Covers of acts over monoids and pure epimorphisms

Abstract

In 2001 Enochs' celebrated flat cover conjecture was finally proven and the proofs (two different proofs were presented in the same paper [5]) have since generated a great deal of interest among researchers. In particular the results have been recast in a number of other categories and in particular for additive categories (see for example [2], [3], [23] and [24]). In 2008, Mahmoudi and Renshaw considered a similar problem for acts over monoids but used a slightly different definition of cover. They proved that in general their definition was not equivalent to Enochs', except in the projective case, and left open a number of questions regarding the 'other' definition. This 'other' definition is the subject of the present paper and we attempt to emulate some of Enochs' work for the category of acts over monoids and concentrate, in the main, on strongly flat acts. We hope to extend this work to other classes of acts, such as injective, torsion-free, divisible and free, in a future report.

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ORCID for James Renshaw: orcid.org/0000-0002-5571-8007

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