# Simplex category

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{{Short description|Category of non-empty finite ordinals and order-preserving maps}}
In [mathematics](/source/mathematics), the '''simplex category''' (or '''simplicial category''' or '''nonempty finite ordinal category''') is the [category](/source/category_theory) of [non-empty](/source/Empty_set) finite [ordinals](/source/ordinal_number) and [order-preserving map](/source/order-preserving_map)s. It is used to define [simplicial](/source/simplicial_set) and cosimplicial objects.

==Formal definition==

The '''simplex category''' is usually denoted by <math>\Delta</math>. There are several equivalent descriptions of this category. <math>\Delta</math> 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](/source/morphisms). The objects are commonly denoted <math> [n] = \{0, 1, \dots, n\} </math> (so that <math> [n] </math> is the ordinal <math> n+1 </math>). The category is generated by coface and codegeneracy  maps, which amount to inserting or deleting elements of the orderings. (See [simplicial set](/source/simplicial_set) for relations of these maps.)

A [simplicial object](/source/simplicial_object) is a [presheaf](/source/Presheaf_(category_theory)) on <math>\Delta</math>, that is a contravariant functor from <math>\Delta</math> to another category. For instance, [simplicial set](/source/simplicial_set)s are contravariant with the codomain category being the category of sets. A '''cosimplicial object''' is defined similarly as a covariant functor originating from <math>\Delta</math>.

==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 structure](/source/monoidal_category). 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](/source/monoidal_category) freely generated by a single [monoid object](/source/monoid_object), given by <math>[0]</math> with the unique possible unit and multiplication. This description is useful for understanding how any [comonoid](/source/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 [monads](/source/Monad_(category_theory)) (and hence [adjoint functors](/source/adjoint_functors)) since monads can be viewed as monoid objects in [endofunctor categories](/source/functor_category).

== See also ==

* [Simplicial category](/source/Simplicial_category_(disambiguation))
* [PROP (category theory)](/source/PROP_(category_theory))
* [Abstract simplicial complex](/source/Abstract_simplicial_complex)

==References==
* {{Cite book | last1=Goerss | first1=Paul G. | last2=Jardine | first2=John F. |author2-link=Rick Jardine| title=Simplicial Homotopy Theory | publisher=Birkhäuser|location=Basel–Boston–Berlin  | series=Progress in Mathematics | isbn=978-3-7643-6064-1 | year=1999 | volume=174 |doi=10.1007/978-3-0348-8707-6| mr=1711612}}

==External links==
*{{nlab|id=simplex+category|title=Simplex category}}
*[https://mathoverflow.net/q/171920 What's special about the Simplex category?]

{{Category theory}}

Category:Algebraic topology
Category:Homotopy theory
Category:Simplicial sets
Category:Categories in category theory
Category:Free algebraic structures

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Adapted from the Wikipedia article [Simplex category](https://en.wikipedia.org/wiki/Simplex_category) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Simplex_category?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.