fundamental localic groupoid?

fundamental localic groupoid?

[David Roberts]

Hi all,

it is (or should be) well known that the fundamental groupoid of a locally
connected, semilocally 1-connected space X can be given a topology such that
it is a topological groupoid (composition continuous etc). Without these
assumptions on X there are counterexamples to the continuity of composition
(c.f. misguided attempts to build a topological fundamental group). I was
wondering, though, if there is a localic fundamental groupoid of an
arbitrary space. One could presumably pass the the topos Sh(X) and then
consider the fundamental groupoid of that, but I was wondering if there was
a way to pass directly from the description of the arrows of Pi_1(X) as a
set of classes of paths to a locale of classes of paths, and thence to a
localic groupoid. My only 'evidence' that this might be the case is that the
product in Loc is different to the product in Top, and so this may provide a
work around the non-continuity of composition.

Thanks,

David Roberts


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Re: fundamental localic groupoid?

[Ronnie Brown]

David Roberts writes about topologising the fundamental groupoid that
`this should be well known'.

Here are some references on that:
1.  (R.BROWN with G. DANESH-NARUIE), ``The fundamental groupoid as a
topological groupoid'', {\em Proc. Edinburgh Math. Soc.} 19 (1975) 237-244.

2. various editions of my book published now as `Topology and
Groupoids', see 10.5.8, p.385 (the result is more general since it deals
with topologising (\pi X)/N  ).

The method relies on the result that if X is reasonably nice, then a
covering morphism of groupoids G \to \pi_1 X determines a `lifted'
topology on Y = Ob(G) for which \pi_1(Y) \cong G.

Presumably these methods cannot be adapted or modified for locales???

Ronnie Brown


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Re: fundamental localic groupoid?

[Steven Vickers]

Dear David,

I don't know the full answer, but here are some thoughts.

Let I be the (localic) compact real interval [0,1]. I is locally compact
and so for any locale X the locale exponential X^I exists and is the locale
of paths in X. The endpoints of I give two maps d0 and d1: X^I -> X and we
have the groupoid operations in the obvious way. (For composition this
needs that I is a pushout of two copies of itself, glued end to end. This
is not obvious, but I believe I once proved it to myself and perhaps it is
known anyway. Exponentiation then transforms that pushout into a pullback,
showing that X^I is in fact homeomorphic to the locale of composable pairs
of paths.)

The groupoid laws cannot hold until we have factored out homotopy, which
must therefore be the next step. If gamma and delta are two paths agreeing
at the endpoints, then a homotopy from gamma to delta, relative to the
endpoints, can be expressed as a map D -> X where D is the closed Euclidean
2-ball (a closed disc). This uses two maps I -> D, taking I to the upper
and lower half boundaries of D, and giving two maps du and dl: X^D -> X^I.
Hence the fundamental groupoid Pi_1(X) should be the coequalizer of du and
dl.

It's not obvious to me that the path composition can be transferred to that
coequalizer. I think we would like the coequalizer to be stable under
pullback, but that is not always true for locales. Perhaps it is in this
case.

Once the groupoid operations have been established on the coequalizer, I
conjecture it's not going to be too hard to prove the groupoid laws, using
the usual constructions of homotopy theory.

Regards,

Steve Vickers.

On Wed, 28 Apr 2010 11:54:29 +0930, David Roberts
<droberts@maths.adelaide.edu.au> wrote:
> Hi all,
> 
> it is (or should be) well known that the fundamental groupoid of a
locally
> connected, semilocally 1-connected space X can be given a topology such
> that
> it is a topological groupoid (composition continuous etc). Without these
> assumptions on X there are counterexamples to the continuity of
composition
> (c.f. misguided attempts to build a topological fundamental group). I was
> wondering, though, if there is a localic fundamental groupoid of an
> arbitrary space. One could presumably pass the the topos Sh(X) and then
> consider the fundamental groupoid of that, but I was wondering if there
was
> a way to pass directly from the description of the arrows of Pi_1(X) as a
> set of classes of paths to a locale of classes of paths, and thence to a
> localic groupoid. My only 'evidence' that this might be the case is that
> the
> product in Loc is different to the product in Top, and so this may
provide
> a
> work around the non-continuity of composition.
> 
> Thanks,
> 
> David Roberts
> 
> 
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]


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Re: fundamental localic groupoid?

