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/ ]