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\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \newcommand{\groupMul}[1] { \cdot_{\small{#1}}} \newcommand{\groupAdd}[1] { +_{\small{#1}}} \newcommand{\inv}[1] {#1^{-1} } \newcommand{\bm}[1] { \boldsymbol{#1} } \require{physics} \require{ams} \require{mathtools} \)
Math and science::Algebra::Aluffi

Slice category

A slice category is an example of a category whose objects are [something] and whose morphisms are also [one of those something]. 

The objects of a slice category are ambient morphisms [to or from?] an object in an ambient category, and the morphisms of a slice category are ambient morphisms from one slice category object to another. The precise definition is as follows:

Slice category

Let \( \cat{C} \) be a category and let \( A \) be an object of \( \cat{C} \). Then we define \( \cat{C_A} \) to be the category whose objects and morphisms are as follows:

  • \( \catobj{C_A} = \) the set of all morphisms from any object in \( \cat{C} \) to the object \( A \). Thus, [\( f \in \catobj{C_A} \iff \; ? \; \text{ for some object } Z \in \catobj{C} \)].
  • For any two objects \( f_1: Z_1 \to A \) and \( f_2: Z_2 \to A \) in \( \cat{C_A} \), \( \cathom{C_A}(f_1, f_2) \) contains any morphism [\( \sigma \in \; ? \)] such that [\( ? = \; ? \) ].

The back side has a diagram of a slice category. Can you remember what it looks like? There is also info on co-slice categories. Can you remember the definition?