Re: pullback of locally presentable categories

[Steve Lack <s.lack@uws.edu.au>]

Dear Philippe,

If you mean the literal pullback then no. But perhaps you mean the
pseudopullback or the iso-comma objects (where one askes for commutativity
of the square only up to isomorphism) in which case things look better.

Greg Bird proved in his 1984 thesis that:

(i) the 2-category of locally presentable categories, left adjoint functors,
and natural transformations has all flexible limits;

(ii) the 2-category of locally presentable categories, right adjoint
functors, and natural transformations has all flexible limits.

These flexible limits include pseudopullbacks and iso-comma objects. They
also imply the existence of all bilimits (where everything is done up to
isomorphism of 1-cells, and the universal property involves just a
pseudonatural equivalence). In the thesis, flexible limits were called
"limits of retract type".

Makkai and Pare proved in their monograph on accessible categories that:

(iii) the 2-category of accessible categories, accessible functors, and
natural transformations has bilimits.

Bilimits were there called "Limits" (capital L).

Adamek and Rosicky also consider limits of accessible categories. Their
approach is different to Makkai and Pare, which allows them to consider
flexible limits rather than bilimits. They consider the same 2-category Acc
as Makkai and Pare, and show that for various types of flexible limit,
if a diagram lives in Acc, then its (flexible) limit in Cat actually lies in
Acc. They do not show that it is the limit in Acc (and since Acc is not a
full sub-2-category of Cat this is not automatic) but this is not too hard
to do.

In fact if this detail is filled in then the existence of all flexible
limits in Acc would follow once one proved that Acc is closed in Cat under
the splitting of idempotents: if A is an accessible category and
e:A->A an accessible idempotent functor, then the splitting B of e is an
accessible category and the functors r:A->B and i:B->A are accesible
functors. I guess this is probably true, but haven't thought too much about
it.

Regards,

Steve Lack.


On 14/06/10 11:48 PM, "Gaucher Philippe" <Philippe.Gaucher@pps.jussieu.fr>
wrote:

> Dear categorists,
>
> Is a pullback of locally presentable categories locally presentable ? All
> involved functors are accessible in my case.
>
> Thanks in advance. pg.
>


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