• Korean Journal of Mathematics
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• v.27 no.2
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• pp.535-545
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• 2019

The main idea of this paper is to introduce the notion of $cat^1$-monoids and to prove that the category of crossed semimodules ${\mathcal{C}}=(A,B,{\partial})$ where A is a group is equivalent to the category of $cat^1$-monoids. This is a generalization of the well known equivalence between category of $cat^1$-groups and that of crossed modules over groups.