Is there algebra of simplicial sets? For example, symbolic representation of simplicial sets and operations on those representations that allow to construct new simplicial set from existing one – join, disjoin and more involved operations. There is an idea (https://arxiv.org/abs/2106.14587 and other emerging sources) that neural manifolds (manifolds whose each point is some vector of concrete neural activations of some neural network layer in particular inference of learning state) can be described as simplicial sets and so, the going from one neural layer to next and back (inference and learning) can be described as some transformation of neural manifolds or – respectively – simplicial sets. So, it would be nice to do those more complex operations on the symbolic representations of simplicial sets.
Introduction to the simplicial sets https://arxiv.org/abs/0809.4221 states, that each simplicial set can be represented as the canonical sequence of face and degeneration maps that builds this simplicial set. And https://arxiv.org/abs/2212.06937 details the simplicial set semantics of type calculus, that can act as the syntax of such algebra of simplicial sets.
So – I feel that there should be algebra, combinatorial algebra of some other symbolic representation and manipulation of simplicial sets, some calculi, but I don’t know how they are named and where to seek for them. Is it possible to get some hint of guidance in the direction of symbolic manipulation, transformation of simplicial sets?