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 $C$ be a category and let $A$ be an object of $C$ . Then we define $C_{A}$ to be the category whose objects and morphisms are as follows:

$Obj (C_{A}) =$ the set of all morphisms from any object in $C$ to the object $A$ . Thus, [ $f \in Obj (C_{A}) ⟺ ? for some object Z \in Obj (C)$ ].
For any two objects $f_{1} : Z_{1} \to A$ and $f_{2} : Z_{2} \to A$ in $C_{A}$ , ${Hom}_{C_{A}} (f_{1}, f_{2})$ contains any morphism [ $σ \in ?$ ] 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?

01.10.2020