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 [4]) 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], [22] and [23]). 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.

A short note on strongly flat covers of acts over monoids

“…The converse however is not true, the infinite monogenic monoid being a counterexample. It was also shown in[2, Corollary 5.6] that S being right cancellative is sufficient for every S−act to have an SF -cover, and as we can see from the next Lemma, right cancellativity and M L implies Condition (A).Lemma 3.2 A right cancellative monoid with M L is a group.…”

“…We would like to know if there exists a monoid S satisfying M L but not Condition (A) for which every S−act has an SF −cover. Another class of monoids known to have SF −covers are those monoids having weak finite geometric type (see [2,Proposition 5.4]). This would seem to be a good place to look for a counterexample, although the main example of a monoid having weak finite geometric type, that is not right cancellative, is the Bicyclic monoid which does not have M L .…”

“…In 2008, Mahmoudi and Renshaw [13] initiated a study of flat covers of acts over monoids. Their definition of cover proved to be different to that given by Enochs and in 2012, Bailey and Renshaw [2] initiated a study of Enochs' definition of cover. We give here both definitions but note that we use a slightly different terminology to that used in [13].…”

Weak factorization systems for S-acts

“…In this section we provide a number of examples of weak factorization systems, some of which are related to the existence of covers of S−acts, and refer the reader to [4] and [14] for more details of some of the concepts and results used.…”

“…We shall have occasion to consider directed colimits of S−acts where the index set is regarded as an ordinal. For details of directed colimits in general see [2], [4] or [10]. We shall (informally) consider a class as a collection of sets, or viewed another way, a set is a class that is a member of another class.…”