How to allow simplicial maps with "degeneracies" in the case of simplicial complex

Question

Let $f:K\to L$ be a simplicial map, where $K,L$ are simplicial complexes.

Suppose $f(v_0)=w_0$, $f(v_1)=w_1$, and $f(v_2)=w_1$.

Then $f([v_0,v_1,v_2])=[w_0,w_1]$. Is this correct? (where $[v_0,v_1,v_2]$ means the simplicial complex spanned by the vertices $v_0,v_1,v_2$.)

However, suppose I want to define $f([v_0,v_1,v_2])=[w_0,w_1,w_1]$ instead, where there is repeated vertices. How do I write this properly in mathematical language? I am ok with redefining the definition of simplicial map (is there such a definition already?)

I am aware there is something called simplicial sets, which allows for degeneracies. Ideally, I would try not to use terminology from simplicial set theory, unless it is something really simple.

Thanks a lot. Hope it makes sense.

If it helps, the main thing I wish to do is just to have a nice (rigorous) notation to ensure that the simplicial maps commute with face maps: Simplicial maps commute with face maps (that delete vertex)?. I'm not trying to do anything radically new from simplicial maps.

Answer 1

Just define a "generalized $n$-simplex" of $K$ to be an $(n+1)$-tuple of vertices of $K$ such that the set they form is a simplex of $K$. Then any simplicial map $f:K\to L$ induces a map from the generalized $n$-simplices of $K$ to the generalized $n$-simplices of $L$, by applying $f$ to each vertex. The notation $f([v_0,v_1,v_2])=[w_0,w_1,w_1]$ is then as good as any, with the understanding that you are talking about generalized simplices.

This definition is a bit ad hoc and nonstandard. To match up with the more standard definitions involving simplicial sets, you need to impose an order on everything. Specifically, if $K$ is an ordered simplicial complex (i.e., you have a chosen total order on the vertices of $K$), you can define an "ordered generalized $n$-simplex" of $K$ to be an $(n+1)$-tuple of vertices in non-decreasing order such that the set they form is a simplex of $K$. Then, if $f:K\to L$ is an order-preserving simplicial map, it induces a map on ordered generalized $n$-simplices.

The connection with simplicial sets is then that any ordered simplicial complex can be thought of as a simplicial set in a canonical way, and the "$n$-simplices" of that simplicial set are exactly the ordered generalized $n$-simplices of the simplicial complex.