Simplicial Map and n-simplex

A simplicial map f:X\to Y is a family of functions f:X_n\to Y_n that commutes with d_i and s_i. If each X_n is a subset of Y_n such that the inclusions X_n\hookrightarrow Y_n is a simplicial map, then X is said to be a simplicial subset of Y.

The n-simplex \Delta[n] is defined as follows:
\Delta[n]_k:=\{(i_0,i_1,\dots,i_k)\mid 0\leq i_0\leq i_1\leq\dots\leq i_k\leq n\}
where k\leq n.
The face d_j:\Delta[n]_k\to\Delta[n]_{k-1} is defined by d_j(i_0,i_1,\dots,i_k)=(i_0,i_1,\dots\i_{j-1},i_{j+1},\dots,i_k), i.e. deleting i_j. The degeneracy s_j:\Delta[n]_k\to\Delta[n]_{k+1} is given by s_j(i_0,i_1,\dots,i_k)=(i_0,i_1,\dots,i_j,i_j,\dots,i_k), i.e. repeating i_j. Let \sigma_n=(0,1,\dots,n)\in\Delta[n]_n. Any element in \Delta[n] can be written as iterated compositions of faces and degeneracies of \sigma_n.


About mathtuition88
This entry was posted in 2016 and tagged . Bookmark the permalink.

2 Responses to Simplicial Map and n-simplex

  1. davescarthin says:

    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).

    Liked by 1 person

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