On the decomposition of a strong epimorphism into regular epimorphisms

TL;DR

This paper explores how strong epimorphisms can be decomposed into regular epimorphisms in locally presentable categories, providing syntactic methods to determine the number of steps needed.

Contribution

It introduces two syntactic methods using partial Horn theory and generalized algebraic theory to analyze such decompositions.

Findings

Decomposition of strong epimorphisms into regular epimorphisms is possible in locally presentable categories.

Two methods are proposed to determine the number of regular epimorphisms in the decomposition.

The methods also apply to decomposing adjoint functors into monadic functors.

Abstract

Strong epimorphisms and regular epimorphisms are two important classes of morphisms, and they do not coincide in general. Yet, in a locally presentable category, it is known that any strong epimorphism can be decomposed into a transfinite composite of regular epimorphisms. In this paper, we provide two syntactic methods to determine how many regular epimorphisms are needed in such a decomposition, using partial Horn theory and generalized algebraic theory. We start by discussing a general problem of decomposing a morphism into a transfinite composite of morphisms in a given class, which also covers the decomposition of an adjoint functor into monadic functors.