Skew-monoidal semigroup structures on simple rings with several objects
Consider the category \({\mathfrak M}(L)^{\otimes}\) defined as the category \({\mathfrak M}(L)\) of all matrices over a ring \(L=L/K\) with identity over K (objects: finite non-empty sets; arrows: matrix multiplication), enriched with the 'skew-monoidal' Kronecker product \(\otimes\). This is almost a monoidal category; what is missing is that \(a\otimes b=(1_ P\otimes b)(a\otimes 1_ U)\)- except if L is commutative - i.e., \(\otimes\) is not functorial. This is the paradigm that the author axiomatizes under the name ''skew-monoidal categories''. He then shows that every skew-monoidal simple category is of the paradigmatic form getting a Wedderburn-Artin like theorem without recourse to Mitchell's rings with several objects. Reviewer's comment: There seems to exist a trade-off between functoriality and associativity, as seen by the reviewer's definition of a tensor product that is functorial but not associative.
scientific article; zbMATH DE number 3987531