Subject: categories: higher order category theory and homotopy theory

From: Ronnie Brown <[email-masked]>
Date: Thu, 19 Nov 1998 19:06:47 +0000 (GMT)

There is a more down to earth and historically rooted approach. 

A concern of the early topologists (Dehn, Hopf, Alexandroff,...) 
was to find a higher dimensional version of the fundamental group, since 
the were irritated by the facts that 
(i) \pi_1 gave good geometric information (e.g. for complex variable 
theory) which often involved the non commutativity, 
(ii) H_1 was \pi_1 made abelian, for connected spaces, 
(iii) H_n existed for all n \ge 0 (and was abelian). 

There was also the intuition of big things made out of composing little 
bits (simplices in the Poincare formulation). The `solution' was to use 
formal sums (homology), and this gave abelian results. 

Cech's 1932 submission on Higher homotopy groups to the Zurich ICM 
was withdrawn at the request of Alexandroff and Hopf once they had proved 
these groups abelian for n \ge 2. 

The whole aim was to get immediate geometric results and computations. 

My suggestion in 1967 was to use `higher homotopy groupoids' since these 
did not have to be abelian. It was only in 1974 that Philip Higgins and I 
found a definition of a homotopy double groupoid of a pair; 
 with this and the work with Chris Spencer we could prove a 2-D Van 
Kampen theorem. This is still giving actual computations of homotopy 
2-types and invariants not previously known. For an exposition see my 
notes
Groupoids and crossed objects in algebraic topology
from the 1997 School in Algebraic Topology at Grenoble 
http://www.bangor.ac.uk/~mas010/brownpr.html
which also go on briefly to the higher dimensional case (for strict higher 
groupoids) parts of which were announced in 1977 (with Philip Higgins, 
Compte 
Rendue Acad. Sci. Paris S\'er. A. 285 (1977) 997-999, 286 (1978) 91-93. 

Also, with the use of n-fold groupoids one can recover all n-types 
(Loday) a result Grothendieck described to me in 1986 (in Montpellier) as  
`absolutely 
beautiful'. Again, the intricate structure of these can give new 
computations of homotopy invariants and homotopy types, while studying 
pushouts of these (strict again!) one can find new algebraic 
constructions, such as a non abelian tensor product of groups (which act 
on each other).  

We have to develop new methods to compute even with 
these strict objects, so there is still a lot to be done in getting 
explicit information on specific examples using the non strict ones. 

One overall setting is to have functors

                             \Pi
                          ---------->
       (topological data) <--------- (Algebraic data)
                              IB
               |                       |
             U |                       | B 
               |                       |
               V                       V
               ======== (spaces)======                            

where IB (\mathhbb{B}) and B are classifying data or space functors, U  
is forget, and \Pi is a functor which should satisfy Van Kampen Theorem 
(preserve certain colimits), and also \Pi  IB is equivalent to 1. 

At the level of 3-types we have such a set up with topological data = 
squares of (pointed) spaces, algebraic data = crossed squares (a kind of 
triple groupoid). We do not (so far) have such a setup when (algebraic 
data) is say the candidate proposed by Andre (which are equivalent to 
2-crossed modules). Graham Ellis has shown how to compute 3-types and 
homotopy classes of maps using the above set up (Math. Z., 461 (1993) 
93-110. ). The existence of the functor \Pi (satisfying VKT, so one 
can apply free algebraic objects) helps a lot. 

So I find it interesting that even strict 3-fold groupoids have an 
intricate structure (they model 3-types!). The relations and uses of all 
these models needs lots of work. 

The tensor product of crossed complexes (corresponding to a Gray tensor 
product of strict \infinity groupoids) yields interesting models for 
loop spaces, by considering monoids for this tensor:
(see for example Baues and Tonks
On the twisted cobar construction. Mathematical Proceedings of the 
Cambridge Philosophical Society 121 (1997) pp.229-245 ). 

But the nice thing about crossed complexes is one can write down explicitly 
the Eilenberg-Zilber theorem (Tonks' thesis), which I have not seen for 
infinity groupoids! 

Ronnie Brown