Final objects in 2-categories

Final objects in 2-categories

[Adam Gal]

Hi all,

Has someone studied the notion of final objects in a 2-category? I
know that we can define it using the classifying space, and that in
this sense Quillen's theorem A holds and tells us this is equivalent
to some fibers being contractible.
This seems to be a bit too coarse though. For instance in our recent
paper (with E. Gal) we wanted to prove that something is final, and
what we showed is that these fibers have an initial and final object.
So they definitely have contractible classifying spaces, but it seems
that we can say something more precise than this.

The question is if this fits into some finer notion of final object in
2-categories which has been studied.

Thanks,
Adam


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Re: Final objects in 2-categories

[David Roberts]

Hi Adam,

There are three definitions one could think of for a final object x in a
2-category C:

1) C(a,x) is equivalent to * for all a (the usual notion)

2) C(a,x) has a terminal object for all a (and perhaps a condition like the
obvious functors induced by b -> a preserve it). I vaguely recall seeing
this before, but not where. Perhaps others on the list know.

3) C(a,x) has a contractible nerve for all a.

Clearly 1)=>2)=>3), but it depends on your application as to what you use.
Given what you wrote, the last two don't seem inappropriate.

Best regards,
David


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Re: Final objects in 2-categories

[Jonathan CHICHE 齊正航]

Dear Adam, 

I don't have much time right now and have not read your paper carefully. However, to elaborate on David Roberts's answer, in my work about the homotopy theory of 2-categories, the property which I have found the most useful (and which shows up in many natural circonstances) is the following. Given a 2-category A, let us say that an object z of A has a terminal object if Hom(a,z) has a terminal object for every object a of A. This terminology was suggested to me by Jean Bénabou. It is of course compatible with the usual definition if the 2-category happens to be Cat. It can be shown that, if a small 2-category A admits such an object, then the map from A to the point is a weak equivalence, i.e. its nerve is a simplicial weak equivalence. I have some papers around related stuff, which I could communicate when they are in their final version. 

This property of having such an object has already been considered in the literature, for instance in Bunge's "Coherent extensions and relational algebras", Jay's "Local adjunctions" and Betti-Power's "On local adjointness of distributive bicategories". Thanks to Steve Lack for having pointed out my attention to this last paper. You may also find useful this question and its answers on Math Overflow: http://mathoverflow.net/questions/160765/whats-an-initial-object-in-a-poset-enriched-category/161118#161118. 

Best wishes, 

Jonathan

Le 24 juil. 2014 à 16:52, Adam Gal a écrit :

> Hi all,
> 
> Has someone studied the notion of final objects in a 2-category? I
> know that we can define it using the classifying space, and that in
> this sense Quillen's theorem A holds and tells us this is equivalent
> to some fibers being contractible.
> This seems to be a bit too coarse though. For instance in our recent
> paper (with E. Gal) we wanted to prove that something is final, and
> what we showed is that these fibers have an initial and final object.
> So they definitely have contractible classifying spaces, but it seems
> that we can say something more precise than this.
> 
> The question is if this fits into some finer notion of final object in
> 2-categories which has been studied.
> 
> Thanks,
> Adam


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Re: Final objects in 2-categories

[Eduardo J. Dubuc]

Perhaps an adequate definition should be that of being a pseudocolimit
of the identity (or bicolimit ?). Or that the pseudocolimit (or
bicolimit ?) of any 2-diagram X_a indexed by C should be A_x.


On 25/07/14 23:04, David Roberts wrote:
> Hi Adam,
>
> There are three definitions one could think of for a final object x in a
> 2-category C:
>
> 1) C(a,x) is equivalent to * for all a (the usual notion)
>
> 2) C(a,x) has a terminal object for all a (and perhaps a condition like the
> obvious functors induced by b ->  a preserve it). I vaguely recall seeing
> this before, but not where. Perhaps others on the list know.
>
> 3) C(a,x) has a contractible nerve for all a.
>
> Clearly 1)=>2)=>3), but it depends on your application as to what you use.
> Given what you wrote, the last two don't seem inappropriate.
>
> Best regards,
> David
>



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Re: Final objects in 2-categories

[Adam Gal]

Dear Jonathan and David,

Thanks for your answers  and the references. This notion is what
appeared naturally in our setting so it's nice to know that people
have worked on it. We would very much like to hear more about what you
(Jonathan) are working on in this respect.

Sorry for the late letter but I had very bad internet access in the last week.
Best,
Adam

On Sat, Jul 26, 2014 at 3:40 PM, Jonathan CHICHE 齊正航
<chichejonathan@gmail.com> wrote:
> Dear Adam,
>
> I don't have much time right now and have not read your paper carefully. However, to elaborate on David Roberts's answer, in my work about the homotopy theory of 2-categories, the property which I have found the most useful  (and which shows up in many natural circonstances) is the following. Given  a 2-category A, let us say that an object z of A has a terminal object if Hom(a,z) has a terminal object for every object a of A. This terminology was suggested to me by Jean Bénabou. It is of course compatible with the  usual definition if the 2-category happens to be Cat. It can be shown that, if a small 2-category A admits such an object, then the map from A to the  point is a weak equivalence, i.e. its nerve is a simplicial weak equivalence. I have some papers around related stuff, which I could communicate when  they are in their final version.
>
...

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