In
mathematics, the simplex category (or simplicial category or nonempty finite ordinal category) is the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
of
non-empty
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other t ...
finite
ordinals and
order-preserving map
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...
s. It is used to define
simplicial
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
and cosimplicial objects.
Formal definition
The simplex category is usually denoted by
. There are several equivalent descriptions of this category.
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
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
. The objects are commonly denoted
(so that
is the ordinal
). The category is generated by coface and codegeneracy maps, which amount to inserting or deleting elements of the orderings. (See
simplicial set
In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined ...
for relations of these maps.)
A
simplicial object
In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined a ...
is a
presheaf
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
on
, that is a contravariant functor from
to another category. For instance,
simplicial set
In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined ...
s are contravariant with the codomain category being the category of sets. A cosimplicial object is defined similarly as a covariant functor originating from
.
Augmented simplex category
The augmented simplex category, denoted by
is the category of ''all finite ordinals and order-preserving maps'', thus