categories with several compositions?

categories with several compositions?

[John Stell]

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



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Re: categories with several compositions?

[Prof. Peter Johnstone]

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


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RE: categories with several compositions?

[John Stell]

Thanks for pointing that out.
I should have been asking for each composition to have its own identity

John 

-----Original Message-----
From: Prof. Peter Johnstone [mailto:P.T.Johnstone@dpmms.cam.ac.uk] 
Sent: 02 February 2011 15:18
To: John Stell
Cc: 'categories@mta.ca'
Subject: Re: categories: categories with several compositions?

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


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RE: categories with several compositions?

[N.Bowler]

> 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 [each composition having its own identity] and where
> (x *i y) *j z = x *i (y *j z) for all i,j
We can certainly relate these structures to things we already understand
reasonably well (though I can't see how to say `they are just wombats').

First of all, here's a way to generate lots of examples. Pick any monoid M,
and let (e_i | i is in I) be any family of invertible elements of M. For
each i in I, define a new operation *i on M by a *i b = a * (e_i^{-1}) * b.
M is a monoid under each *i, with identities the s_i, and (x *i y) *j z = x
*i (y *j z) for all i,j.

In fact, all examples arise in this way. Pick one of the operations, which
we will treat differently from the others: call it * and call its identity
e. So the structure is a monoid with respect to * and e. I'll call this
monoid M. Now pick some other operation *i and let e_i be the identity for
*i.

e_i * (e *i e) = (e_i * e) *i e = e_i *i e = e

and similarly (e *i e) * e_i = e, so e_i is an invertible element of M with
inverse (e *i e). Now note that for any a and b,

a *i b = (a * e) *i (e * b) = a * (e *i e) * b = a * (e_i^{-1}) * b

so that each *i arises as above.

On the basis of this analysis, it looks like the structures you asked about
bear the same sort of relation to monoids with a designated family of
invertible elements that torsors do to groups.

Nathan


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RE: categories with several compositions?

[Francisco Lobo]

Hi!


Your permutability axiom for different compositions is reminiscent of the  interchange law, so I wonder if the structures you mean are the n-fold categories introduced by Charles Ehresmann in "Catégories structurées" cf.

http://www.numdam.org/item?id=ASENS_1963_3_80_4_349_0

which is possibly the first article on higher-order category theory.


An n-fold category C is just a class C equipped with n composition structures (giving composition operations *_0, ..., *_{n-1} on C) that for all i,j<n satisfy the interchange law

(f *_i g) *_j (u *_i v) = (f *_j u) *_i (g *_j v)

whenever f,g,u,v in C are such that both sides are defined. The notion of  composition structure for a class C coincides with the so-called "arrows  only" definition of a category. It consists of a source operation s:C->C, a target operation t:C->C, and a composition operation

*: (C x_{s,t} C) -> C

where (C x_{s,t} C) is the collection of consecutive arrows with respect to the source and target operations (i.e. the vertex of the pullback of s  and t), such that for all f,u,g in C

1. s( s(f) ) = s(f) = t( s(f) )  and  s( t(f) ) = t(f) = t( t(f) )
2. (f * u) * g = f * (u * g)  whenever both sides are defined
3. s(f) * f = f = f * t(f)

The 1st condition says that a fixed point of s is also fixed point of t and vice-versa, and that the range of these operations contains only their  shared fixed points: the objects of the category. The 2nd condition states that * is associative, and the 3rd that the source and target of an arrow f are respectively left and right units for composition with f (so the objects are used as identity arrows). From these axioms it follows that

s(f * u) = s(f)  and t(f * u) = t(u)

as usual. (Note that f*u means "first do f then u" as is common in semigroup theory.) It has already been pointed out that an Eckmann-Hilton argument shows that under the interchange axiom two composition structures i and j will coincide whenever s_i = s_j and t_i = t_j.


Each entity f in an n-fold category C is an arrow in n different ways. This may be written

f :_{n-1}  s_{n-1}(f) -> t_{n-1)(f)
   ...
   :_{0}    s_{0}(f)   -> t_{0}(f)

These are distinct from the cells of (strict) n-categories. The latter notion is often defined inductively using enrichment, but its single-sorted  (or arrows only) counterpart is precisely an n-fold category such that for all f in C

s_i( s_j(f) ) = s_i(f) = s_i( t_j(f) )  and  t_i( s_j(f) ) = t_i(f)  = t_i( t_j(f) )

whenever i<j<n. These conditions ensure that objects of the structure i will also be objects of the structure i+1, etc. In this case the source and target operations make C a globular set.


The theory of n-fold categories was further developed by Ehresmann et al.  in a series of articles called "Multiple Functors". These were written in English and are also available at http://www.numdam.org/


Hope this information is useful.
Francisco


On Wed, 2 Feb 2011 16:11:26 +0000
John Stell <J.G.Stell@leeds.ac.uk> wrote:

> Thanks for pointing that out.
> I should have been asking for each composition to have its own identity
> 
> John 
> 

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RE: categories with several compositions?

[N.Bowler]

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'.


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Re:categories with several compositions?

[N.Bowler]

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


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