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Semi-biproducts of monoids

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arxiv 2109.06278 v1 pith:QH6PC5AT submitted 2021-09-13 math.CT

classification math.CT
keywords monoidscategorynotionbiproductcorrectionemphsemi-biproductsemi-biproducts
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It is shown that the category of \emph{semi-biproducts} of monoids is equivalent to the category of \emph{pseudo-actions}. A semi-biproduct of monoids is a new notion, obtained through generalizing a biproduct of commutative monoids. By dropping commutativity and requiring some of the homomorphisms in the biproduct diagram to be merely identity-preserving maps, we obtain a semi-biproduct. A pseudo-action is a new notion as well. It consists of three ingredients: a pre-action, a factor system and a correction system. In the category of groups all correction systems are trivial. This is perhaps the reason why this notion, to the author's best knowledge, has never been considered before.

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  1. The cohomology objects of a semi-abelian variety are small

    math.CT 2024-11 accept novelty 6.0 of 10

    In semi-abelian varieties of algebras, all Yoneda n-step extension collections between fixed objects admit a set of representatives, so cohomology objects are small.

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