Dear Sarah, The first thing that ought to be pointed out is that, given a particular 2-category, there may be several natural double categories extending it, so there is often some choice when deciding upon the appropriate definition. If you could say a few words about the application you have in mind (not necessarily in response to the entire mailing list), I could give a more tailored response. That said, I do think there is a natural answer to your question as stated, which should be straightforward to construct directly. Just as fibred categories are defined as functors that admit certain cartesian lifts of morphisms in the base category, one may define the notion of fibred distributor/profunctor as a cell in the double category of categories and distributors that admits certain cartesian lifts of (hetero)morphisms in the base distributor. (Assuming the functors F and F' in the frame of the cell are fibrations, this condition should amount to asking for the induced functor Col(phi) : Col(P) -> Col(P') between the collages of the distributors to be a fibration; see the attached image <https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXG1hdGhjYWwgRSJdLFswLDEsIlxcbWF0aGNhbCBCIl0sWzEsMCwiXFxtYXRoY2FsIEUnIl0sWzEsMSwiXFxtYXRoY2FsIEInIl0sWzAsMSwiRiIsMl0sWzIsMywiRiciXSxbMCwyLCJQIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiYmFycmVkIn19fV0sWzEsMywiUSIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImJhcnJlZCJ9fX1dLFs0LDUsIlxcdmFycGhpIiwxLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwLCJ0YXJnZXQiOjIwfSwic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoibm9uZSJ9LCJoZWFkIjp7Im5hbWUiOiJub25lIn19fV1d> for my notation.) There is a corresponding definition of natural transformation between fibred distributors, preserving the cartesian lifts (again, these natural transformations may be defined in terms of morphisms of fibrations between the collages.) There is one subtlety. It does not seem clear to me how one should go about composing fibred distributors (in such a way that the composite is also fibred). However, this is not a problem: one may instead form a virtual double category <https://tac.mta.ca/tac/volumes/24/21/24-21abs.html> of fibrations, which obviates the need for composites. (Virtual double categories may look a little intimidating at first glance, but are in many ways simpler than double categories, in the same way that multicategories are in many ways simpler than monoidal categories.) The multiary cells in the virtual double category of fibrations are defined analogously to the ordinary, unary natural transformation between fibred distributors, except that they satisfy a compatibility condition with respect to chains of heteromorphisms, rather than with respect to single heteromorphisms. (I hope that I have provided enough detail that you can reconstruct the definition, but let me know if not.) While this virtual double category doesn't obviously have general composites, it does have nullary composites (i.e. identity loose morphisms), which in particular means that it has an underlying 2-category (which ought to coincide with the 2-category Fib). A few further comments: - If you are interested primarily in fibrations over a fixed base category, the obstruction to composition is lifted, and one does indeed obtain a double category. - Instead of distributors, you could work with the equivalent notion of two-sided discrete fibration, or the equivalent notion of two-sided codiscrete cofibration. - Instead of fibrations, one could work with indexed categories; there is a corresponding notion of indexed distributor. - As far as I am aware, the virtual double category described above does not appear in the literature, though related notions have been considered. For instance, in section 6 of Conspectus of variable categories <https://www.sciencedirect.com/science/article/pii/0022404981900219>, Ross Street constructs a Yoneda structure on the 2-category of C-indexed categories (a Yoneda structure being a notion closely related to double categories); and in Example 1.20 of Algebraic weighted colimits <https://arxiv.org/abs/1304.4079>, Roald Koudenburg defines a double category of strict C-indexed categories. - While it may be possible to construct the desired double category via an abstract construction of a double category of two-sided discrete fibrations in the 2-category Fib, as suggested by Henning, this will only work if fibred distributors can be composed. (Even then, I expect it is substantially simpler to construct the double category directly.) Best, Nathanael [image: image.png] On Sun, 27 Sept 2026 at 22:57, Dukic, S.M. (Sarah) via Categories < categories-list@categories.org.au> wrote:
Dear all,
I am wondering about a double category of fibrations. In particular, whether the usual 2-category *Fib* (fibrations, cartesian functors, and natural transformations) extends to a double category, and if so what the notion of promorphisms and cells would be. I'm thinking that the promorphisms might be something like 2-sided fibrations (as described by e.g. Loregian and Riehl). Has anyone worked this out?
I am very much looking forward to your responses!
All the best,
Sarah _______________________________________________ Categories mailing list -- categories-list@categories.org.au To unsubscribe send an email to categories-list-leave@categories.org.au
