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.

Comments

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.