Functors arising from a relational Grothendieck construction

Functors arising from a relational Grothendieck construction

[Luc Pellissier]

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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Re: Functors arising from a relational Grothendieck construction

[David Yetter]

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/ ]


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Re: Functors arising from a relational Grothendieck construction

[Thomas Streicher]

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


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Re: Functors arising from a relational Grothendieck construction

[Ross Street]

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



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Re: Functors arising from a relational Grothendieck construction

[Thomas Streicher]

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


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Re: Functors arising from a relational Grothendieck construction

[Luc Pellissier]

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

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Re: Re: Functors arising from a relational Grothendieck construction

[Thomas Streicher]

> a variant of the one in (Nielsen 2004, TAC 12(7), pp 248???261), but
> consider

You mean Niefield and not Nielsen. This paper makes clear the relation
to Giraud-Conduch'e functors. But they just study the Weak
Factorization Lifting Property (WFLP) which in terms of distributors
means that all components of the natural transformation corresponding
to lax preservation of composition are surjective.

Faithful functors to B reflecting identities correspond to "relational
variable sets" on B as described in the Niefield paper. But they are NOT
Conduch'e fibrations since they just validate WFLP and not FLP.

p : E -> B is a Conduch'e fibration (i.e. validatates FLP) iff it is
exponentiable in Cat/B but p validates WLFP iff it is exponentiable in
Cat_f/B (Cor.4.2 in Niefield paper).

What Niefield calls Grothendieck construction is an instance of the
transition from a lax normalised functor from B^op to Dist to a functor to B
(due to Benabou).
But this has nothing to do with what you describe as Grothendieck construction
which rather is chanke of base along a functor B* -> B.

Thomas



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