Re: fundamental localic groupoid?

[David Roberts <droberts@maths.adelaide.edu.au>]

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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