Editing Simplex category (section)

==Augmented simplex category==
The '''augmented simplex category''', denoted by <math>\Delta_+</math> is the category of ''all finite ordinals and order-preserving maps'', thus <math>\Delta_+=\Delta\cup [-1]</math>, where <math>[-1]=\emptyset</math>. 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 <math>\Delta_+</math> is called an '''augmented simplicial object''' and a covariant functor out of <math>\Delta_+</math> 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 category|monoidal structure]]. The monoidal product is given by concatenation of linear orders, and the unit is the empty ordinal <math>[-1]</math> (the lack of a unit prevents this from qualifying as a monoidal structure on <math>\Delta</math>). In fact, <math>\Delta_+</math> is the [[monoidal category]] freely generated by a single [[monoid object]], given by <math>[0]</math> 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 <math>\Delta_+^\text{op}</math> 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 [[Monad (category theory)|monads]] (and hence [[adjoint functors]]) since monads can be viewed as monoid objects in [[functor category|endofunctor categories]].