On the tensor product of modules over skew monoidal actegories

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• 作者：Korné ; l Szlachá ; nyi ^{szlachanyi.kornel@wigner.mta.hu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：16T10 ; 18D10
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：221
• 期：1
• 页码：185-221
• 全文大小：659 K

文摘

This paper is about skew monoidal tensored d6c3b0fbac26c027bbcbe1e654f0607" title="Click to view the MathML source">V$\mathcal{V}$-categories (= skew monoidal hommed d6c3b0fbac26c027bbcbe1e654f0607" title="Click to view the MathML source">V$\mathcal{V}$-actegories) and their categories of modules. A module over 6d235802f569de9a54c142f870" title="Click to view the MathML source">〈M,⁎,R〉$〈\mathcal{M},⁎,R〉$ is an algebra for the monad $T=R⁎\text{\hspace{0.17em}\_}$ on d6e950da3fd6cc5afc255fa0d2aebe" title="Click to view the MathML source">M$\mathcal{M}$. We study in detail the skew monoidal structure of M^T${\mathcal{M}}^T$ and construct a skew monoidal forgetful functor ${\mathcal{M}}^T\to {}_E\mathcal{M}$ to the category of E  -objects in d6e950da3fd6cc5afc255fa0d2aebe" title="Click to view the MathML source">M$\mathcal{M}$ where 6d6337d082237d" title="Click to view the MathML source">E=M(R,R)$E=\mathcal{M}(R,R)$ 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 M^T${\mathcal{M}}^T$ 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]$[\mathcal{C},\mathcal{M}]$.