Fundamental adjunctions for animae and ∞-categories

Example 1.8.14. Let us state as black boxes two fundamental examples of adjunctions we will use:

(1)

(Path components) The fully faithful inclusion \(\Set \hookrightarrow \An \) admits a left adjoint \(\pi _0\colon \An \to \Set \). This functor preserves arbitrary products.

We call \(\pi _0(X)\) the set of path components of the anima \(X\). We often abuse terminology and refer to an anima as a set if it is in the essential image of the inclusion \(\Set \hookrightarrow \An \) (cf. Example 1.1.4).

(2)

(Homotopy category) We say that an \(\infty \)-category \(C\) is a 1-category if the hom anima \(\Hom _C(X,Y)\) is a set for all \(X,Y \in C\). We denote by \[ \Cat _1 \hookrightarrow \Cat _{\infty } \] the full subcategory spanned by the 1-categories. Then this inclusion admits a left adjoint \[ \Ho \colon \Cat _{\infty } \to \Cat _1. \] Given an \(\infty \)-category \(C\), we refer to the 1-category \(\Ho (C)\) as its homotopy category; it has the same set of isomorphism classes of objects as \(C\), but has hom sets given by the set of path components of the hom anima of \(C\): \[ \Hom _{\Ho (C)}(X,Y) \cong \pi _0(\Hom _C(X,Y)). \]