[David Roberts]

Steven Vickers wrote:

>
> It's not obvious to me that the path composition can be transferred to that
> coequalizer. I think we would like the coequalizer to be stable under
> pullback, but that is not always true for locales. Perhaps it is in this
> case.
>
>
This is precisely where the construction falls down for a non-semilocally
1-connected space. If we restrict to looking at the automorphisms as a point
(which supposedly form a topological group) the multiplication in the group
pi_1 x pi_1 -> pi_1 (i.e. concatenating equivalence classes of loops) is not
continuous in both variables, even though it is separately continuous (*).
This can be traced back to the simple fact that the product of a pair of
quotient maps in Top is not necessarily a quotient map. Note that this is
the usual product in Top, not the compactly generated product. I presume
there is a similar hitch with locales?

(*) I learned of the non-continuity of the multiplication map from, among
other places, the article

J. Brazas, The topological fundamental group and hoop earring spaces, 2009,
arXiv:0910.3685

which dashes optimistic earlier constructions.


Going back to more general thoughts, given an arbitrary space X, the topos
Sh(X) is the classifying topos of some localic groupoid G_X (Joyal-Tierney),
which I thought (!) could be interpreted as the fundamental groupoid in nice
situations. Here I suppose is my real question: what relation is there
between G_X and the (ordinary) fundamental groupoid Pi_1(X)? I suppose a
test case could be the Warsaw circle W: Pi_1(W) is equivalent to the
discrete groupoid on the set {a,b}, and surely G_W is more than this.

One last question: has anyone studied the relation between (strong) shape of
a space X and this localic groupoid G_X?

David Roberts

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Re: fundamental_localic_groupoid?

[Joyal, André]

David Roberts wrote:


>Going back to more general thoughts, given an arbitrary space X, the topos
>Sh(X) is the classifying topos of some localic groupoid G_X (Joyal-Tierney),
>which I thought (!) could be interpreted as the fundamental groupoid in nice
>situations. Here I suppose is my real question: what relation is there
>between G_X and the (ordinary) fundamental groupoid Pi_1(X)? I suppose a
>test case could be the Warsaw circle W: Pi_1(W) is equivalent to the
>discrete groupoid on the set {a,b}, and surely G_W is more than this.

I would like to make a few observations.
If my recollection is right,  during the 1980's the dream of some topos theorists 
(including myself) was to use atomic toposes as generalised K(pi,1)-spaces.
The dream has not materialised and it maybe foolish.

Another avenue is to extend Grothendieck Galois theory, since the
fundamental groupoid of a space classifies the covers of that space.
Eduardo Dubuc is presently developing a general theory of Galois toposes.
The fundamental Galois topos of a topos E could be defined as the 
reflection E--->E(1) of E into the full subcategory of Galois toposes
over the base topos S. I am not supposing that S is the category of sets.
I am writing E(1) because I want to suggest that it is a stage of
a Postnikov tower of S-toposes: 

E(-1)<---E(0)<---E(1)<----E(2)<----

The topos E(-1) is of course the image of the 
geometric morphisme E-->S. The construction of E(-1) 
depends on the factorisation system 

(epimorphisms of toposes, sub-toposes)

The topos E(0) is the totally disconnected localic reflection of E
over the base topos S. Its construction
depends on the factorisation system 

(connected morphisms of toposes, totally disconnected localic morphisms)

The topos E(1) is the Galois topos reflection of E 
over the base topos S. Its construction
depends on the factorisation system 

(1-connected morphisms of toposes, Galois morphisms)


The topos E(2) does not exists in general for the
simple reason that it is a 2-topos, not an ordinary
topos. A 2-topos is enriched over groupoids.
 
