Simplexes, Abstract Map, Simplicial Map

Abstract Map, Simplicial Map

A self-consistent map between abstract complexes defines, and is defined by, a function from one set of vertices into another. (This is called an abstract map.) Yet the same can be said for a piecewise linear map from one topological complex into another, that carries simplexes to simplexes. (This is called a simplicial map.) One direction is obvious. The simplicial map carries vertices to vertices, building an abstract map. For the converse, assume an abstract map carries vertices to vertices, and concentrate on a simplex U. Pull U back to barycentric space, then map the preimage (i.e. the symmetric simplex) onto V, as directed by the abstract map. The result is a continuous map from U into V. This in turn is a continuous map from U into the second complex. Since our map is continuous on every simpleex U, it is continuous from one complex into another. The result is a simplicial map, induced by our abstract map.

If the abstract map is a bijection, the map from each U onto V carries an n-simplex onto an n-simplex. Dimension is preserved, giving a homeomorphism. Therefore the two topological complexes are equivalent.

If an abstract complex has two topological instantiations, the identity map on vertices (a bijection) proves the spaces are homeomorphic. There is but one topological space, up to isomorphism, for each abstract complex. The embedding of K into Ej is a default representation.

A simplicial map that carries a copmlex into itself is sometimes called a pasting map, as it pastes vertices together.

If you like category theory, the geometric realization of an abstract complex is a functor from sets of finite sets (closed under subset) into topological spaces, where functions on vertices become continuous maps. If we restrict topological spaces to simplicial complexes, and if functions are simplicial maps, the categories are equivalent. In other words, the functor admits an inverse functor, which focuses on the vertices of a topological complex.