Factorization theorems for morphisms of ordered groupoids and inverse semigroups

Adapting the theory of the derived category to ordered groupoids, we prove that every ordered functor (and thus every inverse and regular semigroup ho momorphism) factors as an enlargement followed by an ordered fibration. As an application, we obtain Lawson's version of Ehresmann's Maximum Enlargeme nt Theorem, from which can be deduced the classical theory of idempotent-pu re inverse semigroup homomorphisms and E-unitary inverse semigroups.