Alternative closed structure on Cat
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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/ ]
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Foltz, Lair, and Kelly prove that those are the only monoidal biclosed structures on CAT in ?http://www.sciencedirect.com/science/article/pii/0022404980900821 Carl Futia -----Original Message----- From: Peter Selinger <selinger@mathstat.dal.ca> To: ondrej.rypacek <ondrej.rypacek@gmail.com> Cc: Categories List <categories@mta.ca> Sent: Thu, Jul 5, 2012 2:17 pm Subject: categories: Re: Alternative closed structure on Cat 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/ ] > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (7)
-
Mark Weber -
Omar Antolín Camarena -
Ondrej Rypacek -
Ross Street -
selinger＠mathstat.dal.ca -
Topos8 -
Vaughan Pratt