## 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 $\sigma_n$.

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### 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).

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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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