Skew-closed categories
arXiv:1205.6522
Abstract
Spurred by the new examples found by Kornel Szlachányi 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. Szlachányi 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.
Version 2 corrects a mistake in axiom (2.4) noticed by Ignacio Lopez Franco. Only the corrected axiom was used later in the paper so no other consequential change was needed. A few obvious typos have been corrected. Some material on weighted colimits, composite modules and skew-promonoidal categories has been added. Version 3 adds Example 23 and corrects a few typos./