The whole theory depends on three factorisation systems 

(epimorphisms of toposes, sub-toposes)
(connected morphisms of toposes, totally disconnected localic morphisms)
(1-connected morphisms of toposes, Galois morphisms)

The first factorisation system is well known and 
was constructed a long time ago.

Excuse my ignorance, but I do not know if the 
second factorisation system has been fully constructed. 
Can someone tell me?

The third factorisation system was partially
constructed by Dubuc: he can factor
the morphism E-->S when S is the category of sets.

My observations concerning your problems
could be off the mark.


Best,
André




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RE : categories: Re: fundamental localic groupoi

[Joyal, André]

David Roberts wrote:


>Going back to more general thoughts, given an arbitrary space X, the topos
>Sh(X) is the classifying topos of some localic groupoid G_X (Joyal-Tierney),
>which I thought (!) could be interpreted as the fundamental groupoid in nice
>situations. Here I suppose is my real question: what relation is there
>between G_X and the (ordinary) fundamental groupoid Pi_1(X)? I suppose a
>test case could be the Warsaw circle W: Pi_1(W) is equivalent to the
>discrete groupoid on the set {a,b}, and surely G_W is more than this.

I would like to make a few observations.
If my recollection is right,  during the 1980's the dream of some topos theorists 
(including myself) was to use atomic toposes as generalised K(pi,1)-spaces.
The dream has not materialised and it maybe foolish.

Another avenue is to extend Grothendieck Galois theory, since the
fundamental groupoid of a space classifies the covers of that space.
Eduardo Dubuc is presently developing a general theory of Galois toposes.
The fundamental Galois topos of a topos E could be defined as the 
reflection E--->E(1) of E into the full subcategory of Galois toposes
over the base topos S. I am not supposing that S is the category of sets.
I am writing E(1) because I want to suggest that it is a stage of
a Postnikov tower of S-toposes: 

E(-1)<---E(0)<---E(1)<----E(2)<----

The topos E(-1) is of course the image of the 
geometric morphisme E-->S. The construction of E(-1) 
depends on the factorisation system 

(epimorphisms of toposes, sub-toposes)

The topos E(0) is the totally disconnected localic reflection of E
over the base topos S. Its construction
depends on the factorisation system 

(connected morphisms of toposes, totally disconnected localic morphisms)

The topos E(1) is the Galois topos reflection of E 
over the base topos S. Its construction
depends on the factorisation system 

(1-connected morphisms of toposes, Galois morphisms)


The topos E(2) does not exists in general for the
simple reason that it is a 2-topos, not an ordinary
topos. A 2-topos is enriched over groupoids.
 
The whole theory depends on three factorisation systems 

(epimorphisms of toposes, sub-toposes)
(connected morphisms of toposes, totally disconnected localic morphisms)
(1-connected morphisms of toposes, Galois morphisms)

The first factorisation system is well known and 
was constructed a long time ago.

Excuse my ignorance, but I do not know if the 
second factorisation system has been fully constructed. 
Can someone tell me?

The third factorisation system was partially
constructed by Dubuc: he can factor
the morphism E-->S when S is the category of sets.

My observations concerning your problems
could be off the mark.


Best,
André





-------- Message d'origine--------
De: categories@mta.ca de la part de David Roberts
Date: jeu. 29/04/2010 10:09
À: categories@mta.ca
Objet : categories: Re: fundamental localic groupoid?
 
Steven Vickers wrote:

>
> It's not obvious to me that the path composition can be transferred to that
> coequalizer. I think we would like the coequalizer to be stable under
> pullback, but that is not always true for locales. Perhaps it is in this
> case.
>
>
This is precisely where the construction falls down for a non-semilocally
1-connected space. If we restrict to looking at the automorphisms as a point
(which supposedly form a topological group) the multiplication in the group
pi_1 x pi_1 -> pi_1 (i.e. concatenating equivalence classes of loops) is not
continuous in both variables, even though it is separately continuous (*).
This can be traced back to the simple fact that the product of a pair of
quotient maps in Top is not necessarily a quotient map. Note that this is
the usual product in Top, not the compactly generated product. I presume
there is a similar hitch with locales?

