nLab
interval category

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

$$\stackrel{0}{•}\to \stackrel{1}{•}$$\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

• the first oriental

• the 1-globe.

• the object $[1]$ in the simplex category when that is thought of as the full subcategory in Cat on finite (nonemty) linear quivers.

• accordingly, the nerve of the interval category is the simplicial set ${\Delta }^1$ that is represented by $[1]\in \Delta$.

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 $0\to 1$ is an isomorphism. Hence, there is a second morphism, namely it’s inverse $1\to 0$.

$$\stackrel{0}{•}\leftrightarrows \stackrel{1}{•}$$\overset{0}{\bullet}\leftrightarrows\overset{1}{\bullet}

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