We can describe a space gluing from simplices as simplicial set.
When space is gluing from cubes there is notion of cubical set.
What is about some other class of polyhedra? What is known in this direction?
So my space is somehow glued from convex polyhedra. Can I describe it as a functor from some category to the category of sets?
I want to describe what is the problem.
In the case of simplicial set we choose some canonical representative of a simplex in any direction. Then we choose canonical identification of a face of n-dimensional simplex with standard (n-1)-th dimensional simplex. There are degeneracy maps as well. And all these maps should satisfy some identities.
However it is not clear how to identify a face with some standard polyhedra, this identification in general even cannot be affine.