On the tensor product of modules over skew monoidal actegories

This paper is about skew monoidal tensored V-categories (= skew monoidal hommed V-actegories) and their categories of modules. A module over 〈M,⁎,R〉 is an algebra for the monad T=R⁎ _ on M. We study in detail the skew monoidal structure of MT and construct a skew monoidal forgetful functor MT→ME to...

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Bibliographic Details
Published in:Journal of pure and applied algebra 2017-01, Vol.221 (1), p.185-221
Main Author: Szlachányi, Kornél
Format: Article
Language:English
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Summary:This paper is about skew monoidal tensored V-categories (= skew monoidal hommed V-actegories) and their categories of modules. A module over 〈M,⁎,R〉 is an algebra for the monad T=R⁎ _ on M. We study in detail the skew monoidal structure of MT and construct a skew monoidal forgetful functor MT→ME to the category of E-objects in M where E=M(R,R) is the endomorphism monoid of the unit object R. Then we give conditions for the forgetful functor to be strong monoidal and for the category MT of modules to be monoidal. In formulating these conditions a notion of ‘self-cocomplete’ subcategories of presheaves appears to be useful which provides also some insight into the problem of monoidality of the skew monoidal structures found by Altenkirch, Chapman and Uustalu on functor categories [C,M].
ISSN:0022-4049
1873-1376
DOI:10.1016/j.jpaa.2016.06.003