Isomorphisms of categories

Isomorphisms of categories

[Peter May]

DeTeXing an exercise I routinely assign, here is
an example of an isomorphism of categories that is
not `accidental' in Peter Johnstone's sense and is
always used in practice as an isomorphism and not
merely an equivalence.


The fundamental theorem of Galois theory:

Let G = Gal(E/F) be the Galois group of a finite
Galois extension E/F.  Define an isomorphism of
categories between the category of intermediate
fields F\subset K\subset E and field maps
K >--> L that fix F pointwise and the category
of orbits G/H and G-maps between them.



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Re: Isomorphisms of categories

[Toby Bartels]

Peter May phrased the fundamental theorem of Galois theory as:

>Let G = Gal(E/F) be the Galois group of a finite
>Galois extension E/F.  Define an isomorphism of
>categories between the category of intermediate
>fields F\subset K\subset E and field maps
>K >--> L that fix F pointwise and the category
>of orbits G/H and G-maps between them.

Thanks, Peter, for that excellent example.

For a moment, I was going to object that it is uninteresting
because the categories in question are simply posets,
until I looked again and saw that they are not what I was expecting.

I want to point out that, to those of us who "speak no evil"
(I can only really speak for myself, but I expect some others to agree),
this example is very different from the concrete isomorphisms,
such as that between the categories of boolean rings and boolean algebras.
In the latter case, I argued before that what really matters
is that we have a concrete equivalence and that the 2-category Conc
(whose objects are the faithful functors to the category of sets)
and that Conc is a 2-poset: parallel concrete natural transformations
are equal (and equality makes sense to us in that context).

But in this new example, we have an isomorphism of *strict* categories.
That is, there really is a notion of equality of objects in each category,
even if our foundation does not provide equality as identity by fiat,
because we can define what it means for two objects to be equal:
K = L iff, for every element x of F, x in K if and only if x in L
(and similarly for the other category).  In material set theory,
the axiom of extensionality shows that this matches equality as identity;
in a structural foundation like ETCS, the two equalities do not match
and it is really the defined one that we want, not identity at all.

(I should also remark that Peter's categories are small categories,
at least if your foundations are sufficiently impredicative
to have small power sets, as is usual.)


--Toby


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Re: Isomorphisms of categories

[jim stasheff]

Peter May wrote:
> DeTeXing an exercise I routinely assign, here is
> an example of an isomorphism of categories that is
> not `accidental' in Peter Johnstone's sense and is
> always used in practice as an isomorphism and not
> merely an equivalence.
>
>
> The fundamental theorem of Galois theory:
>
> Let G = Gal(E/F) be the Galois group of a finite
> Galois extension E/F.  Define an isomorphism of
> categories between the category of intermediate
> fields F\subset K\subset E and field maps
> K >--> L that fix F pointwise and the category
> of orbits G/H and G-maps between them.
>
>
>
and an isomorphic category of coverings spaces such that...

jim



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Re: Isomorphisms of categories

[Peter May]

On 5/30/10 12:50 PM, jim stasheff wrote:
> Peter May wrote:
>> DeTeXing an exercise I routinely assign, here is
>> an example of an isomorphism of categories that is
>> not `accidental' in Peter Johnstone's sense and is
>> always used in practice as an isomorphism and not
>> merely an equivalence.
>>
>>
>> The fundamental theorem of Galois theory:
>>
>> Let G = Gal(E/F) be the Galois group of a finite
>> Galois extension E/F.  Define an isomorphism of
>> categories between the category of intermediate
>> fields F\subset K\subset E and field maps
>> K >--> L that fix F pointwise and the category
>> of orbits G/H and G-maps between them.
>>
>>
> and an isomprhic category of coverings spaces such that...
>
> jim
No-no, that is in fact the very next exercise:

Covering space theory:  Requiring covering spaces of a (well-behaved)
connected
topological space B to be connected, let \sC ov(B) be the category
of covering spaces of B and maps over B.  If G is the fundamental group
of B,
then the orbit category of G is {\em equivalent}, not {\em isomorphic},
to \sC ov(B).
Sketch the proof. (Hint: use a universal cover of B to construct a skeleton
of the category \sC ov(B).)




