Ranges, restrictions, partial maps, and fibrations

In this thesis, we study range restriction categories and their properties. Range restriction categories with split restriction idempotents are shown to be equivalent to the partial map categories of M -stable factorization systems. The notions of a restriction fibration, a range restriction fibration, a stable meet semilattice fibration, and a range stable meet semilattice fibration are introduced and it is shown that (range) stable meet semilattice fibrations provide a bridge between the category of (range) restriction categories and the category of categories and that (range) restriction fibrations are the same as (range) restriction categories so that these fibrations provide a useful setting for studying (range) restriction categories. Finally, we construct the free range restriction structures over directed graphs using deterministic trees.