categories with several compositions?
Can anyone tell me whether these structures have been studied anywhere? A kind of generalized monoid with two or more compositions *1, *2, etc with a single identity that works for both and where (x *i y) *j z = x *i (y *j z) for all i,j More generally, a kind of category with several compositions: for each object y there is a set Dy and instead of the usual C(x,y) x C(y,z) -> C(x,z) we have Dy -> [C(x,y) x C(y,z), C(x,z)] So you have a family of compositions at each object which "associate with each other" in the manner of the above equation, and where there is a single identity for each object. thanks John Stell [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On Wed, 2 Feb 2011, John Stell wrote:
Can anyone tell me whether these structures have been studied anywhere?
A kind of generalized monoid with two or more compositions *1, *2, etc with a single identity that works for both and where (x *i y) *j z = x *i (y *j z) for all i,j
Substituting the common identity for y in this equation yields x *j z = x *i z, so the compositions all coincide. Similarly in the multiple-object case. Peter Johnstone [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
I've just noticed there was a bit more to your question:
More generally, a kind of category with several compositions: for each object y there is a set Dy and instead of the usual
C(x,y) x C(y,z) -> C(x,z)
we have Dy -> [C(x,y) x C(y,z), C(x,z)]
So you have a family of compositions at each object which "associate with each other" in the manner of the above equation, and where there is a single identity for each object. I assume you would now want an identity at each object for each composition. Then exactly the same argument as in my last email shows that structures like this can be analysed in terms of categories C with a designated family of (assignments to each object a of C of an invertible endomorphism of a).
Nathan PS There's a small typo in my last email. Replace `s_i' by `e_i'. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
I apologise for cluttering the list up further, but I misunderstood your second question, so my answer wasn't quite right. The objects you asked about can be analysed in terms of categories with a designated collection of invertible endomorphisms. Nathan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (3)
-
John Stell -
N.Bowler＠dpmms.cam.ac.uk -
Prof. Peter Johnstone