Slice categories

Construction 3.7.1 (Slice category). Let \(p_C\colon C\to \Gamma \) and \(p_Y\colon Y\to \Gamma \) be categories over \(\Gamma \), let \(\psi \colon Y\to C\) be a functor over \(\Gamma \), and assume that \(p_Y\) is exponentiable. Relative currying of \(\psi \) gives a section \(\bar \psi \colon \Gamma \to \ulFun _\Gamma (Y,C)\). Relative currying of the constant-family functor \((\pr _C,\pr _C)\colon C\times _\Gamma Y\to C\times _\Gamma C\) gives a functor \(p_Y^*\colon C\to \ulFun _\Gamma (Y,C)\) over \(\Gamma \). We define the slice \(C_{/\psi }\to \Gamma \) by the pullback square of categories over \(\Gamma \)

Here \([1]_\Gamma :=\Gamma \times [1]\to \Gamma \). The relative functor category in the upper right is constructed by taking the dependent product along \(p_Y\) of the arrow category of \(C\times _\Gamma Y\) over \(Y\); thus its existence uses only the assumed exponentiability of \(p_Y\).

Remark 3.7.2. We will mostly be interested in the case \(Y=\Gamma \) and \(p_Y=\id _\Gamma \). For a section \(x\colon \Gamma \to C\), we write \(C_{/x}\) for the resulting slice over \(\Gamma \).

Lemma 3.7.3. For every category \(X\) and every functor \(\psi \colon Y \to C\), there is a pullback square of the form

Remark 3.7.4. Informally speaking, the previous lemma says that a functor \(X \to C_{/\psi }\) into the slice consists of a map \(g \colon X \star Y \to C\) fitting in a commutative diagram as follows:

Construction 3.7.5. Let \(p_C\colon C\to \Gamma \) and \(p_X\colon X\to \Gamma \) be categories over \(\Gamma \), let \(\varphi \colon X\to C\) be a functor over \(\Gamma \), and assume that \(p_X\) is exponentiable. Let \(\bar \varphi \colon \Gamma \to \ulFun _\Gamma (X,C)\) be its relative currying. Define the coslice \(C_{\varphi /}\to \Gamma \) by the pullback square of categories over \(\Gamma \)