Re: categories with several compositions?
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On Wed, 02 Feb 2011 09:20:11 AM EST, John Stell <J.G.Stell@leeds.ac.uk> asked:
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 ...
What am I missing when I think I see, using y = 1 (an identity map, as suggested by the additional remarks Stell makes below), that x *j z = (x *i 1) *j z = x *i (1 *j z) = x *i z ? Cheers, -- Fred --
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/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (4)
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Francisco Lobo -
Fred E.J. Linton -
John Stell -
N.Bowler＠dpmms.cam.ac.uk