Alternative closed structure on Cat

Alternative closed structure on Cat

[Ondrej Rypacek]

Dear All

Is there a name for and what is known about the tensor product of ordinary categories which looks like the underlying 1-category of Gray's (Gray) tensor product? 
Explicitly, roughly: 
- objects of C \otimes D are pairs (c,d) , c object of C , d an object of D
- arrows alternating lists of arrows from C and D, i.e. they are generated by 
 	(f,d) : (c,d) -> (c',d) for f : c -> c',
 	(c,g) : (c,d) -> (c,d') for g : d -> d'

and modulo the equations:   (f',d) . (f, d) = (f'f, d), (c,g') . (c,g) = (c,g'g), and identities, left and right unit laws and associativity in each component separately.
	

And is the category of categories with respect to this tensor closed ? 


Thank you!
Ondrej



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Re: Alternative closed structure on Cat

[Peter Selinger]

Dear Ondrej,

Power and Robinson state in [1, Section 2] that the tensor you
describe is indeed part of a monoidal closed structure: the function
category has as objects all functors, and as morphisms the (not
necessarily natural) transformations. Moreover, Power and Robinson
state that this is the unique other symmetric monoidal closed
structure on Cat, i.e., there are no others besides this one and the
"usual" one. I have never seen a proof of this last fact.

[1] J. Power and E. Robinson. "Premonoidal categories and notions of
computation." Mathematical Structures in Computer Science 7(5):
445-452, 1997. (www.eecs.qmul.ac.uk/~edmundr/pubs/mscs97/premoncat.ps)

-- Peter

Ondrej Rypacek wrote:
>
> Dear All
>
> Is there a name for and what is known about the tensor product of =
> ordinary categories which looks like the underlying 1-category of Gray's =
> (Gray) tensor product?=20
> Explicitly, roughly:=20
> - objects of C \otimes D are pairs (c,d) , c object of C , d an object =
> of D
> - arrows alternating lists of arrows from C and D, i.e. they are =
> generated by=20
>  	(f,d) : (c,d) -> (c',d) for f : c -> c',
>  	(c,g) : (c,d) -> (c,d') for g : d -> d'
>
> and modulo the equations:   (f',d) . (f, d) =3D (f'f, d), (c,g') . (c,g) =
> =3D (c,g'g), and identities, left and right unit laws and associativity =
> in each component separately.
> =09
>
> And is the category of categories with respect to this tensor closed ?=20=
>
>
>
> Thank you!
> Ondrej
>
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>



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Re: Alternative closed structure on Cat

[Ross Street]

On 05/07/2012, at 11:38 PM, Peter Selinger wrote:

> Power and Robinson state in [1, Section 2] that the tensor you
> describe is indeed part of a monoidal closed structure: the function
> category has as objects all functors, and as morphisms the (not
> necessarily natural) transformations. Moreover, Power and Robinson
> state that this is the unique other symmetric monoidal closed
> structure on Cat, i.e., there are no others besides this one and the
> "usual" one. I have never seen a proof of this last fact.
> 
> [1] J. Power and E. Robinson. "Premonoidal categories and notions of
> computation." Mathematical Structures in Computer Science 7(5):
> 445-452, 1997. (www.eecs.qmul.ac.uk/~edmundr/pubs/mscs97/premoncat.ps)

Yes, this is what I call the "funny" tensor product on Cat.

Categories enriched in Cat with the funny tensor product are
called "sesquicategories": they are less than 2-categories as they
have whiskering but only ambiguous horizontal composition of 2-cells.

There is a bit of literature on all this. For example, it is mentioned in

Categorical structures, Handbook of Algebra Volume 1 (editor M. Hazewinkel; Elsevier Science, Amsterdam 1996; ISBN 0 444 82212 7) 529-577.

and/or

Higher categories, strings, cubes and simplex equations, Applied Categorical Structures 3 (1995) 29- 77 & 303; MR96b:18009.

As to finding all the symmetric monoidal closed structures on a locally finitely presentable category, 
the object-in-two-categories technique is provided by

F. Foltz, GM. Kelly and C. Lair, Algebraic categories with few monoidal biclosed structures or none, 
J. Pure and Applied Algebra 17 (1980) 171–177.

