Skew Monoidal Monoids

Skew monoidal categories are monoidal categories with non-invertible "coherence" morphisms. As shown in a previous article, bialgebroids over a ring R can be characterized as the closed skew monoidal structures on the category Mod-R in which the unit object is R R . This offers a new appro...

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Published in:Communications in algebra 2016-06, Vol.44 (6), p.2368-2388
Main Author: Szlachanyi, K
Format: Article
Language:English
Subjects:
Algebra
Bialgebroid
Categories
Coherence
Equivalence
Monoids
Skew monoidal category
Symmetry
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description Skew monoidal categories are monoidal categories with non-invertible "coherence" morphisms. As shown in a previous article, bialgebroids over a ring R can be characterized as the closed skew monoidal structures on the category Mod-R in which the unit object is R R . This offers a new approach to bialgebroids and Hopf algebroids. Little is known about skew monoidal structures on general categories. In the present article, we study the one-object case: skew monoidal monoids (SMMs). We show that they possess a dual pair of bialgebroids describing the symmetries of the (co)module categories of the SMM. These bialgebroids are submonoids of their own base and are rank 1 free over the base on the source side. We give various equivalent definitions of SMM, study the structure of their (co)module categories, and discuss the possible closed and Hopf structures on a SMM.
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subjects Algebra
Bialgebroid
Categories
Coherence
Equivalence
Monoids
Skew monoidal category
Symmetry
title Skew Monoidal Monoids
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