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Wikipedia / 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.

Formal definition

The simplex category 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. (See simplicial set for relations of these maps.)

A simplicial object is a presheaf to another category. For instance, simplicial sets are contravariant with the codomain category being the category of sets. A cosimplicial object.

Augmented simplex category

The augmented simplex category is the category of all finite ordinals and order-preserving maps. Accordingly, this category might also be denoted FinOrd. The augmented simplex category is occasionally referred to as algebraists' simplex category and the above version is called topologists' simplex category.

is called an augmented simplicial object 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 is the monoidal category freely generated by a single monoid object with the unique possible unit and multiplication. This description is useful for understanding how any comonoid 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.

See also

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