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arXiv · 1201.4981

Skew-monoidal categories and bialgebroids

Kornel Szlachanyi
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Abstract

Skew-monoidal categories arise when the associator and the left and right units of a monoidal category are, in a specific way, not invertible. We prove that the closed skew-monoidal structures on the category of right R-modules are precisely the right bialgebroids over the ring R. These skew-monoidal structures induce quotient skew-monoidal structures on the category of R-R-bimodules and this leads to the following generalization: Opmonoidal monads on a monoidal category correspond to skew-monoidal structures with the same unit object which are compatible with the ordinary monoidal structure by means of a natural distributive law. Pursuing a Theorem of Day and Street we also discuss monoidal lax comonads to describe the comodule categories of bialgebroids beyond the flat case.

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BibTeXRIS

Kornel Szlachanyi. 2012-08-31. Skew-monoidal categories and bialgebroids. https://doi.org/10.1016/j.aim.2012.06.027

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