# Simplex (abstract)

A topological space $| A |$ whose points are non-negative functions $\phi : A \rightarrow \mathbf R$ on a finite set $A$ satisfying $\sum _ {a \in A } \phi ( a) = 1$. The topology on $| A |$ is induced from $\mathbf R ^ {A}$, the space of all functions from $A$ into $\mathbf R$. The real numbers $\phi ( a)$ are called the barycentric coordinates of the point $\phi$, and the dimension of $| A |$ is defined as $\mathop{\rm card} ( A) - 1$. In case $A$ is a linearly independent subset of a Euclidean space, $| A |$ is homeomorphic to the convex hull of the set $A$( the homeomorphism being given by the correspondence $\phi \mapsto \sum _ {a \in A } \phi ( a) \cdot a$). The convex hull of a linearly independent subset of a Euclidean space is called a Euclidean simplex.

For any mapping $f: A \rightarrow B$ of finite sets, the formula $(| f | \phi ) ( b) = \sum _ {f ( a) = b } \phi ( a)$, $b \in B$, defines a continuous mapping $| f |: | A | \rightarrow | B |$, which, for Euclidean simplices, is an affine (non-homogeneous linear) mapping extending $f$. This defines a functor from the category of finite sets into the category of topological spaces. If $B \subset A$ and $i: B \rightarrow A$ is the corresponding inclusion mapping, then $| i |$ is a homeomorphism onto a closed subset of $| A |$, called a face, which is usually identified with $| B |$. Zero-dimensional faces are called vertices (as a rule, they are identified with the elements of $A$).

A topological ordered simplex is a topological space $X$ together with a given homeomorphism $h: \Delta ^ {n} \rightarrow X$, where $\Delta ^ {n}$ is a standard simplex. The images of the faces of $\Delta ^ {n}$ under $h$ are called the faces of the topological ordered simplex $X$. A mapping $X \rightarrow Y$ of two topological ordered simplices $X$ and $Y$ is said to be linear if it has the form $k \circ F \circ h ^ {-} 1$, where $k$ and $h$ are the given homeomorphisms and $F$ is a mapping $\Delta ^ {n} \rightarrow \Delta ^ {n}$ of the form $| f |$.

A topological simplex (of dimension $n$) is a topological space $X$ equipped with $( n + 1)!$ homeomorphisms $\Delta ^ {n} \rightarrow X$( that is, with $( n + 1)!$ structures of a topological ordered simplex) that differ by homeomorphisms $\Delta ^ {n} \rightarrow \Delta ^ {n}$ of the form $| f |$, where $f$ is an arbitrary permutation of the vertices. Similarly, a mapping of topological simplices is called linear if it is a linear mapping of the corresponding topological ordered simplices.

Elements of simplicial sets (cf. Simplicial set) and distinguished subsets of simplicial schemes (cf. Simplicial scheme) are also referred to as simplices.