(*) I learned of the non-continuity of the multiplication map from, among
other places, the article

J. Brazas, The topological fundamental group and hoop earring spaces, 2009,
arXiv:0910.3685

which dashes optimistic earlier constructions.


Going back to more general thoughts, given an arbitrary space X, the topos
Sh(X) is the classifying topos of some localic groupoid G_X (Joyal-Tierney),
which I thought (!) could be interpreted as the fundamental groupoid in nice
situations. Here I suppose is my real question: what relation is there
between G_X and the (ordinary) fundamental groupoid Pi_1(X)? I suppose a
test case could be the Warsaw circle W: Pi_1(W) is equivalent to the
discrete groupoid on the set {a,b}, and surely G_W is more than this.

One last question: has anyone studied the relation between (strong) shape of
a space X and this localic groupoid G_X?

David Roberts


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Re fundamental localic groupoid?

[Joyal, André]

Dear Marta,

I thank you for the information.
It seems that you have constructed the factorisation system:

(connected morphisms of toposes, totally disconnected localic morphisms)

It depends on identifying correctly the internal notion
of totally disconnected locale. I will read your papers:

> Marta Bunge, On two non-discrete localic generalizatins of \Pi_0, Cahiers Issue on the celebration of the 100th anniversary of the birth of Charles Ehresmann.  

>Marta Bunge, Fundamental Pushout Toposes,Theory and Applications of Categories 20 (2008) 186-214.

Best,
André


-------- Message d'origine--------
De: martabunge@hotmail.com de la part de Marta Bunge
Date: dim. 02/05/2010 12:17
À: Joyal, André; David Roberts
Cc: Robert Rosebrugh; Eduardo Dubuc; Peter Johnstone
Objet : RE: RE : categories: Re: fundamental localic groupoid?
 

Dear Andre, Dear All,
I just sent a message to categories in response to the specific question of Robert Davis, prompted by remarks made by Peter Johnstone. 
It concerned of a locally connected Grothendieck topos, in particular in the localic case. The three papers mentioned in my previous message were from the 1990's. 

More relevant to the message by Andre is the following reference on Galois toposes in the locally connected case.
Marta Bunge, Galois groupoids and covering morphisms in topos theory,Proceedings of the Fields Institute: Workshop on Descent, Galois Theory and Hopf Algebras,Fields Institute Communications, American Mathematical Society, 2004, 131-162.

In the general case (not necessarily locally connected) the the totally disconnected and zero-dimensional reflections are constructed in:

Marta Bunge, On two non-discrete localic generalizatins of \Pi_0, Cahiers Issue on the celebration of the 100th anniversary of the birth of Charles Ehresmann.  

Marta Bunge, Fundamental Pushout Toposes,Theory and Applications of Categories 20 (2008) 186-214.

As I explained in my previous message, I have no time for commnents right now. 
Cordial regards,
Marta





************************************************
Marta Bunge
Professor Emerita
Dept of Mathematics and Statistics 
McGill UniversityBurnside Hall, Office 1005
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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Re: fundamental localic groupoid?

[Marta Bunge]

