Simplicial maps commute with face maps (that delete vertex)?

Question

Let $f: K\to L$ be a simplicial map. And let $d_i([v_0,\dots,v_n]=[v_0,\dots,\widehat{v_i},\dots,v_n]$, i.e. delete the $i$th vertex of the simplex $[v_0,\dots,v_n]$.

Is it true that for any simplex $\sigma\in K$, we have $f(d_i(\sigma))=d_i(f(\sigma))$?

Thanks for any help.

Answer 1

Are you working with simplicial complexes or simplicial sets? With simplicial sets, the answer is certainly yes. With simplicial complexes, the answer is no: let $K=L=1-\text{simplex}$ with vertices $0$ and $1$. Then the map interchanging the vertices is simplicial but does not commute with the face maps.

If you work with simplicial complexes in which the vertices are ordered and the maps must preserve the ordering, then I think $f d_i = d_i f$ for such maps.