Formal category theory in augmented virtual double categories

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 $\mathcal K$.