weak multiple categories

weak multiple categories

[Ronald Brown]

Just been away so it was interesting to see the correspondence on all this,
in the double case.

It is useful first to have some examples of weak double categories. As
usual, these can come from homotopy theory.

Let A,B be subspaces of a space X. R= Let R_2(X:A,B) be the space of maps
I^2 \to X which map the edges in direction 1 to A and the edges in direction
2 to B (so A,B had better have a non empty intersection). Then we get
partially defined compositions in both directions, but no identities, no
associativity. You can do better by considering `Moore rectangles'. There
are also involutions in each directions, giving -_1, -_2.

It seems hard to get a homotopy double groupoid out of this, unless one of
A,B is contained in the other. However Loday showed that the fundamental
group of R at the constant map does inherit the other 2 structures, giving a
cat^2-group (a double groupoid in groups).

It is a nice exercise to generalise the above to an n-ad (n-subspaces).

Curiously, the interchange law is exactly valid in R_2.

I expect that this law is one of those to go, for certain applications, such
as multiple holonomy, and rewriting. The first could be important for
gravity theory (that would be nice!).

There is a way of handling the first (?) obstruction to the interchange law,
by considering cubical multiple categories with connection as defined in

116. (with F.A. AL-AGL and R. STEINER), `Multiple categories: the
equivalence between a globular and cubical approach', Advances in
Mathematics, 170 (2002) 71-118.

The monoidal closed structure allows for the notion of `cubical algebra'.
i.e. with a monoid structure w.r.t. tensor product. C \otimes C \to C. This
includes a whiskering operation of C_0 on the left and right of C_1, and
also a map say b: C_1 \times C_1 \to C_2, which somehow measures the failure
of the interchange law for the each of the two binary operations which can
be defined on C_1 using whiskering. Such a cubical algebra should (?) be
related to Sjoerd Crans' tesis, but the cubical format could be more
convenient: I have always found that format more useful in several ways than
the globular approach, as multiple compositions are easy to understand, but
the relation between the two, when it exists, is important.

Analogues of the map b occur in braided crossed modules, 2-crossed modules,
automorphisms of crossed modules, ....

In the paper

123. (with I.ICEN), `Towards a 2-dimensional notion of holonomy',
Advances in Math, 178 (2003) 141-175.

we tried to get towards a double groupoid associated with 2-holonomy, but
this now seems naive. We would like a map such as the above b to be involved
with some physical phenomena!
Properties of the globular version may not yet have been written down! But
groupoid/crossed module analogues are in

59.  (with N.D. GILBERT), ``Algebraic models of 3-types and
automorphism  structures for crossed modules'', {\em Proc. London
Math. Soc.} (3) 59 (1989)  51-73.

Ronnie Brown
www.bangor.ac.uk/r.brown