Re: Equivalence of pseudo-limits

Re: Equivalence of pseudo-limits

[Dominic Verity]

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
>

Re: Equivalence of pseudo-limits

[Richard Garner]

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
>>
>
>

Equivalence of pseudo-limits

[Alex Hoffnung]

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