Hi Sarah, I think I found what we need using the hints that you received: * Aristote and Tarantino extend improve the result from Carboni et al.: https://arxiv.org/abs/2601.22721v2 The relevant result in theorem 2.13 and the remark just before example 2.14 * Shulman shows in the framed bicategory paper in appendix C how one goes from a proarrow equipment to a fibrant double category. What remains is to fulfil the assumptions of the that theorem: 0. They assume a bicategory K, which will be the 2-category Fib in our case. 1. Finite products, bipullbacks and bicomma objects are presumably constructed similarly as in Cat (discussed earlier in the paper). 2. We need a suitable factorisation system on Fib. They don't have a nice description of that on Cat though. Perhaps the easier route is to apply the Grothendieck construction to two-sided discrete fibrations in Fib and see what comes out. I will have a look if I can obtain a meaningful definition and direct composition operation. Best, Henning On 27/09/2026 15:04, Dukic, S.M. (Sarah) via Categories 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
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