Yoneda lemma and representation theorem for double categories (Q6634625)

This paper studies (vertically) normal lax double functors valued in the weak double category \(\mathbb{C}\mathrm{at}\) of small categories, functors, profunctors and natural transformations, which is referred to as \textit{lax double presheaves}. It is shown that for the theory of double categories they play a similar role as 2-functors valued in \(\mathrm{Cat}\) for 2-categories.\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 2] gives a concise introduction of double category theory.\N\N\item[\S 3] introduces the lax double presheaves as lax double functors \(\mathbb{C}^{\mathrm{op}}\rightarrow\mathbb{C}\mathrm{at}\) with \(\mathbb{C}\) a (strict) double category and \ the weak double category of categories.\N\N\item[\S 4] shows a 2-categorical version of the Yoneda lemma for double categories using the weak double category \(\mathbb{C}\mathrm{at}\). \S 4.1 constructs the Yoneda comparison map, showing that it is 2-natural. \S 4.2 constructs its inverse, establishing the desired Yoneda lemma. \S 4.3 finally deduces that there is a Yoneda embedding from the underlying horizontal 2-category of \ into its 2-category of lax double presheaves \(\mathcal{P} _{\mathbb{C}}\).\N\N\item[\S 5] is concerned with lax double presheaves from a fibrational standpoint, recalling the notion of discrete double fibrations. \S 5.1 recollects the definition and main properties of discrete double fibrations, particularly constructing a 2-category of discrete double fibrations over a double category \(\mathbb{C}\) and showing that it is 2-functorial in \(\mathbb{C}\). \S 5.2 addresses the fibers of a discrete double fibration at both an object and a vertical morphism. \S 5.3 presents a first class of examples of discrete double fibrations given by canonical projections from slice double categories and referred to as \textit{representable discrete double fibrations}.\N\N\item[\S 6] investigates the link between lax double presheaves, establishing (Theorem 6.13) that, given a double category \(\mathbb{C}\), there is a pseudo-natural 2-equivalence, induced by the Grothendieck construction, between the 2-category of lax double presheaves over \(\mathbb{CC}\) and the 2-category of discrete double fibrations over \(\mathbb{C}\).\N\N\item[\S 7] turns to a representation theorem for double categories, establishing (Theorem 7.9) that a lax double presheaf is represented by an object iff the Grothendieck construction has a double terminal object.\N\end{itemize}