2024/01/03 by Ilia Pirashvili, Pirashvili, Ilia
Computer Science · Mathematics · #Logic, programming, and type systems #Homotopy and Cohomology in Algebraic Topology #Multi-Agent Systems and Negotiation
paper · pdf · doi:10.48550/arxiv.2401.01863
It is a well-known fact that the category Cat(C) of internal categories in a category C has a description in terms of crossed modules, when C=Gr is the category of groups. The proof of this result heavily uses the fact that any split epimorphism decomposes as a semi-direct product. An equivalent statement does not hold in the category Mon of monoids. In a previous work on quadratic algebras, I constructed an internal category in the category of monoids, see Section 6. Based on this construction, this paper will introduce the notion of a crossed semi-bimodule and show that it gives rise to an object in Cat(Mon). I will also relate this new notion to the crossed semi-modules introduced earlier by A. Patchkoria.