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Re: Isomorphisms of categories

[David Roberts]

If you take a category connCov_*(X) of _pointed_ connected covering
spaces of a pointed well-behaved space X, then this is a preorder (at
most one arrow between any two objects), and this is equivalent to the
poset Sub(G) of subgroups of G = pi_1(X). However, Sub(G) is
isomorphic to the reflection of connCov_*(X) into the category of
posets.

Without the presence of basepoints, it is, as Ronnie Brown would say,
more natural to use groupoids.

David

> No-no, that is in fact the very next exercise:
>
> Covering space theory:  Requiring covering spaces of a (well-behaved)
> connected
> topological space B to be connected, let \sC ov(B) be the category
> of covering spaces of B and maps over B.  If G is the fundamental group
> of B,
> then the orbit category of G is {\em equivalent}, not {\em isomorphic},
> to \sC ov(B).
> Sketch the proof. (Hint: use a universal cover of B to construct a skeleton
> of the category \sC ov(B).)


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Re: covering spaces and groupoids

[Ronnie Brown]

Peter May wrote:
---------------------------
Covering space theory:  Requiring covering spaces of a (well-behaved)
connected topological space B to be connected, let \sC ov(B) be the
category
of covering spaces of B and maps over B.  If G is the fundamental group
of B, then the orbit category of G is {\em equivalent}, not {\em
isomorphic},
to \sC ov(B). Sketch the proof. (Hint: use a universal cover of B to
construct a skeleton
of the category \sC ov(B).)
--------------------------

I would like to put in a case here for the groupoid approach ( see
`Elements of modern topology' (1968), and subsequent editions; got the
idea from Gabriel/Zisman, so not entirely idiosyncratic). If TopCov(X)
is the category of covering spaces of X, and X admits a universal cover,
then the fundamental groupoid functor \pi induces an equivalence of
categories

\pi: TopCov(X) \to  GpdCov(\pi X)

to the category of groupoid covering morphisms of \pi X. This seems to
me to be the most intuitive version - a covering map is modelled by a
covering morphism. I prefer the proof in this version, since it does not
involve choices of base point, and allows  the non connected case. It
also allows one to discuss the case X is a topological group and to look
at topological group covering maps. (Brown/Mucuk, Math ProcCamb Phil Soc
1994, following up ideas of R.L. Taylor).

The notion of covering morphism of groupoids goes back to P.A. Smith
(Annals, 1951), called a regular morphism, and nowadays a discrete
fibration, I think.

Is there an analogous version for Galois theory?

Ronnie Brown


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Re: covering spaces and groupoids

[Eduardo J. Dubuc]

  > Is there an analogous version for Galois theory?

In SGA 1 A.G. shows an equivalence

                      C ~= GpoidActions(\pi(C))

where C is a category with certain axioms  (that he callls Galoisiene) and

       \pi(C) is the fundamental groupoid of C (its objects are the set I of
fiber functors of C, and its vertice groups are profinite, discrete only in
case of existence of universal covering),

   and  GpoidActions(\pi(C)) is the category of families indexed by I with an
action of \pi(C).

This includes as examples both the case of covering spaces  and classical
Galois Theory (Artin theory with the algebraic closure)

I imagine that the category GpoidActions(\pi(C)) should be equivalent to
GpdCov(\pi X) in a general  abstract setting.

Subsequently, this theory was extended and generalized in a well determined
direction (progroupoids, localic groupoids, localic progroupoids) in several
steps by A.G. himself, Moerdiejk, Bunge, Dubuc and Joyal-Tierney.

Other authors extended the basic theory (presence of universal covering and
discrete groups) in different directions.

