It is used to define simplicial and cosimplicial objects.
can be described as the category of non-empty finite ordinals as objects, thought of as totally ordered sets, and (non-strictly) order-preserving functions as morphisms.
The category is generated by coface and codegeneracy maps, which amount to inserting or deleting elements of the orderings.
A cosimplicial object is defined similarly as a covariant functor originating from
The augmented simplex category, denoted by
is the category of all finite ordinals and order-preserving maps, thus
is called an augmented simplicial object and a covariant functor out of
is called an augmented cosimplicial object; when the codomain category is the category of sets, for example, these are called augmented simplicial sets and augmented cosimplicial sets respectively.
The augmented simplex category, unlike the simplex category, admits a natural monoidal structure.
The monoidal product is given by concatenation of linear orders, and the unit is the empty ordinal
(the lack of a unit prevents this from qualifying as a monoidal structure on
This description is useful for understanding how any comonoid object in a monoidal category gives rise to a simplicial object since it can then be viewed as the image of a functor from
to the monoidal category containing the comonoid; by forgetting the augmentation we obtain a simplicial object.
Similarly, this also illuminates the construction of simplicial objects from monads (and hence adjoint functors) since monads can be viewed as monoid objects in endofunctor categories.