Factorization theorems for morphisms of ordered groupoids and inverse semigroups (Q2758147)

An ordered (and particularly an inductive) groupoid is considered as a set with a partial binary operation and a partial order relation satisfying known axioms. An order functor between (inductive) groupoids is a functor that preserves the order (the meet of identities). Let \(\mathbf{Funct}\) denote the category of all order functors. Let \(G\) and \(H\) be two ordered groupoids. Using the notion of an action of \(H\) on \(G\), the notion of semidirect product \(G\rtimes H\) is defined. All actions form the category \(\mathbf{Act}\), and the notion of semidirect product leads to the functor \(\mathbf{Sd}\colon\mathbf{Act}\to\mathbf{Funct}\). On the other hand, a construction similar to the kernel for groups leads to a functor \(\mathbf{Der}\colon\mathbf{Funct}\to\mathbf {Act}\). It is proved that the last one is left adjoint to \(\mathbf{Sd}\colon\mathbf{Act}\to\mathbf {Funct}\). Various order functors \(\varphi\) are characterized in terms of \(\mathbf{Der}(\varphi)\). A notion of enlargement for order functors is introduced and it is proved that every order functor factors as an enlargement followed by an order fibration.