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 is usually denoted by formula_1. There are several equivalent descriptions of this category. formula_1 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 objects are commonly denoted formula_3 (so that formula_4 is the ordinal formula_5). 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 on formula_1, that is a contravariant functor from formula_1 to another category. For instance, simplicial sets are contravariant with the codomain category being the category of sets. A cosimplicial object is defined similarly as a covariant functor originating from formula_1. Augmented simplex category. The augmented simplex category, denoted by formula_9 is the category of "all finite ordinals and order-preserving maps", thus formula_10, where formula_11. 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. A contravariant functor defined on formula_9 is called an augmented simplicial object and a covariant functor out of formula_9 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 formula_14  (the lack of a unit prevents this from qualifying as a monoidal structure on formula_1). In fact, formula_9 is the monoidal category freely generated by a single monoid object, given by formula_17 with the unique possible unit and multiplication. 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 formula_18 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. The augmented simplex category provides a simple example of a compact closed category.