Cofunctors, monoids, and split epimorphisms

Bryce Clarke – 28 October 2020

It is well-known that small categories are generalisations of monoids, but what is a most suitable way of generalising monoid homomorphisms? While functors between categories provide a familiar solution, Aguiar showed that there is also a fully faithful functor from the category of monoids and monoid homomorphisms to the category of small categories and cofunctors. Furthermore, this fully faithful functor has an interesting right adjoint, which assigns each category to its monoid of ``admissible sections''. In this talk, I will discuss the (2-)category of categories and cofunctors, and provide several characterisations of this right adjoint. I will then explore how this adjunction extends to other contexts, including (Schreier) split epimorphisms between monoids, delta lenses, and split opfibrations.