Re: Functors arising from a relational Grothendieck construction

[David Yetter <dyetter@ksu.edu>]

Dear Luc,

Is that all you want, or would you like k and l to be unique, or unique up to isomorphism in the sense that there is an isomorphism across the diagonal of the 
commutative in C created by two such pairs making the whole diagram commute?

If so, for unique up to isomorphism, such a functor is called a Conduch\'{e} fibration, and for unique, it is called a discrete Conduch\'{e} fibration.  There is a discussion of these and related notions in the n-Lab article on Conduch\'{e} functors:

https://ncatlab.org/nlab/show/Conduch%C3%A9+functor

Best Thoughts,
David Yetter
Professor of Mathematics
Kansas State University

P.S. Not in reply to the question.  I'd be interested if anyone knows nice constructions of discrete Conduch\'{e} fibrations.  It turns out that a discrete Conduch\'{e} fibration over a category with all arrows monic satisfying the right Ore condition (all cospans complete to commutative squares) with another lifting property,  all functors induced on slice categories split, are the ingredients for a construction of C*-algebras generalizing the popular graph and k-graph C*-algebras of Raeburn, Kumjian, Pask and their school. 





From: Luc Pellissier <luc.pellissier@lipn.univ-paris13.fr>
Sent: Monday, June 12, 2017 4:37 AM
To: categories@mta.ca
Subject: categories: Functors arising from a relational Grothendieck construction
    
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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