RE: categories with several compositions?

[Francisco Lobo <f.lobo@cs.man.ac.uk>]

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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