Simple query regarding simplicial map

Question

Let $f:K\to L$ be a simplicial map. Let $\sigma=[v_0,v_1,\dots, v_n]$ be a simplex in $K$. By definition of simplicial map, $f(\sigma)$ is a simplex in $L$.

My question is, does $f(\sigma)$ have to be equal to $[f(v_0),f(v_1),\dots,f(v_n)]$? Or can it be something else?

Thanks for any help.

Answer 1

Use the fact that $L$ is a simplicial complex to see that each face $d^i(f(\sigma))$ of $f(\sigma)$ is simplex of $L$. Then use the fact that $f$ is a simplicial map to see that each such face is of the form $f(d^i\sigma)$. Finally use induction to get the answer.

Note that many authors define a simplicial map to be a map of the vertices. Note also that the vertices $f(v_0),\dots,f(v_n)$ are not necessarily distinct.