"Compatible /admissible" maps of $\Delta$-complexes

Question

In the following I'm going to use definitions and constructions of simplicial complexes and $\Delta$-complexes/sets from Greg Friedman's An elementary illustrated Introduction to simplicial sets.

Let $K$ and $L$ be simplicial complexes. An compatible or admissible simplicial map (ie a map which respects the structure of simplicial complexes; in hightbrow terms one could say that's a morphism inside the category of simplicial complexes) from $K$ to $L$ is a set function

$$\phi_V: V(K) \to V(L)$$

between vertex sets $V(K) \to V(L)$ of the corresponing simplicial complexes, such that if $\{v_0, v_1, ..., v_r \}$ is a simplex of $K$, then $\{\phi(v_0), \phi(v_1), ..., \phi(v_r) \}$is a simplex of $L$. Note that here manifests one of the most striking differences between simplicial complexes and $\Delta$-comlexes, which we are going to discuss next: every simplex of a simplicial complex and $\Delta$-comlexes: every simplex of a simplicial complex is already uniquely determinded by it's vertices. That's hightly wrong for $\Delta$-complexes; cp page 11 in Friedman's notes.

But for simplicial complexes therefore the $\phi_V: V(K) \to V(L)$ has determined already everything: we extend it uniquely to $ \vert \phi \vert: \vert K \vert \to \vert L \vert $ on geometric realisations by linear extension of the vertex map

$$ \vert \phi \vert(\sum_{i=0}^r t_i v_i) := \sum_{i=0}^r t_i \phi_V(v_i) $$

on the geometric realisation $\Delta_r= \sum_{i=0}^r t_i v_i$ of the $r$-vertex $\{v_0,v_1,..., v_r \}$.

Question: If now $T$ and $S$ are $\Delta$-complexes, is there a natural way to define compatible or admissible map in the "world"/category of $\Delta$-complexes in "similar spirit" like done above for simplicial complexes?

By "similar spirit" I mean that the map should be encoded in pure combinatorical terms (ie like in former case for simplicial complexes), but of course, since $\Delta$-complexes allow more "glueing freedom degrees", the combinatorics is expected to become more complicated. But the "keyword" of the desired generalization is that it should have a pure combinatoric description. That's what I'm looking for.

Which conditions it should fullfil to be "compatible"? Obviously, it not suffice to specify it on the set of vertex set only as before. Which "compatibility" conditions should be imposed giving a neccessary and sufficient criterion for a map to respect the $\Delta$-structure?

Answer 1

A less "highbrow" translation of what @FShrike's answer says: a semisimplicial set $X$ consists of a collection of sets $\{ X_n : n \geq 0 \}$: $X_n$ is the set of $n$-simplices. These sets are connected by face maps $d_i: X_n \to X_{n-1}$, $0 \leq i \leq n$, satisfying the simplicial identity $$ d_i d_j = d_{j-1} d_i \text{ for all } i<j. $$ Hatcher's $\Delta$-complexes are the geometric realizations of these. Therefore you should look at morphisms $f: X \to Y$ of semisimplicial sets, and these are just collections of functions $f_n: X_n \to Y_n$ which are compatible with the face maps. These are purely "combinatorial" in the sense that this definition just deals with sets and functions between sets.

Answer 2

I don't know what you're looking for, but consider this.

In "highbrow terms", simplicial complexes and delta complexes are really topological instances of the more powerful concept of (semi)simplicial sets/objects. Delta complexes are (fat) geometric realisations of semisimplicial sets, and there is a distinguished category of semisimplicial sets.

I might call a map of delta-complexes $f:X\to Y$ 'good' if, for some choice of $S,S'$ semisimplicial sets with $X\cong\|S\|,Y\cong\|S'\|$, there is a natural map (i.e. of functors, if you know what that is) $\alpha:S\implies S'$ such that $\|\alpha\|:\|S\|\to\|S'\|$ is taken to $f$ via the homeomorphisms.

Concrete properties of $\|\alpha\|$ are that it takes every face of a 'simplex' $\sigma$ (a horribly ambiguously used word) to the corresponding face of $\alpha(\sigma)$. So $f$ would have to have this property. Whether or not that's sufficient, I don't know.