> 
> Dear Andre, Dear All,
> I just sent a message to categories in response to the specific question of Robert Davis, prompted by remarks made by Peter Johnstone. 
> It concerned of a locally connected Grothendieck topos, in particular in the localic case. The three papers mentioned in my previous message were from the 1990's. 
> 
> More relevant to the message by Andre is the following reference on Galois toposes in the locally connected case.
> Marta Bunge, Galois groupoids and covering morphisms in topos theory,Proceedings of the Fields Institute: Workshop on Descent, Galois Theory and Hopf Algebras,Fields Institute Communications, American Mathematical Society, 2004, 131-162.
> 
> In the general case (not necessarily locally connected) the the totally disconnected and zero-dimensional reflections are constructed in:
> 
> Marta Bunge, On two non-discrete localic generalizatins of \Pi_0, Cahiers Issue on the celebration of the 100th anniversary of the birth of Charles Ehresmann.  
> 
> Marta Bunge, Fundamental Pushout Toposes,Theory and Applications of Categories 20 (2008) 186-214.
> 
> As I explained in my previous message, I have no time for commnents right now. 
> Cordial regards,
> Marta
> 
> 
> 
> 
> 
> ************************************************
> Marta Bunge
> Professor Emerita
> Dept of Mathematics and Statistics 
> McGill UniversityBurnside Hall, Office 1005
> 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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Re: fundamental localic groupoid?

[P.T.Johnstone]

I'm surprised that no-one has yet replied to David's original question
by citing the work of Marta Bunge; she has a paper called (I think)
"Classifying toposes and fundamental localic groupoids" dating from the
early 1990s. (Being away from home at present, I don't have the exact
reference to hand; I was hoping Marta would provide it.)

Peter Johnstone

On Apr 28 2010, David Roberts wrote:

>Hi all,
>
> it is (or should be) well known that the fundamental groupoid of a
> locally connected, semilocally 1-connected space X can be given a
> topology such that it is a topological groupoid (composition continuous
> etc). Without these assumptions on X there are counterexamples to the
> continuity of composition (c.f. misguided attempts to build a topological
> fundamental group). I was wondering, though, if there is a localic
> fundamental groupoid of an arbitrary space. One could presumably pass the
> the topos Sh(X) and then consider the fundamental groupoid of that, but I
> was wondering if there was a way to pass directly from the description of
> the arrows of Pi_1(X) as a set of classes of paths to a locale of classes
> of paths, and thence to a localic groupoid. My only 'evidence' that this
> might be the case is that the product in Loc is different to the product
> in Top, and so this may provide a work around the non-continuity of
> composition.
>
>Thanks,
>
>David Roberts
>

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Re: fundamental localic groupoid?

[Eduardo J. Dubuc]

Well Peter, I imagine that the reason is that the question was asking for
"locale of path", and not for the "local of automorphisms of the fiber", which
  is completely developed and solved.

I copy and paste the question:

"I was wondering if there was a way to pass directly from the description of
the arrows of Pi_1(X) as a set of classes of paths to a locale of classes
of paths, and thence to a localic groupoid."

As you know, there are many papers on the fundamental localic groupoid, but
all of then deal (with variations) with the locale of automorphism.

The works I know (of course they may be also other authors I ignore) were made
(in a cronological order by first contribution, but later mixed in time) by:

Grothendieck-Tierney (Tierney first observed that the actions of a grothendiek
progroup were the same thing that the actions of the localic group inverse
image in the category of locales) Moerdijk, Kennison, Bunge and Dubuc. In the
case of topological spaces there is work done by Hernandez Paricio. There is
also the often cited memoir of Joyal-Tierney.

However, there is one paper i know which in some sense deals with paths and
which may have some relevance to the question asked:

Bunge M., Moerdijk I.,On the construction of the Grothendieck
fundamental group of a topos by paths, J. Pure Appl. Alg. 116 (1997).


P.T.Johnstone@dpmms.cam.ac.uk wrote:
> I'm surprised that no-one has yet replied to David's original question
> by citing the work of Marta Bunge; she has a paper called (I think)
> "Classifying toposes and fundamental localic groupoids" dating from the
> early 1990s. (Being away from home at present, I don't have the exact
> reference to hand; I was hoping Marta would provide it.)
>
> Peter Johnstone
>

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Re: fundamental localic groupoid?

[Eduardo J. Dubuc]

David Roberts raised some questions that gave rise to several answers which,
being interesting in themselves, "miss the mark" (as Joyal said) concerning
the original questions.

