The defect functor of a homomorphism and direct unions

TL;DR

This paper investigates the properties of the defect functor associated with a homomorphism in finitely presented categories and characterizes objects for which Ext^1 commutes with direct unions, under specific conditions.

Contribution

It introduces a study of the defect functor's commuting properties and characterizes objects with Ext^1 commuting with direct unions in certain categories.

Findings

Characterization of objects with Ext^1 commuting with direct unions.

Analysis of the defect functor's properties in finitely presented categories.

Conditions under which Ext^1 commutes with direct unions.

Abstract

We will study commuting properties of the defect functor $\mathrm{Dev}_\beta=\mathrm{Coker}\mathrm{Hom}_\mathcal{C}(\beta,-)$ associate to a homomorphism $\beta$ in a finitely presented category. As an application, we characterize objects $M$ such that $\mathrm{Ext}^1_\mathcal{C}(M,-)$ commutes with direct unions (i.e. direct limits of monomorphisms), assuming that $\mathcal{C}$ has a generator which is a direct sum of finitely presented projective objects.