Simplicial space

A topological space $X$ equipped with a covering by topological simplices (called a triangulation) such that the faces of every simplex belong to the triangulation, the intersection of any two simplices is a face of each (possibly empty), and a subset $F\subset X$ is closed if and only if its intersection with every simplex is closed. Every simplicial space is a cellular space. The specification of a triangulation is equivalent to the specification of a homeomorphism $|S|\to X$, where $|S|$ is the geometric realization of some simplicial complex. Simplicial spaces are also called simplicial complexes or simplicial decompositions. Simplicial spaces are the objects of a category whose morphisms $X\to Y$ are mappings such that every simplex of the triangulation of $X$ is mapped linearly onto some simplex of the triangulation of $Y$. The morphisms are also called simplicial mappings.

#### Comments[edit]

The term "simplicial space" is not often used in this sense; the more usual name for a space which admits a triangulation is a polyhedron (cf. Polyhedron, abstract). The term "simplicial space" more commonly means a simplicial object in the category of topological spaces (cf. Simplicial object in a category).

#### References[edit]

[a1] |  E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. 113ff

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[a2] |  B. Gray, "Homotopy theory. An introduction to algebraic topology" , Acad. Press (1975) pp. §12