I copy and paste from Davis Roberts postings:

"but I was wondering if there was a way to pass directly from the description
of the arrows of Pi_1(X) as a set of classes of paths to a locale of classes
of paths, and thence to a localic groupoid."

"Going back to more general thoughts, given an arbitrary space X, the topos
Sh(X) is the classifying topos of some localic groupoid G_X (Joyal-Tierney),
which I thought (!) could be interpreted as the fundamental groupoid in nice
situations. Here I suppose is my real question: what relation is there between
G_X and the (ordinary) fundamental groupoid Pi_1(X)? I suppose a test case
could be the Warsaw circle W: Pi_1(W) is equivalent to the discrete groupoid
on the set {a,b}, and surely G_W is more than this."

The fundamental group, progroup, localic group, groupoid, localic groupoid,
etc etc are all required (fundamentally !!) to represent first degree
cohomology. This of course characterize them. Their classifying toposes should
be the categories of "covering projections" (= locally constant sheaves in the
locally-connected case).

Classically  Pi_1(X) was a set of classes of paths. Grothendieck started a
departure (of course, out of facts already known at the time) from this, and
constructed Pi_1(X) via fiber functors, where a topology appears for Pi_1.
This Pi_1  can also be done combinatorialy by associating groupoids to family
covers, and with several variations yields pro-things or localic things or
even pro-localic things.

Now, the equivalence with the Poincare Groupoid of paths holds (for
topological spaces) strictly for the case of "locally connected, semilocally
1-connected space X" and no more. The proof does not work in any other case.
Notice that this case correspond to the existence of a universal covering,
that is, the fiber functor is representable.

Bunge-Moerdijk studied paths in the general topos case, (the topos of sheaves
on the unit interval playing the role of the unit interval), but their
approach yields the good Pi_1 only in the representable case.

So, it seems that under the present ideas concerning what a path is or should
be, paths can not be used to construct the correct Pi_1 outside the
representable case, where only discrete groupoids are pertinent, and no
topology (or locales) is necessary.

Some new ideas on a correct generalization of the notion of path ?



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Re: fundamental localic groupoid?

[David Roberts]

Hi Eduardo, Marta, Andre, Peter and everyone else.

Thank you all for your replies.

I knew Marta's work would be applicable, but wasn't sure how much.
This was the sort of clarification I was looking for

> Bunge-Moerdijk studied paths in the general topos case, (the topos of
> sheaves on the unit interval playing the role of the unit interval), but
> their approach yields the good Pi_1 only in the representable case.

It was my lack of knowledge of things localic that was hindering me.

But I am now intrigued by Andre's mention of the 2-topos E(2). Given
the number of approaches to 2-topoi (Street, Weber, Shulman) I know
this is a little bit more involved than a "groupoid-enriched topos"
(although I imagine Andre was just emphasising that it is a
(2,1)-category, not a general bicategory).

The reason I am interested is that in my thesis I extend the
(geometric*) fundamental bigroupoid Pi_2 from spaces to topological
groupoids, and construct a topological groupoid over a 'nice enough'
space which is, according Pi_2 2-connected. The construction is
functorial, and will also work for manifolds to give a (Frechet) Lie
groupoid (this joint work with Andrew Stacey). There is a subtlety in
that 'nice enough' means locally simply connected and locally
relatively 2-connected -- 2-well connected in the terminology of my
thesis -- but for this to be locally trivial, like the universal
covering space, 'nice enough' means having a basis of relatively
contractible open sets. The construction mimics the construction of
the universal covering space, in that it is the source fibre of the
fundamental bigroupoid with an appropriate topology.

(*geometric, in the sense that it isn't some sort of composition of
functors through (bi)simplicial sets, or using geometric realisation,
or an adjoint to N:2Gpd -> sSet; but using paths and surfaces and so
on)

Now I'm trying to imagine what relation the tower of topoi that Andre
sketched has to the usual Whitehead tower in the case of a
Grothendieck topos Sh(X) for a space X. It is probably the case that
the 'Joyal tower' corresponds to the classical notion up to the
maximum dimension n that X is locally n-connected (or more generally:
n-well connected = 'locally (n-1)-connected and locally relatively
n-connected') - and then one needs to use techniques more along the
lines of Marta and coworkers.