e.d.
Ronnie Brown wrote:
> Peter May wrote:
> ---------------------------
> Covering space theory:  Requiring covering spaces of a (well-behaved)
> connected topological space B to be connected, let \sC ov(B) be the
> category
> of covering spaces of B and maps over B.  If G is the fundamental group
> of B, then the orbit category of G is {\em equivalent}, not {\em
> isomorphic},
> to \sC ov(B). Sketch the proof. (Hint: use a universal cover of B to
> construct a skeleton
> of the category \sC ov(B).)
> --------------------------
>
> I would like to put in a case here for the groupoid approach ( see
> `Elements of modern topology' (1968), and subsequent editions; got the
> idea from Gabriel/Zisman, so not entirely idiosyncratic). If TopCov(X)
> is the category of covering spaces of X, and X admits a universal cover,
> then the fundamental groupoid functor \pi induces an equivalence of
> categories
>
> \pi: TopCov(X) \to  GpdCov(\pi X)
>
> to the category of groupoid covering morphisms of \pi X. This seems to
> me to be the most intuitive version - a covering map is modelled by a
> covering morphism. I prefer the proof in this version, since it does not
> involve choices of base point, and allows  the non connected case. It
> also allows one to discuss the case X is a topological group and to look
> at topological group covering maps. (Brown/Mucuk, Math ProcCamb Phil Soc
> 1994, following up ideas of R.L. Taylor).
>
> The notion of covering morphism of groupoids goes back to P.A. Smith
> (Annals, 1951), called a regular morphism, and nowadays a discrete
> fibration, I think.
>
> Is there an analogous version for Galois theory?
>
> Ronnie Brown
>

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Re: covering spaces and groupoids

[Peter May]

I'm not against coverings of groupoids: 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.

Peter

On 6/1/10 3:56 AM, Ronnie Brown wrote:
> Peter May wrote:
> ---------------------------
> Covering space theory:  Requiring covering spaces of a (well-behaved)
> connected topological space B to be connected, let \sC ov(B) be the
> category
> of covering spaces of B and maps over B.  If G is the fundamental group
> of B, then the orbit category of G is {\em equivalent}, not {\em
> isomorphic},
> to \sC ov(B). Sketch the proof. (Hint: use a universal cover of B to
> construct a skeleton
> of the category \sC ov(B).)
> --------------------------
>
> I would like to put in a case here for the groupoid approach ( see
> `Elements of modern topology' (1968), and subsequent editions; got the
> idea from Gabriel/Zisman, so not entirely idiosyncratic). If TopCov(X)
> is the category of covering spaces of X, and X admits a universal
> cover, then the fundamental groupoid functor \pi induces an
> equivalence of categories
>
> \pi: TopCov(X) \to  GpdCov(\pi X)
>
> to the category of groupoid covering morphisms of \pi X. This seems to
> me to be the most intuitive version - a covering map is modelled by a
> covering morphism. I prefer the proof in this version, since it does
> not involve choices of base point, and allows  the non connected case.
> It also allows one to discuss the case X is a topological group and to
> look at topological group covering maps. (Brown/Mucuk, Math ProcCamb
> Phil Soc 1994, following up ideas of R.L. Taylor).
>
> The notion of covering morphism of groupoids goes back to P.A. Smith
> (Annals, 1951), called a regular morphism, and nowadays a discrete
> fibration, I think.
>
> Is there an analogous version for Galois theory?
> Ronnie Brown
>
>
>
>



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Re: covering spaces and groupoids

[Ronnie Brown]

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









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Re: covering spaces and groupoids

[Peter May]

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
>

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Re: covering spaces and groupoids

[Ronnie Brown]

Dear Bill,

You ask:

In the applications of algebraic topology to topology, where do the 
‘basepoints’ originate?

I'd like to give a different type of answer to that suggested in your 
list. (yet another is : whoopee, we have a group!)

After I first thought of the van Kampen theorem for the whole 
fundamental groupoid I realised one wanted *computation*, and the whole 
fundamental groupoid of a space was too big for that. The fundamental 
group at a base point was too small in may cases, such as the circle. 
The solution that seemed right to me, and which took a while to find, 
was the fundamental groupoid on a set of base points chosen conveniently 
according to the geometry of a given situation. Eldon Dyer said I ought 
to take a hard line on this: if the connected space X is the union of 
127 open sets whose intersections have 3,272 components, you do not want 
to take a single base point! Such situations (even with infinitely many 
components) occur in group theory applications, and analogously in 
topology in connected covering spaces over a union of 2 open sets.

I advocated the fundamental groupoid on a set of base points in my 1968 
book `Elements of Modern Topology', but this concept has I think not 
been mentioned in any later date algebraic topology text in English by 
other authors. Also group theorists are happy with graphs and free 
groups, but are very wary of the free groupoid on a graph! The only new 
result from that book that has been taken up is the gluing theorem for 
homotopy equivalences, but its origin is usually unacknowledged.

