Every category gives rise to an arrow category such that the objects of are the morphisms (or arrows, hence the name) of .
For any category, its arrow category is the category such that:
- an object of is a morphism of ;
- a morphism of is a commutative square
- composition in is given simply by placing commutative squares side by side to get a commutative oblong.
Up to equivalence, this is the same as the functor category
for the interval category . is also written , , , or , since and (for the -simplex) are common notations for the interval category.