Todd has a “do not tamper” request up on the page at the moment, so I haven’t actually added this (and perhaps Todd intends to add precisely what I am about to say) but I think that the page needs to mention and prove the following remark: that the graph minors of $G$ are precisely the graphs $H$ such that there exists a monomorphic functor $\Pi_1 H\to\Pi_1 G$, where $\Pi_1$ is the “groupoid of paths” functor.

]]>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:

- Every 1-overlap of opens in the cover generates an element in the groupoid.
- Every 2-overlap generates an edge (i.e. isomorphism) in the groupoid. This edge goes between the elements generated from the corresponding 1-overlaps we get if we take out 1 of the opens.
- Every 3-overlap generates a composition relation between the three edges we get by looking at the edges generated from the corresponding 2-overlaps we get if we take out 1 of the opens.

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.

]]>It would be nice if somebody more competent in this area expanded it.

The above equivalence of categories can in fact be lifted to an equivalence between the bicategory of localic groupoids, complete flat bispaces, and their morphisms and the bicategory of Grothendieck toposes, geometric morphisms, and natural transformations. The equivalence is implemented by the classifying topos functor, as explained in

Ieke Moerdijk, The classifying topos of a continuous groupoid II,

Cahiers de topologie et géométrie différentielle catégoriques 31, no. 2 (1990), 137–168. ]]>