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The following was a private email to Marta Bunge over some discussion on the category theory mailing list recently. She kindly gave me permission to post it to the list, and I’m putting it here so I can then later make an nlab page or two out of it
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 $\Omega X /\sim$ 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
I’ve also got some notes about generalising quasitopological groupoids to quasi-C groupoids for C either a concrete category or a (subcanonnical) site. The latter is much more conjectural, and I don’t know if it is good for anything, but people here may be able to make something of it. The question of sheaves on a quasi-C groupoid seems to be a mildly interesting one, perhaps, as I mention above.
One thing which intrigues me is whether sheaves on a quasitopological or quasi-C groupoid form a topos. This would have great bearing on the problem considered by Marta and myself (we aren’t working together on this, it’s just Zeitgeist I imagine).
Let $X$ be a totally path-disconnnected Hausdorff space, $X_+$ the same with a disjoint basepoint, and then consider the suspension $\Sigma X_+$
What would be a simple nontrivial example of this?
You don’t want too trivial, I guess, otherwise any discrete space is an example. My guess is $X = \{ 1/n | n \ge 1\}$ and then I think $\Sigma X_+$ is the Hawaiian earring. The real trick is to come up with a non-regular totally path-disconnected Hausdorff space. I’ve just asked at MathOverflow.
Whose terminology is ’quasitopological’ in the present instance?
Bourbaki: General Topology Vol 1.
OK, got an answer: take the real line with the K-topology (whatever that is) and then $X = \mathbb{Q}$ with the subspace topology from this.
Edit: from Wikipedia
… let $K = \{1/n | n \ge 1 \}$. Generate a topology on $\mathbb{R}$ by taking as basis all open intervals $(a, b)$ and all sets of the form $(a,b)-K$.
This topology is the K-topology.
This topology is the K-topology.
What about K-topology?
Done. Thanks for the prod, I’d meant to do it, honestly, but was trying to get ready for work this morning and missed my train anyway :)
Thanks, David!
I added (guess what…) a toc. And also linked to K-topology from real number. From a new stub section Topologies. Please improve that subsection.
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