Twisted double functors and their applications

In a seminal paper on Yoneda theory for double categories [1], Paré begins by noticing that lax double functors are the right kind of mapping to generalize the Hom functor from categories to double categories. Specifically, the Hom functor on a double category $𝔻$ is a lax double functor ${\mathrm{Hom}}_{𝔻}:{𝔻}^{\mathrm{op}}\times 𝔻\to 𝕊\mathrm{𝖾𝗍}$ that sends a pair of objects $x$ and $y$ in $𝔻$ to the set of tight morphisms $f:x\to y$ between them and acts with a pair of tight morphisms by pre- and post-composition. Alternatively, the Hom functor can be described as a normal lax double functor ${\mathrm{Hom}}_{𝔻}:{𝔻}^{\mathrm{op}}\times 𝔻\to ℂ\mathrm{𝖺𝗍}$ and a Yoneda theory developed along these lines [2].

Properly speaking, this is a tight Yoneda theory for double categories. What, then, could be a loose Yoneda theory? In attempting to construct a loose Hom functor, an obstacle is immediately encountered: the loose Hom should send a pair of objects $x$ and $y$ to the set (or category) of loose morphisms $m:x\begin{array}{c}\mapsto \\ \to \hfill \end{array}y$ and act with a pair of loose morphisms by pre- and post-composition. But that is generally not possible when the receiving double category is $𝕊\mathrm{𝖾𝗍}$ or $ℂ\mathrm{𝖺𝗍}$ since it requires sending loose morphisms to tight morphisms (functions or functors). It is only possible in the special case that $𝔻$ is a strict double category, so that the transpose ${𝔻}^{\top }$ makes sense.

We develop a theory of twisted double functors, a new kind of double functor that sends loose morphisms to tight morphisms and vice versa. That is less straightforward than it might initially seem since twisted functors must allow composition comparisons in both the loose-to-tight and tight-to-loose directions, even though are former are usually invertible. Our prime example is the twisted Hom functor on a double category $𝔻$, a twisted normal lax functor denoted $𝔻(-,=):{𝔻}^{\mathrm{co}}\times 𝔻\looparrowright ℂ\mathrm{𝖺𝗍}$. From this example many others can be constructed by precomposition with ordinary double functors, particularly the twisted representables $𝔻(a,-):𝔻\looparrowright ℂ\mathrm{𝖺𝗍}$ for each object $a\in 𝔻$. Twisted representables encompass the familiar (bi-)indexed categories; for example, when $𝖢$ is a category with finite limits, the twisted representable $𝕊\mathrm{𝗉𝖺𝗇}(𝖢)(1,-):𝕊\mathrm{𝗉𝖺𝗇}(𝖢)\looparrowright ℂ\mathrm{𝖺𝗍}$ contains the canonical self-indexing of $𝖢$.

Having defined twisted double functors, as well as their morphisms and higher morphisms, we introduce loosely discrete double opfibrations as a notion of discrete opfibration internal to $\mathrm{𝐂𝐚𝐭}$. Our main result is an elements construction establishing, for any double category $𝔻$, an equivalence between twisted copresheaves on $𝔻$, defined to be twisted normal lax functors $𝔻\looparrowright ℂ\mathrm{𝖺𝗍}$, and loosely discrete opfibrations over $𝔻$. We also introduce a collage construction establishing an equivalence between twisted bimodules, defined to be twisted normal lax functors ${𝔼}^{\mathrm{co}}\times 𝔻\looparrowright ℂ\mathrm{𝖺𝗍}$, and the double barrels (aka loose bimodules) recently proposed as a key concept of double-operadic systems theory [3]. As a result, we obtain several equivalent descriptions of what we believe to be a significant new concept in double category theory.

The foregoing is work in progress that is nearly complete. In future projects, we hope to utilize the concepts introduced here to develop a loose Yoneda theory for double categories, as well as to axiomatize compact double categories from the perspective of adjunctions between twisted Homs.