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

[Ross Street <ross.street@mq.edu.au>]

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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