TL;DR
This paper introduces skew-closed categories, extending the concept of closed categories to a laxer context inspired by new examples related to bialgebroids and quantum structures, with foundational lemmas and cocompletion analysis.
Contribution
It reworks Eilenberg-Kelly's theory for lax monoidal contexts, defines skew closed categories, and explores their properties and applications in quantum algebra.
Findings
Defined skew closed categories and proved Yoneda lemmas for them.
Connected skew monoidal and skew closed structures with quantum groupoids.
Analyzed closed cocompletion in the context of skew categories.
Abstract
Spurred by the new examples found by Kornel Szlach\'anyi of a form of lax monoidal category, the author felt the time ripe to publish a reworking of Eilenberg-Kelly's original paper on closed categories appropriate to the laxer context. The new examples are connected with bialgebroids. With Stephen Lack, we have also used the concept to give an alternative definition of quantum category and quantum groupoid. Szlach\'anyi has called the lax notion {\em skew monoidal}. This paper defines {\em skew closed category}, proves Yoneda lemmas for categories enriched over such, and looks at closed cocompletion.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Advanced Topics in Algebra