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A simplicial map is by definition a map $f:K\to L$ between simplicial complexes that sends each simplex of $K$ to a simplex of $L$ by a linear map taking vertices to vertices.

A cellular map is by definition a map $f:X\to Y$ between cellular complexes such that $f(X^n)\subset Y^n$ for all $n$, where $X^n$ is the $n$-th skeleton of $X$.

Is it true that every simplicial map is cellular? I think it seems true but I'm not sure.

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  • $\begingroup$ Your definition does not seem standard to me. Take a simplicial complex and then make it finer. Is the inclusion from the finer complex to the first complex simplicial? $\endgroup$ – s.harp Mar 12 at 21:20
  • $\begingroup$ @s.harp: presumably by "making a simplicial complex finer" you mean "subdividing simplices". If so, then there is no "inclusion" from the finer complex to the coarser complex. (However, there is a simplicial map.) $\endgroup$ – Rob Arthan Mar 12 at 22:21
  • $\begingroup$ @s.harp This definition is from Hatcher's. $\endgroup$ – probably123 Mar 12 at 23:20
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Hint: by definition, a simplicial map $f : K \to L$, maps the vertices of an $i$-simplex $\sigma$ of $K$ onto a set of vertices in $L$ that span a $j$-simplex $\tau$ of $L$, where necessarily $j \le i$. So $f$ maps the $i$-skeleton of $K$ (which comprises the union of the $k$-simplices in $K$ with $k \le i$) into a union of $l$-simplices in $L$ with $l \le i$, which is contained in the $i$-skeleton of $L$ (which comprises the union of all the $l$-simplices in $L$ with $l \le i$).

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