Re: covering spaces and groupoids

["Joyal, André" <joyal.andre@uqam.ca>]

Dear Bill,

The base point is important in algebraic topology for many reasons.
The category of pointed spaces can be understood as the first step toward 
"embedding" the category of spaces into something like an additive category 
(for example, into the category of spectra).
An additive category is pointed. The base point of a space is playing the
role of the null element. A wedge of pointed spaces
is their "sum", and a smash product of two pointed
spaces is their tensor product.
By a theorem of Kan, the model category of pointed connected spaces is 
Quillen equivalent to the model category of (simplicial) groups.
The category of groups is not additive, since a group
is not abelian, but it is nearly so.
The category of (simplicial) groups is a homotopy
variety of algebras, hence also the category of pointed
connected spaces. The pointed circle is the natural
generator of the category of pointed connected spaces.
Every pointed connected space is a homotopy sifted
colimit of bouquets of circles.

The category of pointed n-connected spaces is a
homotopy variety of algebras for every n geq 0.
The pointed (n+1)-sphere is the natural generator.

Best wishes,
André






-------- Message d'origine--------
De: categories@mta.ca de la part de F. William Lawvere
Date: jeu. 03/06/2010 14:14
À: may@math.uchicago.edu; ronnie.profbrown@btinternet.com
Cc: jds@math.upenn.edu; categories
Objet : categories: Re: covering spaces and groupoids
 



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




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