Regular simplicial sets (Q847596)

A simplicial set is finite if it contains a finite number of non-degenerate simplices. It is finitely dimensional if the degree of its non-degenerate simplices is bounded. So finite implies finitely dimensional. In general, the simplicial space of maps between two finite simplicial sets is not finite. The author proves that for a class of simplicial sets that he calls regular, the simplicial space from any finite simplicial set to any regular finitely-dimensional simplicial set is finitely dimensional as well. In particular, if the target simplicial set is regular and finite, the space of maps is finite as well (with the source simplicial set still finite). The author also proves some stability properties of the full subcategories of regular and finite regular simplicial sets. The former is closed under limits, sums, sub-objects and a nerve is always regular. The latter is closed under finite limits, cartesian closed, closed under sub-objects and finite sums.