Dear Luc, Yes, these functors have been studied before: they are what I called `discrete Conduch\'e fibrations' in a paper published in TAC 5 (1999), 1--11. The name derives from a paper by Francois Conduch\'e in C.R. Acad. Sci. Paris 275 (1972), A891--894, where he characterized the exponentiable objects in slices of Cat as those functors F such that any factorization of Ff lifts to a factorization of f, uniquely up to morphisms sent by F to identities. (Actually, Jean-Luc Verdier had found this condition first, but I first encountered it through Conduch\'e's paper.) Since these functors include both fibrations and opfibrations, I got into the habit of calling them `Conduch\'e fibrations'. The discrete case which cropped up in my 1999 paper is obtained by adding the condition that F reflects identities. Peter Johnstone On Mon, 12 Jun 2017, Luc Pellissier wrote:
Dear Category Theorists,
with my adviser Damiano Mazza and his other student Pierre Vial, we are looking for a name – or even better, a reference – for the following kind of functors:
Let C and B be two categories, F : C ---> D a functor satisfying, for all morphisms f:c -> c' in C: - if Ff = g \circ h, then there exists two morphisms k,l such that + f = k \circ l + Fk = g + Fl = h - if Ff = id_a for a certain object a, then f itself is an identity.
These functors arise when applying the Grothendieck construction to relational presheaves: P : B ---> Rel. Indeed, the category of relational presheaves on B is equivalent (through the Grothendieck construction) to a category whose objects are such functors over B.
If anyone could point us in a right direction, it would be much appreciated.
Best,
— Luc
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