Can I add to Steve's excellent summary of the situation the observation that there are circumstances under which the strict pullback of locally presentable categories will also be locally presentable. I know of three such. The first is when one of the functors being pulled back is an isofibration: that is, a functor admitting (necessarily cartesian) liftings of isomorphisms. In this case, it has been observed by Joyal and Street that the strict pullback also enjoys the universal property of the pseudopullback, and hence is locally presentable as in Steve's message. The second situation is when one of the functors in question is monadic. For then the vertex of the strict pullback can be constructed from inserters and equifiers in Acc, and hence is accessible; moreover, it is necessarily complete (since it projects onto a complete category via a limit-creating functor, namely the pullback of the monadic one), and hence locally presentable. The third situation is when one of the functors in question is comonadic: in which case the same argument pertains, but with completeness now substituted by cocompleteness. Note in particular that these last two circumstances include the situation of pulling back a reflective or coreflective subcategory. Richard --On 16 June 2010 10:51 Steve Lack wrote:

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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