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From: Steve Lack <steve.lack@mq.edu.au>
To: Jamie Vicary <jamievicary@gmail.com>
Cc: Categories list <categories@mta.ca>
Subject: Re: The tricategory of bicategories
Date: Mon, 24 Oct 2011 09:59:04 +1100	[thread overview]
Message-ID: <E1RIJdI-00027x-1i@mlist.mta.ca> (raw)
In-Reply-To: <CANr23v1ppcxeT89bNopp=SpX2xEyyru05cgQnLbOgvKFf4qvhg@mail.gmail.com>

Dear Jamie,

I agree that there is no canonical choice of horizontal composition of
pseudonatural transformations, but that the various possible choices
are, in a suitable sense, equivalent. 

Whether or not you should be bothered by this I can't really say. But 
perhaps it's worth pointing out that there are various different possible
descriptions of the structure of weak 3-category, and not all of them include
a chosen horizontal composition of pseudonatural transformations. Some,
particularly, the simplicial approaches, include *no* choices of compositions.
Others include some choices of composition, but not this particular one. 
For example, the notion of Gray-category does include chosen composition
of 1-cells, and vertical composition of 2-cells, but does not include a chosen
horizontal composition of 2-cells. 

Best wishes,

Steve Lack.

 
On 21/10/2011, at 9:17 PM, Jamie Vicary wrote:

> Dear categorists,
> 
> Suppose you have categories A, B, C, and functors S,S': A-->B, T,T':
> B-->C, and natural transformations alpha: S==>S', beta: T==>T'.
> Suppose we want to see these as part of a 2-category of categories;
> then we had better know the horizontal composite of alpha and beta.
> There are two possible ways to evaluate this composite: as the natural
> transformation having components beta_{S'X}.T(alpha_X), and as the
> natural transformation having components T'(alpha_X).beta_{S(X)}. But
> these are equal, since beta is a natural transformation. So we have no
> difficulty uniquely defining our horizontal composite, and obtaining a
> canonical 2-category of categories.
> 
> But now suppose that A, B, C are bicategories, S,S',T,T' are
> pseudofunctors, and alpha and beta are pseudonatural transformations.
> Then the two possible definitions for the horizontal composite of
> alpha and beta will not necessarily be equal, although of course they
> will be related by an invertible modification. But then we have a
> problem forming the tricategory of bicategories, pseudofunctors,
> pseudonatural transformations and modifications: there is no longer a
> canonical choice available for horizontal composition of pseudonatural
> transformations.
> 
> Presumably this choice can be made, and a tricategory is the result,
> and different choices yield equivalent tricategories. But it bothers
> me that there seems to be no canonical tricategory of bicategories.
> Should it? Or is my reasoning flawed?
> 
> Jamie.
> 



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  reply	other threads:[~2011-10-23 22:59 UTC|newest]

Thread overview: 3+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2011-10-21 10:17 Jamie Vicary
2011-10-23 22:59 ` Steve Lack [this message]
2011-10-24 23:13   ` Richard Garner

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