About the cartesian closedness of the category of all small diagrams

About the cartesian closedness of the category of all small diagrams

[gaucher]

Dear categorists,

I have three questions, the first one is a mathematical question, the
second one a bibliographical question and the last one is a speculative
question.

1) Let K be a complete, cocomplete and cartesian closed category.
Consider the category DK of all small diagrams over K. The objects are
all small diagrams F:I-->K from a small category I to K. And a map from
(F:I-->K) to (G:J-->K) is a functor f:I-->J together with a natural
transformation mu:F-->Gf. DK is complete and cocomplete and I would like
to know if it is cartesian closed as well.

2) My question was initially posted in
https://mathoverflow.net/q/266597/24563. From MathOverflow, I now know
that the functor DK-->Cat forgetting K is a fibred category. Since then,
I browsed the Borceux book's chapter devoted to fibred categories (Vol.2
Chap.8). Is there other reference you could recommend me ?

3) I also would like to know what is known about the link between
locally presentability and fibred category. Googling these terms or
looking them up in MathSciNet together gives nothing relevant. Actually,
my motivation is to know whether D(DeltaTop) and D(SimplicialSet) are
locally presentable and cartesian closed (DeltaTop is the category of
Delta-generated spaces, and SimplicialSet the category of simplicial
sets). Therefore I would like to conclude this email with a speculative
question: is there a general philosophy to deduce from the properties of
the fibers of a fibred category E-->B the same property on E ? In the
case of DK-->Cat, the fiber over I is the well-known category of
I-shaped diagrams over K...



Philippe Gaucher.


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Re: About the cartesian closedness of the category of all small diagrams

[Ross Street]

Dear Philippe

On 14 Apr 2017, at 12:13 AM, gaucher <gaucher@irif.fr<mailto:gaucher@irif.fr>> wrote:

1) Let K be a complete, cocomplete and cartesian closed category.
Consider the category DK of all small diagrams over K. The objects are
all small diagrams F:I-->K from a small category I to K. And a map from
(F:I-->K) to (G:J-->K) is a functor f:I-->J together with a natural
transformation mu:F-->Gf. DK is complete and cocomplete and I would like
to know if it is cartesian closed as well.

Yes it is. The internal hom of F : I --> K and G : J --> K is H : [I,J] -->  K defined by
Hr = end over i in I of [Fi,Gri]
where [I,J] is internal hom in Cat and, for h, k in K, [h,k] is internal hom in K.

3) I also would like to know what is known about the link between
locally presentability and fibred category.

I suspect that, if T : K^{op} --> LFP is a functor, where K is locally finitely presentable (lfp)
and LFP is the 2-category of lfp categories and right adjoint functors which preserve filtered colimits,
then the ``Grothendieck fibration construction'' El(T) --> T gives an lfp category El(T).
I can't think of a quick reason but others may think of one, a reference, or a counterexample.

Best wishes,
Ross


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Re: About the cartesian closedness of the category of all small diagrams

[Thomas Streicher]

For a fibration of ccc's over a ccc one knows that the total category
is again ccc and this structure is preserved (could be in Bart Jacob's book).

As to (1) even if K = Set we know that the Set^C are ccc's (actually toposes)
but reindexing in general doesn't preserve the ccc's structure since
in Set^C the exponentials are not computed pointwise (unless C is discrete).
It already goes wrong when C is the ordinal 2.

Thomas


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Re: About the cartesian closedness of the category of all small diagrams

[Thomas Streicher]

As follows from Ross Street's argument a fibration between ccc's may
be cartesian closed even if it is not a fibration of cartesian closed
categories (instantiating K by Set).

Thomas


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Re: About the cartesian closedness of the category of all small diagrams

[Thomas Streicher]

> For a fibration of ccc's over a ccc one knows that the total category
> is again ccc and this structure is preserved (could be in Bart Jacob's book).

This is wrong! Let EE be the free topos (with nno). Then the
fibration Fam(EE) over Set is a fibered ccc but if it were a cartesian
closed functor between toposes then EE would have small sums which it hasn't.

It is not clear at all which fibrations of ccc's over ccc's have the
property that they are ccc-preserving functors between ccc's. It seems
as if such fibrations have to be internally complete. That's what I
definitely overlooked.

Sorry for that,
Thomas




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Re: About the cartesian closedness of the category of all small diagrams

[Ronnie]

This is something of an answer to question 2).  I was very influenced by

Streicher, T. ‘Fibred categories `a la B´enabou’.
http://www.mathematik.tu-darmstadt.de/∼streicher/ (1999) 1–85.
to see how useful fibrations and cofibrations of categories for giving 
the abstract background to constructions that occurred commonly in my 
work with Higgins and Loday on nonabelian colimit constructions in  
homotopy theory. So Sivera and I put these in Appendix B of the book 
"Nonabelian Algebraic Topology" (EMS 2011) (NAT).  Note that Higgins 
pioneered in effect the use of the functor Ob: Groupoids \to Sets as a 
cofibration of categories, (see his Notes on Categroies and Groupoids" 
TAC Reprint) and Section B.3 of NAT gives suitable general results as 
background to say colimits of groupoids, knowing them for sets.

There is more sophisticated material in the above lectures which I have 
not managed to use.  Any advice on this could be useful!

Ronnie Brown

------ Original Message ------
From: "gaucher" <gaucher@irif.fr>
To: categories@mta.ca
Sent: 13/04/2017 15:13:46
Subject: categories: About the cartesian closedness of the category of 
all small diagrams

>Dear categorists,
>
>I have three questions, the first one is a mathematical question, the
>second one a bibliographical question and the last one is a speculative
>question.
>
>1) Let K be a complete, cocomplete and cartesian closed category.
>Consider the category DK of all small diagrams over K. The objects are
>all small diagrams F:I-->K from a small category I to K. And a map from
>(F:I-->K) to (G:J-->K) is a functor f:I-->J together with a natural
>transformation mu:F-->Gf. DK is complete and cocomplete and I would 
>like
>to know if it is cartesian closed as well.
>
>2) My question was initially posted in
>https://mathoverflow.net/q/266597/24563. From MathOverflow, I now know
>that the functor DK-->Cat forgetting K is a fibred category. Since 
>then,
>I browsed the Borceux book's chapter devoted to fibred categories 
>(Vol.2
>Chap.8). Is there other reference you could recommend me ?
>
>3) I also would like to know what is known about the link between
>locally presentability and fibred category. Googling these terms or
>looking them up in MathSciNet together gives nothing relevant. 
>Actually,
>my motivation is to know whether D(DeltaTop) and D(SimplicialSet) are
>locally presentable and cartesian closed (DeltaTop is the category of
>Delta-generated spaces, and SimplicialSet the category of simplicial
>sets). Therefore I would like to conclude this email with a speculative
>question: is there a general philosophy to deduce from the properties 
>of
>the fibers of a fibred category E-->B the same property on E ? In the
>case of DK-->Cat, the fiber over I is the well-known category of
>I-shaped diagrams over K...
>
>
>
>Philippe Gaucher.
>

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