Perhaps they even give the Cat example.

Best wishes,
Ross

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Re: Alternative closed structure on Cat

[Mark Weber]

Dear Ondrej and Peter

The fact to which Peter referred, that the tensor product in question

> is the unique other symmetric monoidal closed
> structure on Cat

was proved in the paper
[1] F. Foltz, G.M.Kelly, and C. Lair, "Algebraic categories with few biclosed monoidal structures or none", JPAA 17:171-177, 1980

As for the name, this tensor product has been called the "funny tensor product" by some authors. But as I argued in my paper

[2] Free products of higher operad algebras
http://arxiv.org/abs/0909.4722

in which such a tensor product is defined for any structure definable by a "normalised higher operad" in the sense of Batanin, the name "free product" is a better choice of terminology.

Mark Weber

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Re: Alternative closed structure on Cat

[Ondrej Rypacek]

Thanks for all answers and references. It's much appreciated!

Before I tuck in, am I likely to find a definition in terms of a colimit? 

Ondrej



On 6 Jul 2012, at 00:42, Mark Weber wrote:

> Dear Ondrej and Peter
> 
> The fact to which Peter referred, that the tensor product in question
> 
>> is the unique other symmetric monoidal closed
>> structure on Cat
> 
> was proved in the paper
> [1] F. Foltz, G.M.Kelly, and C. Lair, "Algebraic categories with few biclosed monoidal structures or none", JPAA 17:171-177, 1980
> 
> As for the name, this tensor product has been called the "funny tensor product" by some authors. But as I argued in my paper
> 
> [2] Free products of higher operad algebras
> http://arxiv.org/abs/0909.4722
> 
> in which such a tensor product is defined for any structure definable by a "normalised higher operad" in the sense of Batanin, the name "free product" is a better choice of terminology.
> 
> Mark Weber



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Re: Alternative closed structure on Cat

[Omar Antolín Camarena]

Isn't this funny tensor product of C and D the pushout of the
inclusions of (Ob C x Ob D) into (C x Ob D) and (Ob C x D) --where Ob
C is the discrete category with the same objects as C?

On Fri, Jul 6, 2012 at 5:13 AM, Ondrej Rypacek <ondrej.rypacek@gmail.com> wrote:
> Thanks for all answers and references. It's much appreciated!
>
> Before I tuck in, am I likely to find a definition in terms of a colimit?
>
> Ondrej
>
>
>
> On 6 Jul 2012, at 00:42, Mark Weber wrote:
>
>> Dear Ondrej and Peter
>>
>> The fact to which Peter referred, that the tensor product in question
>>
>>> is the unique other symmetric monoidal closed
>>> structure on Cat
>>
>> was proved in the paper
>> [1] F. Foltz, G.M.Kelly, and C. Lair, "Algebraic categories with few biclosed monoidal structures or none", JPAA 17:171-177, 1980
>>
>> As for the name, this tensor product has been called the "funny tensor product" by some authors. But as I argued in my paper
>>
>> [2] Free products of higher operad algebras
>> http://arxiv.org/abs/0909.4722
>>
>> in which such a tensor product is defined for any structure definable by  a "normalised higher operad" in the sense of Batanin, the name "free product" is a better choice of terminology.
>>
>> Mark Weber
>
>

-- 
Omar Antolín Camarena


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Re: Re: Alternative closed structure on Cat

[Ondrej Rypacek]

Shoot, I overlooked that. 
Thanks, Ondrej


On 6 Jul 2012, at 14:00, Jeff Egger <jeffegger@yahoo.ca> wrote:

>> Before I tuck in, am I likely to find a definition in terms of a colimit? 
> 
> 
> Sure.  It's the pushout of C_0xD <--- C_0xD_0 ---> CxD_0, where C_0 denotes 
> the discrete category with the same class of objects as C.  Or did you have 
> something else in mind?  Mark Weber's paper is likely contain a much more 
> profound answer to this question.
> 
> Cheers,
> Jeff.


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Re: Alternative closed structure on Cat

[Vaughan Pratt]

On 7/6/2012 2:13 AM, Ondrej Rypacek wrote:
  > Before I tuck in, am I likely to find a definition in terms of a colimit?

Not as likely as after you wake up the next morning.  :)

Vaughan


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