Re: covering spaces and groupoids

["F. William Lawvere" <wlawvere@hotmail.com>]

Dear Ronnie and Peter,

 

    In the applications of algebraic topology to topology,
where do the ‘basepoints’ 
originate? From my (regrettably too few) contacts with algebraic
topologists I gleaned the following:

 

1.    
(S. Eilenberg) The base point is the residue of
a collapsed subspace, which results, for example, in constructing a model of
the 2-sphere by collapsing the boundary of a 2-ball. 

 

2.    
(B. Eckmann)  The pairs, space/subspace (whose homology is often studied)  can be usefully generalized
to arbitrary maps as objects, not just inclusion maps.

 

3.    
(R. Swan)  A construction is usually not functorial if one of its steps
involves complementation of subobjects; but collapsing subobjects retains
nearly the same information, yet is functorial.

 

4.    
(M. Artin and G. Wraith)  An important refinement of the morphism
category of 2. above involves ‘gluing’ along a left-exact functor between
two categories, a special ‘comma’ category construction that in fact always
yields a topos if the original categories are toposes. For example, the inverse
image functor   i  of a grounding of one topos over
another yields in this way a topos whose objects are maps  i(S) à E.

 

5.    
(P. Freyd) Under the name of sconing the
geometrical construction of 4. is very useful in case the objects S of the base
topos deserve to be called ‘discrete’. (Ronnie B. points out that this sort of category is the
natural domain of the fundamental groupoid.)

 

6.    
Suppose a topos (of spaces) is locally connected
over another one (of discrete spaces). That means that the inverse image functor i (itself the left adjoint of a
points functor) has its own further left adjoint  p  counting
connected components.  Then the
constructions of 4., 5., yield a result which is again locally connected; the
extended  p  assigns to any  Aà
E the pushout E/A with Aà i p A. In the spirit of 3. I think of E/A as the exterior of A. This
construction is clearly a left adjoint and hence co-continuous in contrast to
the construction which merely collapses any A to a point (with which it agrees
in case A has exactly one component).

  Here is a proposed application of the construction of
4., 5, 6., to geometric analysis, serving e.g. as a refutation to the supposed
ubiquity of rings without unit:

 

The easy notion of support for
covariant quantities like measures is concerned with domain of dependence: An
element of M(E) might come from an element of M(A) via A à E and hence be
supported on A. Also for contravariant quantities we need not make an abusive
use of the properties of minus and zero. A function on X ‘of compact support’ may be interpreted as one that does
not depend on the large part L 
which is remote from some small part K of interest; here K union L = X.
The complements of such  K  are to be inoperative in the variation
of such particular functions. But even the line has two ends so that constancy
on the components of L is a more functorial condition on functions. If the
codomain space R has certain algebraic structure, then R(X,L) = R^(X/L), the
exponential space of functions on the indicated pushout enjoys all the same
algebraic structure, as does the colimit over all large remote L in X (these
being filtered). Of course, this construction R(X/infinity) is functorial  only
for proper maps X à
Y, i.e. those whose inverse image preserves the large remoteness. The covariant
dependency of the dual space Hom (R (X/infinity), R) of functionals is likewise
only along proper maps, in contrast to that of the smaller space M(X)=Hom(R^X,R)
of functionals that have to integrate all functions of the category.
Best wishes,Bill





> Date: Wed, 2 Jun 2010 08:41:58 -0500
> From: may@math.uchicago.edu
> To: ronnie.profbrown@btinternet.com
> CC: jds@math.upenn.edu; categories@mta.ca
> Subject: categories: Re: covering spaces and groupoids
> 
> I'm eclectic, and prefer closer contact with the real world of existing
> applications.
> There the overwhelming majority of the literature uses universal covers
> as usually
> constructed. I didn't go into it, but the dependence of that on the
> basepoint is also
> ephemeral: you get a universal covering space functor from the
> fundamental groupoid
> to such coverings easily enough. That is also used in applications
> (quite recently by
> Kate Ponto in work on fixed point theory). In any case, I don't place
> the emphasis you
> do on this matter, which I regard as minor from the point of view of
> algebraic topology.
> 
> Peter On 6/2/10 2:03 AM, Ronnie Brown wrote:
>> Dear Peter,
>>
>> You wrote:
>> --------------------------------
>> I reworked that theory from scratch when writing ``A concise course in
>> algebraic topology''.
>> Chapter 3 (pp21-32)  does covering spaces, covering groupoids, the orbit
>> category and the various
>> equivalences of categories among them. I like it, but that chapter is
>> maybe the
>> main reason that my book is less popular than others: non-categorical
>> types find
>> it too difficult for young minds to absorb the first time around.
>> --------------------------------
>>
>> It seems to me that you give a complicated route via the universal cover
>> to the inverse equivalence from GpdCov(\pi X) to TopCov(X), which
>> assumes connectivity and so requires a  choice of base points.
>>
>> My account starts with any covering morphism q: G \to \pi_1 X of
>> groupoids and gives precise local conditions on X for there to be a
>> `lifted topology' on Ob(G) which makes it a covering space of X with
>> fundamental groupoid canonically isomorphic to G. No connectivity is
>> assumed, which makes it useful for discussing coverings of fundamental
>> groupoids of non connected topological groups. It has other uses, such
>> as topologising \pi_1 X.
>>
>> I now find something quite unintuitive, even bizarre, in any emphasis on
>> `fundamental groups and change of base point': it is like giving railway
>> schedules in terms of return journeys and change of start points.
>>
>> The later editions of my book also give a full account of orbit spaces
>> and orbit groupoids under the action of a group, giving conditions for
>> the fundamental groupoid of the orbit space to be naturally isomorphic
>> to the orbit groupoid of the fundamental groupoid.
>>
>> Ronnie
>>

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