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Characterizing weakly Schreier extensions of monoids

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It is well known that split extensions of groups H by N correspond to actions of H on N. This is not so for monoids, however actions of monoids do correspond to a certain class of split extensions, called the Schreier extensions. A split extension comprising kernel k: N → G, cokernel e: G → H and splitting s: H → G is said to be Schreier when for each g in G, there is a unique n in N and h in H such that g = k(n)s(h). The uniqueness condition can be relaxed providing the notion of a weakly Schreier extension, of which the Artin glueings of topological spaces provide a natural example.

In this talk we provide a characterization of weakly Schreier extensions between monoids N and H, as a certain quotient of N x H, paired with something that resembles an action. We then use this characterization to construct some new examples of weakly Schreier extensions.

This talk is part of the Category Theory Seminar series.

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