Biclosed bicategories: localisation of convolution

TL;DR

This paper summarizes key results on constructing biclosed categories, providing conditions for extending monoidal structures via dense functors, with a focus on localization and universal properties.

Contribution

It offers new sufficient conditions for extending (pro)monoidal structures along dense functors to cocomplete categories, using localization techniques.

Findings

Conditions for extending monoidal structures are established.

Localization via dense functors is characterized.

Universal property of the extension is described.

Abstract

We give a summary (without proofs) of the main results in the author's thesis entitled ``Construction of biclosed categories'' (University of New South Wales, Australia, 1970). This summary is reprinted directly from Report 81-0030 of the School of Mathematics and Physics, Macquarie University, April 1981. In particular, it gives sufficient conditions for existence of an extension of a (pro)monoidal category structure along a given dense functor to a cocomplete category. The two basic procedures used in the proof turn out to be special cases of the final result, the two respective dense functors then being the Yoneda embedding followed by a localisation. The final result has a standard universal property based on left Kan extension of (pro)monoidal functors along the given dense functor, however this property is not stated explicitly here.