Smooth categories etc

Smooth categories etc

[R Brown]

Dear Bill and Colleagues, 

In reply here only to Bill (4), but great interest in the other comments, I entirely agree with the many pointed approach, which was published in my 1967 Proc LMS article. This is  relevant also to higher homotopy theory. If you define \pi_n(X,P) where P is now a *set* of base points, then you see this has to have the structure of module over \pi_1(X,P). As an example of its use, consider the map 

                  S^n \vee [0,1] \to S^n \vee S^1

which identifies 0 and 1, where the S^n is stuck to [0,1] at 0.  Clearly \pi_n of the first space on the set of base points P consisting of 0 and 1 is the free module on one generator over the indiscrete groupoid \I=\pi_1([0,1],P). The main theorem of Higgins and RB implies that \pi_n of the second space at 0 is the free module on one generator over the infinite cyclic group, which itself is obtained from \I by identifying 0 and 1 in the category of groupoids. 

I know this result can also be obtained using covering space arguments and homology, but I got into groupoids by trying to avoid this detour to covering spaces  to describe the fundamental group of S^1. It is a question of finding algebra which  models geometry. 

One of my joint papers, which calculated (using vKT for crossed modules) some  \pi_2(X,x)  as a module,  was rejected by one journal on the grounds that the calculations were too elaborate when `the interest is in the group and not the module'. So much for homological algebra! 
There is a culture in algebraic topology which neglects the operations of \pi_1;  perhaps it seems an encumbrance when there is only one base point, but Henry Whitehead commented in 1957 that the operations fascinated the early workers in homotopy theory. 

It is amusing to speculate what might be infinite loop space theory, or little cube operads, if you allow many base points! Could it clear up the subjects??!! Answers on a postcard please (joke). 

A standard lesson in mathematics is that you should forget structure at the latest possible moment. In homotopy theory low dimensional identifications can and usually do affect higher dimensional homotopy invariants. To try and cope with this, we need homotopical functors which carry algebraic information in a range of dimensions, to model how spaces are glued together. This has been the aim of the various Higher Homotopy van Kampen theorems. To handle these algebraic structures one needs category theory ; as one example,  I am currently working on a joint paper using fibred and cofibred categories to relate high and low dimensional information on colimits and induced structures.  

The hope also is that because of the wide interest in deformation, i.e. homotopy,  as a means of classification, these tools and methods will have wider implications. 

Ronnie

Dear Ronnie and Colleagues,

Your comments are extremely interesting.  Thank you very much for raising 
in so striking a manner the question of the relation between general 
monoidal structures and cartesian closed structures.
Below are some observations which show, I think,
that everybody should be interested in this relation because it is 
manyfold and fruitful.

(1)    While cartesian closed structures have the virtue of being unique, 
general monoidal closed structures have the virtue of not being unique. 
Thus, for example, the cartesian closed presheaf toposes (with their 
exactness properties and combinatorial truth object) often have a further 
monoidal closed structure given by Brian Day's convolution with respect to 
a pro-co-monoidal structure on the site. Cubical as well as simplicial 
sets have both cartesian and non-cartesian closed structures, and that is 
'true', not merely 'convenient'.

(2)    Another category having both cartesian and non-cartesian monoidal 
structures is the real interval from zero to infinity with 'x dominates y' 
as the morphism from x to y. (Actually, this category is derived by 
collapsing a natural topos of dynamical systems in 'Taking categories 
seriously' TAC Reprints.) Categories enriched with respect to the 
non-cartesian structure here (see 'Metric Spaces' TAC reprints) arise 
every day in analysis and the rich insights of enrichment theory (Functor 
categories, bi-module composition, free categories, etcetera) should be 
systematically applied to the advance of analysis and geometry, while on 
the other hand metric examples inspire further developments of enrichment 
theory. Cauchy (who never worked on idempotent splitting in ordinary 
categories and additive categories in the way that Freyd and Karoubi did) 
does not deserve to have his name brandished as a joke to scare one's 
uncomprehending colleagues in analysis. The kind of completeness that is 
inspired by two-sided intervals (unlike the one-sided intervals 
inaccurately alluded to in common discussions of 'density') indeed reduces 
to the one attributed to Cauchy in the particular example of Metric 
Spaces. The author hoped that observation would contribute to the advance 
of analysis and the development of enrichment theory, not to the supply of 
buzzwords.

    In fact, there is an insufficiently known branch of analysis called 
'Idempotent Analysis', which deals largely with composition of bi-modules, 
or more precisely, with the relation between the two closed structures on 
the infinite interval. Of course, that monoidal category is isomorphic to 
the unit interval under multiplication (still cartesian closed too) which 
induces many of the relations between probablility and entropy.

(3)    Perhaps the most common relation between non-cartesian monoidal 
categories and cartesian categories arises when a structure such as vector 
space is interpreted in a cohesive background. I am sticking to my story 
that cohesive backgrounds are basically cartesian closed, due to the 
ubiquitous role of diagonal maps and also due to the fact that, for 
example, bornological vector spaces have an obvious monoidal closed 
structure, whereas topological vector spaces have none. The rumor that 
topological vector spaces might have a tensor with an adjoint hom is part 
of the disinformation that makes functional analysis look more difficult 
than it is. A more accurate account of the relation between non-Mackey 
convergence and closed structure can be found in C. Houzel's paper on 
Grauert finiteness, Mathematische Annalen, vol. 205, 1973, 13-54:
essentially, the topological categories are merely enriched in the 
genuinely monoidal closed bornological ones. Similarly, the idea that not 
all dual spaces are complete seems to be based on a misguided generality 
in the notion of Cauchy nets (they should be bounded).

(4)    Although pointed spaces are somewhat entrenched in algebraic 
topology, there is an improvement suggested by your own work, Ronnie. 
Consider the category whose objects are arrows S ---> E where E is a space 
(object of a cartesian closed cohesive background category) and S is a 
discrete space. This category is even a topos if the category of E's was, 
as is the larger category of arrows between general pairs of spaces. The 
first category is actually an adjoint retract of the second, correcting 
the discontinuity that arises from the traditional limitation S = 1. 
Intuitively, in the case where the pair of spaces is a subspace inclusion, 
the adjoint collapses the subspace to a point if the subspace is 
connected, but if it is not connected, does not artificially merge its 
components. There are many applications of this corrected construction of 
the space which results from 'neglecting' a subspace, both in algebraic 
topology and in functional analysis, too numerous to discuss here.

Bill