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Skew-monoidal semigroup structures on simple rings with several objects. (English) Zbl 0611.18005

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.
Reviewer: R.Guitart

MSC:

18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
20M50 Connections of semigroups with homological algebra and category theory
18D35 Structured objects in a category (MSC2010)
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References:

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