Slice categories

Definition 21.3.1 (Slice category). Let \(C\) be an \(\infty \)-category and let \(x\) be an object in \(C\). We define the slice categories \(C_{/x}\) and \(C_{x/}\), also known as the over category and the under category, respectively, via the following two pullback squares:

Note that an object of \(C_{/x}\) is a pair \((y,f\colon y \to x)\) of an object \(y\) equipped with a morphism to \(x\), and dually objects of \(C_{x/}\) are pairs \((y,f\colon x \to y)\). As a result of Proposition 1.4.4 morphisms \((y,f) \to (y',f')\) in \(C_{/x}\) may equivalently be encoded by commutative triangles of the form

and dually for morphisms in \(C_{x/}\).

We denote by \[ s\colon C_{/x} \to C \qquadtext { and } t\colon C_{x/} \to C \] the assignments \(s(y,f\colon y \to x) = y\) and \(t(y,f\colon x \to y) = y\), i.e. we first include the slice into the arrow category \(\Ar (C)\) and then apply the source/target functor.