Simplex category

In mathematics, the simplex category (or simplicial category or nonempty finite ordinal category) is the category of non-empty finite ordinals and order preserving maps. It is used to define simplicial and cosimplicial objects.

The simplex category is usually denoted by ${\displaystyle \Delta }$. There are several equivalent descriptions of this category. ${\displaystyle \Delta }$ can be described as the category of non-empty finite ordinals as objects, thought of as totally ordered sets, and order-preserving functions as morphisms. The objects are commonly denoted ${\displaystyle [n]=\{0,1,\dots ,n\}}$ (so that ${\displaystyle [n]}$ is the ordinal ${\displaystyle n+1}$). The category is generated by coface and codegeneracy maps, which amount to inserting or deleting elements of the orderings. (See simplicial set for relations of these maps.)

...
Wikipedia