Re: pullback of locally presentable categories

[Richard Garner <rhgg2@hermes.cam.ac.uk>]

---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:
>
>>
>> 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).
>>>
....

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