Inmathematics, thesimplex category (orsimplicial category ornonempty finite ordinal category) is thecategory ofnon-empty finiteordinals andorder-preserving maps. It is used to definesimplicial and cosimplicial objects.

Thesimplex category is usually denoted by${\displaystyle \mathrm{\Delta }}$. There are several equivalent descriptions of this category.${\displaystyle \mathrm{\Delta }}$ can be described as the category ofnon-empty finite ordinals as objects, thought of as totally ordered sets, and(non-strictly) order-preserving functions asmorphisms. The objects are commonly denoted${\displaystyle [n]=\{0,1,\dots ,n\}}$ (so that${\displaystyle [n]}$ is the ordinal${\displaystyle n+1}$). The category is generated by coface and codegeneracy maps, which amount to inserting or deleting elements of the orderings. (Seesimplicial set for relations of these maps.)

Asimplicial object is apresheaf on${\displaystyle \mathrm{\Delta }}$, that is a contravariant functor from${\displaystyle \mathrm{\Delta }}$ to another category. For instance,simplicial sets are contravariant with the codomain category being the category of sets. Acosimplicial object is defined similarly as a covariant functor originating from${\displaystyle \mathrm{\Delta }}$.

Theaugmented simplex category, denoted by${\displaystyle {\mathrm{\Delta }}_{+}}$ is the category ofall finite ordinals and order-preserving maps, thus${\displaystyle {\mathrm{\Delta }}_{+}=\mathrm{\Delta }\cup [-1]}$, where${\displaystyle [-1]=\mathrm{\varnothing }}$. Accordingly, this category might also be denotedFinOrd. 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${\displaystyle {\mathrm{\Delta }}_{+}}$ is called anaugmented simplicial object and a covariant functor out of${\displaystyle {\mathrm{\Delta }}_{+}}$ is called anaugmented 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 naturalmonoidal structure. The monoidal product is given by concatenation of linear orders, and the unit is the empty ordinal${\displaystyle [-1]}$ (the lack of a unit prevents this from qualifying as a monoidal structure on${\displaystyle \mathrm{\Delta }}$). In fact,${\displaystyle {\mathrm{\Delta }}_{+}}$ is themonoidal category freely generated by a singlemonoid object, given by${\displaystyle [0]}$ with the unique possible unit and multiplication. This description is useful for understanding how anycomonoid object in a monoidal category gives rise to a simplicial object since it can then be viewed as the image of a functor from${\displaystyle {\mathrm{\Delta }}_{+}^{\text{op}}}$ 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 frommonads (and henceadjoint functors) since monads can be viewed as monoid objects inendofunctor categories.

• Simplicial category
• PROP (category theory)
• Abstract simplicial complex

• Goerss, Paul G.;Jardine, John F. (1999).Simplicial Homotopy Theory. Progress in Mathematics. Vol. 174. Basel–Boston–Berlin: Birkhäuser.doi:10.1007/978-3-0348-8707-6.ISBN 978-3-7643-6064-1.MR 1711612.

• Simplex category at thenLab
• What's special about the Simplex category?

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

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