Kan extensions and cartesian monoidal categories

TL;DR

This paper explores how Kan extensions relate to cartesian monoidal categories, generalizing classical results to enriched contexts and providing new insights into algebraic functors and their adjoints.

Contribution

It introduces the notion of cartesian monoidal categories in the enriched setting and generalizes existing results on Kan extensions and algebraic functors.

Findings

Generalization of Day's 1970 result to enriched categories

Introduction of cartesian monoidal categories in the enriched context

Results on left extension along promonoidal modules

Abstract

The existence of adjoints to algebraic functors between categories of models of Lawvere theories follows from finite-product-preservingness surviving left Kan extension. A result along these lines was proved in Appendix 2 of Brian Day's 1970 PhD thesis. His context was categories enriched in a cartesian closed base. A generalization is described here with essentially the same proof. We introduce the notion of cartesian monoidal category in the enriched context. With an advanced viewpoint, we give a result about left extension along a promonoidal module and further related results.