Gray tensor product

Gray tensor product

[Mike Stay]

Has anyone "unpacked" the meaning of the Gray tensor product of strict
2-categories?  I'm looking for something like "the Gray product C
tensor D is the 2-category whose
- objects are pairs (c,d)
- morphisms are ...
- 2-morphisms are ..."

My higher-category-fu isn't strong enough yet to grok the implicit definition
    2Cat(C tensor D, E) ~= 2Cat(C, Ps(D,E)),
where Ps(D,E) is the 2-category of 2-functors D->E, pseudonatural
transformations, and modifications.
-- 
Mike Stay - metaweta@gmail.com
http://www.cs.auckland.ac.nz/~mike
http://reperiendi.wordpress.com


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Re: Gray tensor product

[Mike Stay]

On Fri, May 27, 2011 at 10:16 AM, Finn Lawler <flawler@cs.tcd.ie> wrote:
> Mike Stay wrote:
>> Has anyone "unpacked" the meaning of the Gray tensor product of strict
>> 2-categories?  I'm looking for something like "the Gray product C
>> tensor D is the 2-category whose
>> - objects are pairs (c,d)
>> - morphisms are ...
>> - 2-morphisms are ..."
>
> Gray's book Formal Category Theory: Adjointness for 2-Categories (Springer LNM 391) is the original reference.  Theorem 4.9 constructs the 'lax' tensor product.  A good reference for the 'pseudo' tensor product is chapter 5 of Nick Gurski's 2007 Ph.D. thesis 'An algebraic theory of tricategories'.

Thanks to everyone!  Gurski's exposition is exactly what I was looking for.
-- 
Mike Stay - metaweta@gmail.com
http://www.cs.auckland.ac.nz/~mike
http://reperiendi.wordpress.com


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Re: Gray tensor product

[Ross Street]

Dear Mike

Perhaps my notes on the subject at

http://www.math.mq.edu.au/~street/GrayTensor.pdf

will help a bit.

Ross
www.math.mq.edu.au/~street
Sent from my iPad

On 27/05/2011, at 1:57 AM, Mike Stay <metaweta@gmail.com> wrote:

> Has anyone "unpacked" the meaning of the Gray tensor product of strict
> 2-categories?  I'm looking for something like "the Gray product C
> tensor D is the 2-category whose
> - objects are pairs (c,d)
> - morphisms are ...
> - 2-morphisms are ..."
> 
> My higher-category-fu isn't strong enough yet to grok the implicit definition
>    2Cat(C tensor D, E) ~= 2Cat(C, Ps(D,E)),
> where Ps(D,E) is the 2-category of 2-functors D->E, pseudonatural
> transformations, and modifications.
> -- 
> Mike Stay - metaweta@gmail.com
> http://www.cs.auckland.ac.nz/~mike
> http://reperiendi.wordpress.com
> 
> 


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Re: Gray tensor product

[Finn Lawler]

Mike Stay wrote:
> Has anyone "unpacked" the meaning of the Gray tensor product of strict
> 2-categories?  I'm looking for something like "the Gray product C
> tensor D is the 2-category whose
> - objects are pairs (c,d)
> - morphisms are ...
> - 2-morphisms are ..."

Gray's book Formal Category Theory: Adjointness for 2-Categories (Springer LNM 391) is the original reference.  Theorem 4.9 constructs the 'lax' tensor product.  A good reference for the 'pseudo' tensor product is chapter 5 of Nick Gurski's 2007 Ph.D. thesis 'An algebraic theory of tricategories'.


FL

-- 
Finn Lawler


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Re: Gray tensor product

[Ronnie Brown]

There are quite a few paper on this particularly by Sjoerd  Crans. Part
of the problem is that a definition of tensor product will always be in
terms of a structure generated by c \tens d subject to certain
relations, so it will not necessarily be easy to get at the elements of
the tensor product. Hence the exponential law which you quote shows that
it is more explicit to give the elements of Ps(D,E), and these are in
each dimension families of functions satisfying certain conditions.

It is actually easier, I feel,  to do this cubically, as is done
explicitly for the omega-case in Section 10 of
Al-Agl, F.~A., Brown, R. and Steiner, R.
{Multiple categories: the equivalence of a globular and a  cubical
approach}.
{Adv. Math.} \textbf{170}~(1) (2002) 71--118.
which gives an explicit description of n-fold left homotopy in that
cubical context, and so in principle a translation to the globular case.

In the omega-groupoid case it is possible to say more: see the book
project, and references there,  advertised on
http://pages.bangor.ac.uk/~mas010/nonab-a-t.html


Ronnie Brown




On 27/05/2011 00:57, Mike Stay wrote:
> Has anyone "unpacked" the meaning of the Gray tensor product of strict
> 2-categories?  I'm looking for something like "the Gray product C
> tensor D is the 2-category whose
> - objects are pairs (c,d)
> - morphisms are ...
> - 2-morphisms are ..."
>
> My higher-category-fu isn't strong enough yet to grok the implicit definition
>      2Cat(C tensor D, E) ~= 2Cat(C, Ps(D,E)),
> where Ps(D,E) is the 2-category of 2-functors D->E, pseudonatural
> transformations, and modifications.


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