Re: Function composition of natural transformations?

[Ronnie Brown <mas010@bangor.ac.uk>]

There is another way of looking at the strict multiple globular category
case, which is to use the monoidal closed structure, as ws established via
the cubical case in

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

This monoidal closed structure is fairly clear cubically, but is difficult
to translate into globular formulae in higher dimensions. If A=END(C),
where C is a multiple category (globular or cubical), so that A is one
also, then the `enriched composition' is  a morphism A \otimes A \to A. In
low dimensions this gives left and right whiskering A_0 \times A_1 \to A_1,
A_1 \times A_0 \to A_1, and there is also a function say
{  ,  }: A_1 \to A_1 \to A_2, which measures the lack of agreement of two
possible definitions of compositions, and I think this is what Tom refers
to in his email.
In the cubical formulation, A_2 consists of `squares', and the sides of the
squares are easy to interpret using whiskering. One way round the square is
a.g \circ f.v

and the other is
f.u\circ b.g

if f:a \to b, g:u \to v.

In the groupoid case, ideas of this type are used in

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

and in other papers of Nick Gilbert. The extra structure on a crossed
module M (or 2-groupoid, for that matter) of a monoid morphism M \otimes M
\to M allows the modelling of homotopy 3-types. However, for calculations
of 3-types, crossed squares seem better, because of a Van Kampen Type
theorem, not apparently available for the other structures.

Ronnie Brown




Tom LEINSTER wrote:
>
> This may be `mere' pedagogy for ordinary categories, but if you try the
> same thing for 2-categories then it becomes a `genuine' issue.  To put it
> another way, the two different but equivalent presentations of a concept
> (natural transformation) become, on categorification, significantly
> different.
>

snip...

-- 
 Professor Emeritus R. Brown,
 School of Informatics, Mathematics Division,
 University of Wales, Bangor
 Dean St., Bangor, Gwynedd LL57 1UT,
 United Kingdom
 Tel. direct:+44 1248 382474|office:     382681
 fax: +44 1248 361429
  World Wide Web: home page:
 http://www.bangor.ac.uk/~mas010/
 (Links to survey articles: Higher dimensional group theory
  Groupoids and crossed objects in algebraic topology)

 Centre for the Popularisation of Mathematics:
 http://www.cpm.informatics.bangor.ac.uk/
  (reorganised site with new sculpture animations)