Re: Final objects in 2-categories

["Jonathan CHICHE 齊正航" <chichejonathan@gmail.com>]

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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