There are several different ways to generalize the notion of surjections to general categories. Surjections are the functions that do not factor through any proper subset of the codomain, which leads to the concept of strong epimorphisms. At the same time, surjections are the functions obtained by taking quotients of the domain, which are abstracted as regular epimorphisms. The two classes of morphisms coincide in regular categories, such as the categories of sets, groups, rings, or other algebras, but not in general. Still, it is known that a strong epimorphism in locally presentable categories can be expressed as a transfinite composite of regular epimorphisms.
I will discuss the problem of how close the two classes in a given category are, specifically, how many regular epimorphisms are required for this expression. Our approach is based on two syntactic presentations of locally finitely presentable categories, namely partial Horn theory and generalized algebraic theory, which I will explain in the talk.
This is joint work with Yuto Kawase.