On the tensor product of modules over skew monoidal actegories

详细信息    查看全文

• 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 bbcbe1e654f0607" title="Click to view the MathML source">V$\mathcal{V}$-categories (= skew monoidal hommed bbcbe1e654f0607" title="Click to view the MathML source">V$\mathcal{V}$-actegories) and their categories of modules. A module over 〈M,⁎,R〉$〈\mathcal{M},⁎,R〉$ is an algebra for the monad $T=R⁎\text{\hspace{0.17em}\_}$ on 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 M$\mathcal{M}$ where 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]$[\mathcal{C},\mathcal{M}]$.