Re: Discrete fibrations vs. functors into Set

[Clemens Berger <Clemens.BERGER@univ-cotedazur.fr>]

> Dear Uwe,
>
>   I believe that the answer to questions 2. & 3. is yes. It is
> likewise that both claims follow quite formally from the classical
> ``local'' correspondence between discrete fibrations/opfibrations and
> contra/covariant functors on small categories. This correspondence is
> of course a special case of Grothendieck's correspondence between
> between fibrations/opfibrations and contra/covariant pseudofunctors.
>
>   In the discrete case you are dealing with there is an additional
> link with Lawvere's comprehension schemes and Street-Walter's
> comprehensive factorisation of a functor. I thematized this in joint
> work with Ralph Kaufmann.
>
>   The morphisms in your first category just express the fact that the
> functor P:Cat->CAT which takes a small category A to its diagram
> category PA=[A,Set] comes equipped, for each f:A->B in Cat, with an
> adjoint pair f_!:PA<=>PB:f^* given by left Kan extension along f,
> resp. precomposition with f. Composition in your first category
> amounts to composition of these adjunctions.
>
>   Each of the PA has a terminal object *_A, and one can check that the
> functor Cat/B->PB which takes f:A->B to f_!(*_A) has a fully faithful
> right adjoint. This is closely related to Lawvere's "comprehension
> schemes".
>
>   The unit of the latter adjunction replaces a general f:A->B by a
> discrete opfibration. You get actually the factorisation of f into an
> initial functor followed by a discrete opfibration (this is dual to
> Walter-Street's factorisation into final functor followed by discrete
> fibration), explicitly:
>
> A->el(f_!(*_A))->B
>
> where the category in the middle is the category of elements of the
> diagram f_!(*_A). The functor f is a discrete opfibration iff the
> first arrow is invertible, and an inital functor iff the second arrow
> is invertible. From this you can derive an equivalence between your
> two categories (answering question 3).
>
> The orthogonal factorisation systems (initial, discrete opfibrations)
> and (final, discrete fibration) have nice stability properties,
> nowadays running under the denomination ``modality''.
>
> All the best,
>               Clemens.
>
>   Le 2020-12-02 14:53, Uwe Egbert Wolter a ??crit??:
>> Dear all,
>>
>> We consider two categories. The first category with objects given by a
>> small category B and a functor F:B->Set and morphisms
>> (H,alpha):(B,F)->(C,G) given by a functor H:B->C and a natural
>> transformation alpha:F=>H;G. The second category has as objects
>> discrete
>> fibrations p:E->B and morphisms (H,phi):(E,p)->(D,q:D->C) are given by
>> functors H:B->C and phi:E->D such that phi;q=p;H.
>>
>> 1. Are there any "standard" terms and notations for these categories?
>> 2. For both categories we do have projection functors into Cat! Are
>> these functors kind of (op)fibrations?
>> 3. We know that the Grothendieck construction establishes equivalences
>> between corresponding fibers of the two projection functors into Cat.
>> Do
>> these fiber-wise equivalences extend to an equivalence between the two
>> categories?
>>
>> Thanks
>>
>> Uwe
>>

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