# slice category

Definition from Wiktionary, the free dictionary

## English[edit]

### Noun[edit]

**slice category** (*plural* **slice categories**)

- (category theory) Given a category
*C*and an object*X*∈ Ob(*C*), the*slice category*has, as its objects, morphisms from objects of*C*to*X*, and as its morphisms, morphisms connecting the tails of its own objects in a commutative way (i.e., closed under composition). The category is said to be "over*X*". (More formally, the objects of*C*over*X*are ordered pairs of the form (*A*,*f*) where*A*is an object of*C*and*f*is a morphism from*A*to*X*. Then the morphisms of*C*over*X*have such ordered pairs as their domains/codomains instead of objects of*C*directly.)- If
**slice category***C*over*X*has two objects (*A*,*f*) and (*B*,*g*) and a morphism*h*: (*A*,*f*) → (*B*,*g*), then this morphism would correspond to a like-named morphism*h*:*A*→*B*of*C*such that .

- If