ScalingStacks

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 \(F(c) \simeq d\).