A simplicial map is a family of functions that commutes with and . If each is a subset of such that the inclusions is a simplicial map, then is said to be a simplicial subset of .

The -simplex is defined as follows:

where .

The face is defined by , i.e. deleting . The degeneracy is given by , i.e. repeating . Let . Any element in can be written as iterated compositions of faces and degeneracies of .

Presumably the above can include a translation into everyday language maybe as follows. The sequence: point (zero-simplex?), line connecting two points (1-simplex),equilateral triangle with face enclosed by three lines connecting three points (2-simplex), regular tetrahedron enclosed by four triangles…. (3-simplex) and so on undrawably but without limit into the higher dimensions, with each n-simplex providing the “faces” bounding the (n+1)-simplex. The sequence has very beautiful numerical properties and can be employed to model “real life” situations (as in my Ph.D. thesis long ago).

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Thanks for your comment! I am looking for a PhD topic on this area too (applications of algebraic topology). Can you share some of the “real life” situations that you mentioned?

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