Re: pullback of locally presentable categories

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

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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