Simplicial complexes are useful in proving things about distributed systems. We define and use simplicial maps between two such complexes, and these maps also seem to be standard objects of study in pure math. But in distributed computing there is another type of map we consider: (what we call) the carrier map, which maps a simplex of one complex to a subcomplex of another while preserving set inclusion. Formally, given complexes $\mathcal{A}$ and $\mathcal{B}$, a carrier map $\Xi$ between these two is a function $\Xi : \mathcal{A} \rightarrow 2^\mathcal{B}$ where for each $\sigma \in \mathcal{A}$, $\Xi(\sigma) \subseteq \mathcal{B}$ is a subcomplex, and for each $\sigma \subseteq \tau \in \mathcal{A}$, $\Xi(\sigma) \subseteq \Xi(\tau)$.
I'm looking for a mathematical reference on carrier maps. Do carrier maps as defined above appear anywhere in mathematics literature; are they of any interest to mathematicians/have they been studied by mathematicians?