pullback of locally presentable categories

pullback of locally presentable categories

[Gaucher Philippe]

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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Re: pullback of locally presentable categories

[Steve Lack]

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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Re: pullback of locally presentable categories

[Richard Garner]

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.
>>
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: pullback of locally presentable categories

[Richard Garner]

---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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Re: pullback of locally presentable categories

[Steve Lack]

Dear Richard and others,


On 17/06/10 12:35 AM, "Richard Garner" <rhgg2@hermes.cam.ac.uk> wrote:

>
> ---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;

Yes, I agree.

> - if one of the functors is monadic or comonadic, then the
> pullback is locally presentable.

No, I don't think this is enough for local presentability.

If (as well as both functors being accessible and one being an isofibration)
both functors were continuous then the pullback would be complete and so
locally presentable. If both were cocontinuous, then the pullback would be
cocomplete and so locally presentable. But monadicity of one is not enough.

For example, let A be the full subcategory of Set consisting of all
one-element sets. The inclusion and both A and Set are locally presentable.
Now choose any functor 1-->Set whose image is not a singleton. This is
accessible, and 1 is locally presentable. The pullback of A->Set and 1->Set
is empty, so not locally presentable.

Steve.


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