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 $$\cat{C}$$ tothe 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 an 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?