The interval category – often denoted or or – is the category with two objects and precisely one nontrivial morphism connecting them:
The interval category serves as a combinatorial model for the directed interval. It is the directed canonical interval object in Cat. It is also called the walking arrow. It might also be called the “arrow category” although that term is also used for a category of functors out of .
The notation comes from the fact that the interval category is also the ordinal number regarded as a poset, regarded as a category.
It also appears as
Since every category is also an (n,r)-category for , we may regard also as some -category. For instance regarded as a (∞,1)-category and modeled as a quasi-category, the interval category is the simplicial set .
The interval groupoid is a combinatorial model for the undirected interval.
It is the free groupoid on the interval category, where the morphism is an isomorphism. Accordingly the interval groupoid has a second nontrivial morphism, the inverse .
This is the undirected interval object in Cat and in Grpd.
The interval category is one of those diagram category that are not terribly interesting in themselves, but that serve an important role in category theory as a whole.
For instance a natural transformation between two functors is precisely the same as a strictly commuting diagram
in Cat, where on the left we have the cartesian product of with .
Accordingly, for the interval groupoid, a natural isomorphism is the same as a diagram
This is a left homotopy in .
Dually, forming the functor category
from the interval category produces the arrow category of , and a natural transformation is also the same as a diagram
With replaced by this is again a natural isomorphism, now represented as a right homotopy in Cat.
The analogous statements are true in higher category theory.