nLab
interval category

Often denoted 2 or I, the interval category is the category with two objects and one nontrivial morphism

01\overset{0}{\bullet}\to\overset{1}{\bullet}

The interval category serves as a combinatorial model for the 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 2.

The notation 2 comes from the fact that the interval category is also the ordinal number 2 regarded as a category.

It also appears as

Interval groupoid

In the context not of categories but of groupoids and further that of ∞-groupoids the free groupoid on the interval category is relevant, where the morphism 01 is an isomorphism. Hence, there is a second morphism, namely it’s inverse 10.

01\overset{0}{\bullet}\leftrightarrows\overset{1}{\bullet}

This is the interval groupoid, which is the undirected interval object in Cat and in Grpd.