Re: Re: Functors arising from a relational Grothendieck construction

[Thomas Streicher <streicher@mathematik.tu-darmstadt.de>]

> 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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