nLab
overcategory

Contents

Definition

The slice category or over category C/c of a category C over an object cC has

  • objects that are all arrows fC such that cod(f)=c, and
  • morphisms g:XXC from f:Xc to f:Xc such that fg=f.
C/c={X g X f f c}C/c = \left\lbrace \array{ X &&\stackrel{g}{\to}&& X' \\ & {}_f \searrow && \swarrow_{f'} \\ && c } \right\rbrace

The slice category is a special case of a comma category.

There is a forgetful functor U c:C/cC which maps an object f:Xc to its domain X and a morphism g:XXC (from f:Xc to f:Xc such that fg=f) to the morphism g:XX.

The dual notion is an under category.

Examples

  • If C=P is a poset and pP, then the slice category P/p is the down set (p) of elements qP with qp.

  • If 1 is a terminal object in C, then C/1 is isomorphic to C.

Properties

The assignment of overcategories C/c to objects cC extends to a functor

C/():CCat,C/(-) : C \to Cat \,,

Under the Grothendieck construction this functor corresponds the the codomain fibration

cod:[I,C]Ccod : [I,C] \to C

from the arrow category of C.

Generalizations