From Gs-monoidal to Oplax Cartesian Categories: Constructions and Functorial Completeness

TL;DR

This paper extends gs-monoidal categories to oplax cartesian categories with preorders, explores their relationships with Kleisli and span categories, and proves theorems on Yoneda embeddings and functorial completeness.

Contribution

It introduces oplax cartesian categories by enriching gs-monoidal categories with preorders and analyzes their connections to other categorical structures.

Findings

Gs-monoidal categories naturally arise as Kleisli and span categories.

The paper proves theorems on Yoneda embeddings and functorial completeness.

Functorial completeness induces a completeness result for lax functors to Rel.

Abstract

Originally introduced in the context of the algebraic approach to term graph rewriting, the notion of gs-monoidal category has surfaced a few times under different monikers in the last decades. They can be thought of as symmetric monoidal categories whose arrows are generalised relations, with enough structure to talk about domains and partial functions, but less structure than cartesian bicategories. The aim of this paper is threefold. The first goal is to extend the original definition of gs-monoidality by enriching it with a preorder on arrows, giving rise to what we call oplax cartesian categories. Second, we show that (preorder-enriched) gs-monoidal categories naturally arise both as Kleisli categories and as span categories, and the relation between the resulting formalisms is explored. Finally, we present two theorems concerning Yoneda embeddings on the one hand and functorial completeness on the other, the latter inducing a completeness result also for lax functors from oplax cartesian categories to $\mathbf{Rel}$.