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We know locales are “better” than spaces, and that groupoids are “better” than groups (Edited to remove incorrect statement – see Mike Shulman’s comment below). I wanted to find a way of defining the fundamental groupoid that was natural from this point of view.
I’ve worked this out, and proved the Seifert-van Kampen theorem. Surprisingly, it all seems to work for lattices (with the exception of the preservation of products through the fundamental groupoid functor, which seems to only work with locales)!
I’m not in academia and I’m new to algebraic topology and category theory, so I would really appreciate feedback, and have any errors pointed out. Also, if this is interesting to anyone I would love to hear it! I have lots more ideas for extending this stuff that I haven’t worked out yet.
I’m going to use the language of locales, since that makes things more intuitive – but I will not use the distributivity, or existence of infinite joins.
Let L be a locale. A cover of L is a finite set of opens such that the join is $\top$, and every meet in the cover is the join of opens in the cover.
Covers form a category, where the morphisms are functions taking each open to an open containing it. I call these clumpings.
Opens from a cover C can overlap. In particular, an n-overlap is an open from C in the meet of a (multi-)set of opens from C, where the set has cardinality n. Note: every open in C which is contained in the meet is a distinct n-overlap.
A skeleton of a cover is a groupoid constructed in the following way:
For a cover C, we call this groupoid $Sk_C$.
Sk is a functor.
Finally, we make a diagram in the category of groupoids from all of the covers of L, along with their morphisms, using Sk. Then, we take the limit of this diagram, which is the fundamental groupoid of L. We call this $\pi L$.
$\pi$ is also a functor.
And of course, this functor preserves pushouts! I’m not going to go into the full proof, but the key idea is to use the first isomorphism theorem for frames/lattices (which we get because these are varietal).
A good thing to try all this stuff out on is the pseudocircle – of course we get a groupoid equivalent to $\mathbb{Z}$!
It’s kinda weird that we just stopped with 3-overlaps – the really cool part is that it seems to all still work if we just go to n! The main thing I’ve had trouble with in actually working this out is just understanding composition in n-groupoids enough to make sure everything is all good. My guess is that a Kan complex is the easiest kind of n-groupoid to do this for – but I am open to suggestions!
Anyway I hope this all at least approximately right! I have a paper with the proofs (and pictures!) that I can email to you if you’d like.
Hi, this sounds very interesting and very nice! Thanks for posting your ideas here! I would be very interested to see your paper. My email address is richard (at) rwilliamson-mathematics.info. I will be checking emails/here only intermittently over the next week or two, but will be happy to take a look when I get the chance.
Welcome!
I don’t think I would say that little localic homotopy theory has been done, though. Rather I would say that people tend to do the more general topos-theoretic homotopy theory, regarding locales as localic toposes. The “modern” approach to topos-theoretic homotopy theory is through higher toposes, and the homotopy groups of a topos are defined via its shape. The fundamental group thereby defined agrees with the classical fundamental group of a topos and also, I believe, with the more concretely unraveled versions defined in the papers cited at localic homotopy theory. Have you compared your definition with these?
Hi Richard, thank you! I’ll email a copy to you soon.
Thank you Mike for the clarification – I’ll admit that I haven’t been able to understand the shape theory stuff yet. My definition was partially inspired by Kennison’s paper What is the fundamental group, which is cited in the paper. In particular, the fourth definition uses (IIRC) chains of opens which overlap as a replacement for the traditional paths/loops in the standard definitions - and then takes the limit.
For shape theory, think of the lattice of open sets of a space, X. From an open cover of X one can build a simplicial complex, called the Cech nerve of the cover, that gives a model of the space in terms of the combinatorial information on the intersections etc of the cover. See the nLab entry Cech methods or the short note that I wrote here. Using finer covers one expects ’better’ information on the space. (In fact the individual nerves only give snapshots of the space and it is the system of all the nerevs that tells one much more.)
Thanks for linking me to your note, Tim – I’ve really enjoyed it!
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