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.

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A simplex is also a constituent of a simplicial complex, and a simplicial complex such that all subsets of its underlying subset are simplices is also called a simplex.