Looking at the way this groupoid van Kampen theorem could be used, it 
seemed amazing to me that one could obtain *complete* information on a 
fundamental group by deducing that from knowledge of a larger structure, 
for which one had colimit information, whereas my tries with nonabelian 
cohomology gave only exact sequences. It seems that groupoids have the 
advantages of structure in dimensions 0 and 1, and that this is needed 
for what Grothendieck later called `integration of homotopy types'.

Could one find analogous objects with structure in dimensions 0, 1, 
...,n? This question led (after many years, and with fortunate 
collaborations) to higher dimensional van Kampen theorems, which gave 
quite new information on homotopy invariants, some of them nonabelian, 
e.g. second relative homotopy groups, triad homotopy groups, n-adic 
Hurewicz Theorems. Such results were published in 1978, 1981 (with 
Higgins), 1987 (with Loday). These theorems are, I think, not even 
mentioned in any texts on algebraic topology (except mine). Some current 
writers (Faria Martins, Kauffman, Ellis, Mikhailov,...) are using these 
techniques.

So for me the question is: why are people unwilling to throw off the 
shackles of a single base point? I would welcome enlightenment. It may 
be that it is found just too hard to fit this idea into what is 
currently considered the `real world', and so to obtain new results. For 
example, what happens to iterated loop space theory, with more than one 
base point?

Those interested in the sociology of science might like an excerpt from 
a lecture by Alan MacKay on icosahedral symmetry at the LMS in 1985. He 
said the reaction went through three phases:

Phase 1) It is false.

Phase 2) It is true, but unimportant.

Phase 3) It is true, it is very important; and we have known it for years!

All the best

Ronnie




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Re: covering spaces and groupoids

[F. William Lawvere]

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
>>

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Re: covering spaces and groupoids

[Joyal, André]

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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Re: covering spaces and groupoids

[Ronnie Brown]

Dear all,

André makes a proper and convincing case for the importance of pointed 
spaces in algebraic topology . Part of this importance is the difficulty 
of homotopy theory, which necessitates approximations, and these are 
sometimes adequate and convenient.  Henry Whitehead introduced two 
methods to approximate homotopy theory: one was to describe special 
cases such as the category of polyhedra which were n-dimensional and 
r-connected, e.g. n small, or r near to n. The other was stabilisation. 
However, in introducing CW-complexes in ``Combinatorial Homotopy I'' he 
explains that he does not stick to the single vertex case since he wants 
to include covering spaces. There is often no canonical choice of base 
point in a covering space; there are also advantages in an account which 
does not require connectivity. .

I was told that one of Philip Hall's dictums was that you want the 
algebra to model the geometry, and not try to force it into a previously 
known format.

Grothendieck wrote in part:

Choosing paths for connecting the base points
natural to the situation to one among them, and reducing the
groupoid to a single  group, will then hopelessly destroy the
structure and inner symmetries of the situation, and result in a
mess of generators and relations no one dares to write down, because
everyone feels they won't be of any use whatever, and just confuse
the picture rather than clarify it. I have known such perplexity
myself a long time ago, namely in Van Kampen type situations, whose
only understandable formulation is in terms of (amalgamated sums of)
groupoids.

Cases where the choice of a single base point is inconvenient are when 
there is a group action (as shown by Higgins and Taylor, 1981); in 
non-connected cases; in Nielsen fixed point theory (as shown by Philip 
Heath, and recently by Kate Proto); and for work in algebraic geometry 
by Zoonekynd.

I like the description of the circle as obtained from the unit interval 
[0,1] by identifying 0 and 1 in the category of spaces; and of the 
integers as obtained from the `unit interval groupoid '  cal I  (the 
indiscrete groupoid on {0,1}, and so finite) by identifying 0 and 1 in 
the category of groupoids.

One gets analogous results for \pi_n as modules over \pi_1 in describing 
\pi_n(S^n \vee S^1). Of course this example can be done using covering 
spaces, but that seems a roundabout route.

The Bass-Serre theory of graphs of groups often uses choice of base 
points and of trees; yet Higgins showed in 1976 that there was a nice 
normal form for the fundamental groupoid of a graph of groups.

The action of Z_2 on S^1 by reflection has 2 fixed points, and is better 
described  (look at quotients)  by the action of Z_2 on the fundamental 
groupoid with these as base points rather than on the fundamental group 
on one of them (which one?).

