Abstract
In this paper we study flatness properties (pullback flatness, limit flatness, finite limit flatness) of acts over semigroups. These are defined by requiring preservation of certain limits from the functor of tensor multiplication by a given act. We give a description of firm pullback flat acts using Conditions (P) and (E). We also study pure epimorphisms and their connections to finitely presented acts and pullback flat acts. We study these flatness properties in the category of all acts, as well as in the category of unitary acts and in the category of firm acts, which arise naturally in the Morita theory of semigroups.
Highlights
Acts over monoids have been studied actively since the beginning of 1970s
These are defined by requiring the preservation of diagrams of certain types from the functor of tensor multiplication
Theorem 5.4 in [24] applied to the monoid S1 tells us that AS1 is a pullback flat unitary act over the monoid S1 precisely when every surjection BS → AS is a pure epimorphism in UActS1
Summary
Acts over monoids have been studied actively since the beginning of 1970s. The monograph [16] contains a detailed overview of main properties that have been studied, and a list of publications in this area. These are defined by requiring the preservation of diagrams of certain types from the functor of tensor multiplication. An act is called pullback flat if the functor of tensoring by it preserves pullbacks. Bo Stenström in his article [24] introduced a property which was later called strong flatness and which means preservation of both pullbacks and equalizers. He proved that strong flatness is equivalent to certain easyto-check Conditions (P) and (E). We show that under certain assumptions on a semigroup, the category of firm acts is locally finitely presentable
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