ScalingStacks

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).