Actually (and this is thinking out loud) E(1), being a Galois topos,
is sheaves on some groupoid, so I imagine E(2) is probably stacks (of
groupoids) on a bigroupoid, and hence of course it is a
(2,1)-category.. hmm. In the nice case that there is an initial object
in E(2), I suppose I could conjecture that this initial stack is
presented by my 2-connected cover, but I have no evidence whatsoever
for it beyond a vague analogy.

Food for thought.

Thanks again,

David

PS: Despite being called Robert (or Rob) more times than I can
remember, I have picked up a swag of new names in the past few days!
Very peculiar :-)


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Re: fundamental localic groupoid?

[David Roberts]

Hi all,

the following message was in reply to a private email from Marta,
asking for more details about counterexamples to a topological
fundamental groupoid for all spaces. Note that none of it is my own
work, except the speculative comments at the end.

David

---------- Forwarded message ----------
From: David Roberts <droberts@maths.adelaide.edu.au>
Date: 3 May 2010 17:15
Subject: Re: categories: fundamental localic groupoid?
To: Marta Bunge <marta.bunge@mcgill.ca>


Hi Marta,

The passage I quoted was from Eduardo Dubuc - I was being a bit slack
in not attributing the quote.

> Secondly, in your original message, you wrote something relevant to one of
> the questions I have been trying to give an answer to -- namely, whether
> there is a construction of the paths version of the fundamental localic
> groupoid of a Grothendieck topos in the non locally connected case,
> considering that there is one such for the coverings version of it.

So you are wanting a construction of the localic Pi_1(Sh(x)) from
Joyal-Tierney in terms of paths, whatever "paths" means? If so, that
is about the gist of what I was wondering too.

>You mentioned counterexamples to previous attempts. Could you be more precise
> about such (misguided) attempts and to the counterexamples?

Here goes.

The 'topological fundamental group', pi_1^top(X,x) is the space \Omega
X /~ of loops at x, mod the relation of homotopy as usual for the
fundamental group. The underlying set is that of the ordinary
fundamental group. Various people, including Bliss

D. K. Bliss, The topological fundamental group and generalized
covering spaces, Topology Appl., 124(3) (2002), 355-371

have wrongly assumed that the product given by concatenation of loops
is continuous, and so pi_1^top(X,x) is a topological group. This is
not necessarily the case, as the proof relies on the assumption that a
product of identification maps is again an identification map. This is
not always true in the category of all topological spaces with the
usual product (but I believe it is true in the category of locally
compact Hausdorff spaces - I think this is in Brown's paper 'Ten
topologies for X \times Y'). It does leave open the question as to
whether or not there are spaces where the product is discontinuous,
and in the paper

J. Brazas, The topological fundamental group and hoop earring spaces,
2009, arXiv:0910.3685

the author constructs a class of counterexamples as follows (I haven't
personally checked this):

Let X be a totally path-disconnnected Hausdorff space, X_+ the same
with a disjoint basepoint, and then consider the suspension \Sigma X_+
with basepoint *. Then the author shows that pi_1^top(\Sigma X_+,*) is
T_1 but if X is not a regular space, pi_1^top(\Sigma X_+,*) is not
regular, hence not a topological group.

It is true that pi_1^top is a functor from Top to the category of
quasi-topological groups: that is, topological groups minus the
condition that multiplication is continuous, only that left and right
multiplication L_g, R_g is continuous in each element g.

It seems to me to be immediate that there is a 'quasi-topological
fundamental groupoid', where left and right composition by any path
is continuous, but not the whole composition map

G_1 \times_{G_0^2} G_1 \to G_1.

One could then consider a (suitable) category of sheaves on this
groupoid and see what arises.

Do you mind if I cross post this to the categories mailing list, in
case others are curious about details?

Kind regards,

David


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