Penman equation

In mathematics, the simplex category (or simplicial category or 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 $\mathrm{\Delta }$ and is sometimes denoted by Ord. There are several equivalent descriptions of this category. $\mathrm{\Delta }$ can be described as the category of non-empty finite ordinals as objects, thought of as totally ordered sets, and 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 on $\mathrm{\Delta }$, that is a contravariant functor from $\mathrm{\Delta }$ 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 $\mathrm{\Delta }$. Note that in topology a simplicial object defined in this way would be called an augmented simplicial object because of the presence of an augmentation map. This map can be dropped to yield a traditionally defined simplicial object.

An algebraic definition identifies $\mathrm{\Delta }$ as the freely generated monoidal category on a single monoidal generator. 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 ${\mathrm{\Delta }}^{\text{op}}$ to the monoidal category containing the comonoid. Similarly, this also illuminates the construction of simplicial sets from monads (and hence adjoint functors) since monads can be viewed as monoid objects in endofunctors categories.

The simplex category provides a simple example of a compact closed category.

Augmented simplex category

Augmented simplex category, denoted by ${\mathrm{\Delta }}_{+}$ is the category of all finite ordinals and order preserving maps, thus ${\mathrm{\Delta }}_{+}=\mathrm{\Delta }\cup [-1]$, where $[-1]=\mathrm{\varnothing }$. Augmented simplex category is occasionally referred to as algebraists' simplex category and the above version is called topologists' simplex category.

Augmented simplex category is used to define augmented simplicial sets and augmented cosimplicial sets.

See also

• Simplicial category
• PRO (category theory)

References

• P. G. Goerss and J. F. Jardine, Simplicial Homotopy Theory, Progress in Mathematics Vol. 174, Birkhäuser Basel-Boston-Berlin (1999) ISBN 3-7643-6064-X

External links

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