Re: Equivalence of pseudo-limits

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

In the case singled out by Alex a rather direct proof can
also be given. Let Psd denote the 2-category of 2-functors,
pseudonatural transformations and modifications P -> Cat. The
pseudolimit of a 2-functor F: P -> Cat may be identified with
the hom-category Psd(1,F); and accordingly we have a
pseudolimit 2-functor lim = Psd(1,-): Psd -> Cat, which sends
pseudonatural equivalences F =~ G to equivalences of
categories lim(F) =~ lim(G). The corresponding result for
pseudolimits in other 2-categories now follows by the
Cat-enriched Yoneda lemma.

Richard

--On 11 September 2008 12:27 Dominic Verity wrote:

> Hi Alex,
>
> This property is certainly known to hold for a much larger class of
> 2-categorical limits - the flexible limits, which class includes the classes
> of pseudo, lax and op-lax limits. I believe you will find a proof of this
> result in the Bird, Kelly, Power and Street paper entitled "Flexible Limits
> for 2-Categories". Failing that the Power and Robinson paper "A
> characterisation of PIE limits" probably contains this result is some form.
>
> You can also find explications of the pseudo and lax results in earlier
> works by Street, although obvious candidates for the best one to consult
> elude me at the moment.
>
> The most elementary proof of the flexible limit result starts by observing
> that all flexible limits can be constructed using products, splittings of
> idempotents and a couple of less familiar, exclusively 2-categorical, limits
> called inserters and equifiers. It is then straight forward to verify the
> result you mention for each of these particular limits and then to infer
> that it must therefore hold for all flexible limits.
>
> In the early 1990's Robert Pare introduced a class of limits called the
> persistent limits. These were defined for 2-categories, but made use of his
> double categorical approach to 2-limits. Persistent limits are precisely
> those limits which have the stability property you seek, but with regard to
> a slightly more general class of double categorical diagram transformations
> whose 1-cellular components are all equivalences.
>
> In my thesis (1992), I prove that the class of flexible limits introduced by
> Bird, Kelly, Power and Street is identical to Pare's class of persistent
> limits - thus closing the circle and demonstrating that the flexible limits
> are in a natural sense the largest class of 2-limits which have this
> property.
>
> Regards
>
> Dominic Verity
>
> 2008/9/10 Alex Hoffnung <alex@math.ucr.edu>
>
>> Hi all,
>>
>> Given an indexing 2-category J, a pair of parallel functors
>> F,G : J ----> CAT, and a natural equivalence  f : F ==> G,
>> the pseudo-limits of F and G should be equivalent.
>>
>> I am trying to find out what paper, if any, I can cite for this theorem.
>>  Or
>> maybe this is just the type of thing that nobody has bothered to write
>> down.
>> Any help would be appreciated.
>>
>> best,
>> Alex Hoffnung
>>
>
>