Re: Makkai's suggestion

[Ronnie Brown <ronnie.profbrown@btinternet.com>]

John Baez writes on 03/09/2010:
----------------------------------

Leinster's refinement of Batanin's approach defines weak infinity-categories
as algebras of an "initial globular operad with contractions".

Here "globular" means we're doing infinity-categories in the obvious way,
where given two n-morphisms f,g: x ->  y we can talk about (n+1)-morphisms
from f to g.

"Algebra of a globular operad with contractions" means we can compose these
n-morphisms in all the pictorially obvious ways, and every pictorially
plausible law holds *up to a higher morphism*.

"Initial" means we're doing this in exactly the right way: for example,
there aren't any *extra* ways of composing morphisms, and we're not sticking
in *too many* of these higher morphisms.

I am sure people will eventually come up with better ways to do
infinity-category theory.  Eventually most math majors will learn it in
college (unless our current civilization collapses in less than, say, 150
years).    But the approaches we've got right now are already pretty good.
Learn 'em!

---------------------------------------------------------
I would like to put in a plea for analogous studies of cubical
approaches to weak higher categories. I started looking for higher
groupoids in homotopy theory in the 1960s but all the time was seeking
cubical structures, as these were clearly relevant to my starting point,
the van Kampen Theorem for the fundamental group and groupoid. It was
easy, even nicely childish,  to draw pictures of a square subdivided
into little squares by lines parallel to the sides, and so to seek for
some maths which expressed the big square as the composite of the little
squares. (The further idea needed was `commutative cube', which required
extra structure.) The cubical singular complex of a space was obviously
some kind of `lax or weak infinity-fold groupoid', but I am not clear if
this idea is captured by any of the current expositions. For the notion
of higher groupoid with a van Kampen Theorem yielding  specific and
precise calculations one wanted colimits rather than `lax colimits (??)'
and so needed to take some kind of homotopy classes to get a strict
structure. Over a period of 11 years it was found with Philip Higgins
that this could be done usefully, but non trivially, for filtered spaces.

The argument of the new book on `Nonabelian algebraic topology'
(downloadable from my web page,
www.bangor.ac.uk/r.brown/nonab-a-t.html, planned to appear in 2010 with
the EMS) is that filtered spaces can be taken as a satisfactory starting
point for algebraic topology, at the border between homotopy and
homology, avoiding a direct use of singular homology and simplicial
approximation, and getting for example nonabelian results in dimension
2. The hope is that the book will allow convenient evaluation and
hopefully extension of these methods.

In all this work we noted various globular ideas but could never use
them in our context to get any new results in homotopy theory. So we
could define `algebraic inverse to subdivision' cubically, but not
globularly. We could define classifying spaces simplicially or
cubically, but there seems no evidence that this works globularly.
Monoidal closed structures worked fine cubically, and could be
translated to other contexts, using equivalences of categories. This is
done for example for strict globular omega categories in

(F.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.

following an earlier approach for the groupoid case by Philip Higgins
and me (JPAA, 1987). The paper by Higgins

`Thin elements and commutative shells in cubical {$\omega$}-categories'.
Theory Appl. Categ. 14 (2005) No. 4, 60--74

shows that the globular case is needed to define commutative shells in
this context, so I am not saying one can get away without globular notions.

Is it possible to capture the properties of the cubical singular complex
of a space (which has been around since Kan's work in 1955, and possible
earlier) in terms of these operadic ideas, and make this useful? Even
more naively, does this relate to `little cubes operads'? How to relate
the putative cubical and globular notions?

Ronnie Brown




[For admin and other information see: http://www.mta.ca/~cat-dist/ ]