The success in 1967-8 (as it seemed to me) of groupoids in 1-dimensional 
homotopy theory led me ask for possible uses of groupoids in higher 
homotopy theory, and this  has led with fortunate collaborations  to the 
notions of `higher dimensional group theory' and `higher dimensional 
algebra', and various new algebraic structures with applications;  an 
account of some aspects of these is in press with the EMS, and might be 
thought of as `towards nonabelian algebraic topology'.

I confess not to have found the appropriate many base point version with 
applications of the fundamental cat^n-group of an n-cube of spaces for 
n> 1.

So it is a case of horses for courses.

Ronnie


>   


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Re: covering spaces and groupoids

[jim stasheff]

Ronnie Brown wrote:
> Dear all,
>
> André makes a proper and convincing case for the importance of pointed 
> spaces in algebraic topology . Part of this importance is the 
> difficulty of homotopy theory, which necessitates approximations, and 
> these are sometimes adequate and convenient.  Henry Whitehead 
> introduced two methods to approximate homotopy theory: one was to 
> describe special cases such as the category of polyhedra which were 
> n-dimensional and r-connected, e.g. n small, or r near to n. 
or r about half n - cf the stable range
what Schlessinger and I call `shallow'
in rational homotopy theory, still manageable   for r about 1/3 n
> The other was stabilisation. However, in introducing CW-complexes in 
> ``Combinatorial Homotopy I'' he explains that he does not stick to the 
> single vertex case since he wants to include covering spaces. There is 
> often no canonical choice of base point in a covering space; there are 
> also advantages in an account which does not require connectivity. .

e.g. an n-fold covering as a fibre bundle
cf the Galois correspondence

jim

>
> I was told that one of Philip Hall's dictums was that you want the 
> algebra to model the geometry, and not try to force it into a 
> previously known format.

YES! one of the problems in math bio is to have the math model the 
biology and not try to force it into a previously known format.
>
> Grothendieck wrote in part:
>
> Choosing paths for connecting the base points
> natural to the situation to one among them, and reducing the
> groupoid to a single  group, will then hopelessly destroy the
> structure and inner symmetries of the situation, and result in a
> mess of generators and relations no one dares to write down, because
> everyone feels they won't be of any use whatever, and just confuse
> the picture rather than clarify it. I have known such perplexity
> myself a long time ago, namely in Van Kampen type situations, whose
> only understandable formulation is in terms of (amalgamated sums of)
> groupoids.

...


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Re: covering spaces and groupoids

[Marta Bunge]

Dear Bill and Ronnie,

 

In the context of topos theory it is
very natural to consider groupoids both for Galois toposes as for the van
Kampen theorems. 

 

 

See Marta Bunge, "Galois groupoids
and Covering Morphisms in Topos Theory", Fields Institute Communications,
volume 4 (2004) 131-161. I will be happy to send anyone interested a pdf file
of this paper. 

 

 

I quote from the Introduction:
"Also in section 4 we introduce and study the notion of a Galois topos
over an arbitrary base topos S. Although these relative Galois toposes are not
assumed to be either connected or pointed, they come naturally equipped with a
bag of points indexed by the connected components of a (non-connected)
universal cover; this is in line with the view advocated by Grothendieck and Brown
that, rather than a single base point, one ought to work with a suitable
"paquet des points", for instance, one that is invariant under the
symmetries in the given situation. This idea was naturally and independently
incorporated into topos theory both by Kennison and myself, by discussing  the
fundamental groupoid of an unpointed (and possibly pointless) locally connected
topos." The justification for these statements are given in detail in the
paper.

 

Here is the outline: 

 

1. Introduction. 

2. Locally constant objects in toposes. 

3. Stack completions and the fundamental
groupoid of a topos. 

4. Galois groupoids and Galois toposes. 

5. Locally paths simply connected
toposes over an arbitrary base. 

6. Generalized covering morphisms and a
van Kampen theorem.

References. 

 

With best regards, 

 

Marta

 

 

************************************************ 

 

Marta Bunge 

 

Professor Emerita 

 

Dept of Mathematics and Statistics 

 

McGill University 805 Sherbrooke St. West Montreal,
QC, Canada H3A 2K6 

 

Office: (514) 398-3810/3800 

 

Home: (514) 935-3618 

 

marta.bunge@mcgill.ca
http://www.math.mcgill.ca/~bunge/
************************************************




  		 	   		  

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