Chain map induced by simplicial map

Question

In Munkres' book (page 62 Section 12), the chain map is defined as follows:

Let $f:K\to L$ be a simplicial map. If $v_0\dots v_p$ is a simplex of $K$, then the points $f(v_0),\dots,f(v_p)$ span a simplex of $L$. We define a homomorphism $f_\sharp:C_p(K)\to C_p(L)$ by defining it on oriented simplices as follows:

$$f_\sharp([v_0,\dots,v_p])=\begin{cases} [f(v_0),\dots,f(v_p)] &\text{if $f(v_0),\dots,f(v_p)$ are distinct}\\ 0&\text{otherwise}. \end{cases}$$

Question) My question is what is the main purpose of setting the value to 0 when the $f(v_0),\dots,f(v_p)$ are not distinct?

Will it be wrong to set $f_\sharp([v_0,\dots,v_p])=[f(v_0),\dots,f(v_p)]$ even if $f(v_0),\dots,f(v_p)$ are not distinct?

Thanks.

Answer 1

By the definition of a simplex, all its vertices must be distinct. Otherwise you're not dealing with simplicial complexes anymore (and rather simplicial sets or something like that).