Re: Functors arising from a relational Grothendieck construction

[Luc Pellissier <luc.pellissier@lipn.univ-paris13.fr>]

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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