Division categories and inverse semigroups
Abstract division categories were introduced by Leech as a small category C having finite pushouts, all of whose morphisms are epimorphisms, and with a quasi initial object. The concept of the division category was originally introduced to help study semigroup coextensions(see ). Möbius categories were introduced by Leroux. They were created as a general program to extend the theory of Möbius functions. In  Schwab shows that a reduced standard division category of a combinatorial inverse monoid (with the set of idempotents under natural partial order and locally finite) is a Möbius category. In the last section of Chapter 2 a result of Schwab is detailed using the Leech construction of inverse monoids of fractions. ^
Romero, Efren, "Division categories and inverse semigroups" (2006). ETD Collection for University of Texas, El Paso. AAI1435324.