target of the initial functor \(\mathcal C\rightarrow\mathcal C[\mathbf W^{-1}]\) that carries all morphisms in \(\mathbf W\) to equivalences. As an extreme example, the \(\infty\)-groupoid completion of \(\mathcal C\) is its localization at all of its morphisms; this defines a left adjoint \(\mathrm{Cat}_\infty \xrightarrow{(-)^{\textup{gpd}}} \mathcal S\) to the inclusion. More generally, we can identify the localization at (the morphisms in) a subcategory \(\mathbf W\subseteq \mathcal C\) as the pushout

A special case of localization is given by a reflective localization adjunction, i.e. an adjunction in which the right adjoint is fully faithful. In this case, writing \(\mathbf W\subseteq \mathcal C\) for the subcategory of morphisms in \(\mathcal C\) that are carried to equivalences in \(\mathcal D\), the left adjoint witnesses \(\mathcal D\) as the localization \(\mathcal C[\mathbf W^{-1}]\). In this case, \(R\) can be characterized as the inclusion of the full subcategory of objects of \(\mathcal C\) that are local with respect to the morphisms in \(\mathbf W\), i.e. those \(c \in \mathcal C\) such that for every \(d \rightarrow e\) in \(\mathbf W\) the morphism \(\mathrm{Hom}_\mathcal C(d,c) \leftarrow \mathrm{Hom}_\mathcal C(e,c)\) is an equivalence [Lur09, Prop. 5.5.4.2].⁴⁶ Of course, dual remarks pertain to coreflective localization adjunctions, i.e. adjunctions in which the left adjoint is fully faithful.

Note that we might obtain the same localization even if we change the collection \(\mathbf W\). For instance, it is unnecessary to invert equivalences (since they are already invertible), and for any pair of composable morphisms \(f\) and \(g\) inverting any two of \(f\), \(g\), and \(gf\) automatically inverts the third (since equivalences have the two-out-of-three property). This observation plays a key role in the theory of accessible localizations of presentable \(\infty\)-categories (see Subsection A.7).

A.2.6 Connected categories versus weakly contractible \(\infty\)-categories[00IH]

While certain results in ordinary category theory refer to connected categories (i.e. those whose groupoid completions are connected), their \(\infty\)-categorical analogs generally instead refer to weakly contractible \(\infty\)-categories (i.e. those whose \(\infty\)-groupoid completions are contractible). For instance, the forgetful functor \(\mathcal C_{c/} \rightarrow\mathcal C\) commutes with (and detects) weakly contractible colimits.⁴⁷