Discrete fibrations vs. functors into Set

Discrete fibrations vs. functors into Set

[Uwe Egbert Wolter]

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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Re: Discrete fibrations vs. functors into Set

[streicher]

See pp.16-17 of my notes on fibered cats available from the arxiv.
There is an obvious functor Set^(_) : cat^op -> Cat to which one can apply
the Grothendieck construction.

Moreover, a cartesian functor is a fibered equivalence iff all its fibers
are ordinary equivalences. All this is folklore and just documented in my
notes.

Thomas

> 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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Re: Discrete fibrations vs. functors into Set

[Andrée Ehresmann]

Answer to Uwe,

The notion of a discrete fibration, and its equivalence with a functor
to Sets as well as with the action of a category on a set,?? were
initially introduced by Charles Ehresmann in
"Gattungen von Lokalen Strukturen" (Jahresbericht d. DMV Bd. 60 (1957)
S. 4 9 ??? 7 7,
reprinted in
https://ehres.pagesperso-orange.fr/C.E.WORKS_fichiers/Ehresmann_C.-Oeuvres_II_1.pdf
<https://ehres.pagesperso-orange.fr/C.E.WORKS_fichiers/Ehresmann_C.-Oeuvres_II_1.pdf>

The first category you describe is generally called the category of
diagrams into Sets, Diag(Sets); the second one which is isomorphic, is
called the category of (morphisms between) discrete fibrations.

The category Diag(Sets), and more generally the (2-)category Diag(H) for
any category H, have been extensively studied by Ren?? Guitart in some
1970's papers, in particular in
D??compositions et lax-compl??tions, (avec L. Van den Bril), CTGD XVIII,4,
p. 333-407, 1977.
http://archive.numdam.org/article/CTGDC_1977__18_4_333_0.pdf
<http://archive.numdam.org/article/CTGDC_1977__18_4_333_0.pdf>

In the last years, with Alexandre Popoff, C. Agon and M. Andreatta, we
have studied and applied Diag(H) in papers on Math/Music theory, naming
its objects "Poly-Klumpenhouwer-Nets" (or PK-Net) with values in H, for
instance in
"From Nets to PK-Nets: a categorical approach", /Perspective of new
music /54-2, 2016, 5-68.
For other. references, consult my personal site
https://ehres.pagesperso-orange.fr/ <https://ehres.pagesperso-orange.fr/>
Kind regards
Andr??e



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Re: Discrete fibrations vs. functors into Set

[Clemens Berger]

> 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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Re: Discrete fibrations vs. functors into Set

[Walter P Tholen]

In a non-discrete setting these categories and questions have been
considered in part in our recent paper on ???Diagrams, fibrations, and the
decomposition of colimits??? with George Peschke:
arXiv:2006.10890v1[math.CT]
Among other things, the paper extends results obtained by Rene??? Guitart
who considered categories of diagrams in some of his papers in the Cahiers
of the early 1970s.


Regards,
Walter

From: Uwe Egbert Wolter <Uwe.Wolter@uib.no>
Date: December 2, 2020 at 9:37:06 PM EST
To: categories list <categories@mta.ca>
Subject: categories: Discrete fibrations vs. functors into Set
Reply-To: Uwe Egbert Wolter <Uwe.Wolter@uib.no>

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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