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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ] Categories Home Page www.mta.ca Using the list: Articles for posting should be sent to categories@mta.ca Administrative items (subscriptions, address changes etc.) should be sent to [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Luc, by a theorem of B'enabou (and R. Street independently I guess) functors to BB correspond to lax normalized functors from BB^op to Dist, the bicategory of distributors. (see e.g. "Distributors at Work" on my homepage). This equivalence restricts to one between Conduch'e fibrations over BB and normalized pseudofunctors from BB^op to Dist. Replacing Set by 2 (i.e. {\emptyset,{\emptyset}}) and restricting to discrete guys functors from BB^op to Rel are equivalent to Conduch'e fibrations over BB which are faithful and reflect identities. In this case the Conduch'e condition amounts to a unique lifting property of factorization from the base tot he total category. Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On 16 Jun 2017, at 11:16 PM, Thomas Streicher <streicher@mathematik.tu-darmstadt.de<mailto:streicher@mathematik.tu-darmstadt.de>> wrote: by a theorem of B'enabou (and R. Street independently I guess) functors to BB correspond to lax normalized functors from BB^op to Dist, the bicategory of distributors. ============================== Dear Thomas Thanks for the thought, but, no, I either heard it first from Benabou or from Duskin who attributed it to Benabou. At that stage, to my knowledge, the result had not appeared in print. Best wishes, Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Luc, I have noticed that, obviously, 2-valued distributors are not closed under composition in Set-valued distributors. The reason is that in the latter case the existential quantifier in composition of relations is understood in a proof relevant way. So I really don't understand what you mean by Grothendieck construction applied to a presheaf taking vaues in Rel. Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Thank you all for your answers. I didn't know about Conduché functors, and what I am looking at are indeed a relaxed variant where all unicity conditions are dropped. The equivalence arising from the Grothendieck construction I am interested in is a variant of the one in (Nielsen 2004, TAC 12(7), pp 248–261), but considering more general natural transformations between relational presheaves (and not only functional natural transformations). The conditions I have given in my previous email are the weak factorization lifting property (WFLP) and the discreteness of fibers in this article.
Le 17 juin 2017 à 11:27, Thomas Streicher <streicher@mathematik.tu-darmstadt.de> a écrit :
I have noticed that, obviously, 2-valued distributors are not closed under composition in Set-valued distributors. The reason is that in the latter case the existential quantifier in composition of relations is understood in a proof relevant way. So I really don't understand what you mean by Grothendieck construction applied to a presheaf taking vaues in Rel.
Dear Thomas, I use “Grothendieck construction” – very naively, maybe! – as a shorthand for “pullback of a functor along the forgetful functor of a category of pointed objects to the category of base objects”, that is, given a category BB of base objects, and a category BB* of pointed objects, the pullback of a functor C -> BB in the situation BB* | | | v C ---> BB When BB = Set, and BB* is the category of pointed sets, pullbacks of this form are discrete fibrations; when BB = Cat, pullbacks of this form are Grothendieck fibrations; and I am interested in the case BB = Rel, the category of sets and relations. Is that any clearer? If I am using the term too naively, I would be very interested to have a more correct one. — Luc [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (5)
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David Yetter -
Luc Pellissier -
Peter Johnstone -
Ross Street -
Thomas Streicher