Formal category theory in augmented virtual double categories

TL;DR

This paper develops formal category theory within augmented virtual double categories, formalising key concepts like Kan extensions and Yoneda embeddings, and establishing conditions for small cocompleteness and lifting Yoneda embeddings.

Contribution

It introduces a formal framework for category theory in augmented virtual double categories, formalising classical notions and establishing new conditions for cocompleteness and Yoneda embedding liftings.

Findings

Formalises Kan extension, Yoneda embedding, and other concepts in augmented virtual double categories.

Provides conditions for small cocompleteness of presheaf objects.

Establishes criteria for lifting Yoneda embeddings across functors.

Abstract

In this article we develop formal category theory within augmented virtual double categories. Notably we formalise the classical notions of Kan extension, Yoneda embedding $\text y_A\colon A \to \hat A$, exact square, total category and 'small' cocompletion; the latter in an appropriate sense. Throughout we compare our formalisations to their corresponding $2$-categorical counterparts. Our approach has several advantages. For instance, the structure of augmented virtual double categories naturally allows us to isolate conditions that ensure small cocompleteness of formal presheaf objects $\hat A$. Given a monoidal augmented virtual double category $\mathcal K$ with a Yoneda embedding $\text y_I \colon I \to \hat I$ for its monoidal unit $I$ we prove that, for any 'unital' object $A$ in $\mathcal K$ that has a 'horizontal dual' $A^\circ$, the Yoneda embedding $\text y_A \colon A \to \hat A$ exists if and only if the 'inner hom' $[A^\circ, \hat I]$ exists. This result is a special case of a more general result that, given a functor $F\colon \mathcal K \to \mathcal L$ of augmented virtual double categories, allows a Yoneda embedding in $\mathcal L$ to be "lifted", along a pair of 'universal morphisms' in $\mathcal L$, to a Yoneda embedding in $\mathcal K$.