---and I have further noticed that, although the first situation may encompass the other two, this is by the by, since the argument in this first situation is incorrect. I have only shown the vertex of the pullback to be accessible, but not necessarily locally presentable. I don't see any obvious way of rectifying this. However, the argument in the second and third situations is still valid. So to summarise:-- Given a cospan of locally presentable categories and accessible functors, - if one of the functors is an isofibration, then the pullback is accessible; - if one of the functors is monadic or comonadic, then the pullback is locally presentable. Richard --On 16 June 2010 13:57 Richard Garner wrote:

I have just noticed that the first of the three situations I list below actually encompasses the other two, since a monadic or comonadic functor is necessarily an isofibration.

--On 16 June 2010 13:18 Richard Garner wrote:

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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:

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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).

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