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Stream: community: our work

Topic: Joshua Wrigley

Joshua Wrigley (Feb 22 2024 at 19:34):

Hi all,

As @Morgan Rogers (he/him) mentioned in another thread, I recently defended my PhD thesis, which is now available on my website.

As the title Doctrinal and Groupoidal Representations of Classifying Topoi suggests, the thesis is divided into two parts studying two different species of representing data - Lawverian doctrines and topological groupoids.

• Lawverian doctrines are a functorial generalisation of Lindenbaum-Tarski algebras for propositional theories. In Part A of the thesis, by expositing a classifying topos theory for doctrines, a general framework is constructed for their syntactic completions (thereby recovering, for example, @Davide Trotta's existential completion).
• First-order theories can be reconstructed from certain automorphism groups/isomorphism groupoids of their models (see, for instance, this example from model theory). Topos-theoretically, this can be seen in the representation results of Joyal-Tierney/Butz-Moerdijk. Part B of the thesis provides a model-theoretic characterisation of which open topological groupoids represent a theory, in the sense that its topos of sheaves classifies the theory. This is then used to establish a bi-equivalence between topoi (with enough points) and a localisation of topological groupoids.

It's a tad long, so I'd be happy summarising and answering questions here!

Kevin Arlin (Feb 22 2024 at 20:02):

Looks very cool, Josh! I've been trying to understand a bit about toposes with enough points lately and I've been wondering why it's always about sheaves on topological groupoids. I guess you can define sheaves on a topological category as well--do you know whether you get a topos in the usual way and we just often ignore these presentations of toposes since they're all presented by some topological groupoid anyway? I'm pretty sure this is the right idea since there are also some claims somewhere about the topos of sheaves on something as general as a simplical space but I've been wanting somebody on top of these things to confirm.

John Baez (Feb 22 2024 at 21:15):

As you probably know Kevin - indeed I don't even know why I'm butting in here, I don't know this stuff very well - the famous result is that every Grothendieck topos is equivalent to the topos of (equivariant) sheaves on some localic groupoid. This is not quite the same as a topological groupoid, but I think most topos theorists would argue locales are better behaved than topological spaces. So: while I don't think it's "always about sheaves on topological groupoids", there's a good reason to focus on sheaves on localic groupoids.

Morgan Rogers (he/him) (Feb 22 2024 at 21:19):

@John Baez the result of Butz and Moerdijk is that as soon as your topos has enough points you can use a topological groupoid instead :innocent: this is what Kevin was referring to.

Mike Shulman (Feb 22 2024 at 22:38):

Kevin Arlin said:

I guess you can define sheaves on a topological category as well--do you know whether you get a topos in the usual way

I believe this is true, even for localic categories, but I don't know a reference offhand.

In B3.4.14(a) of Sketches of an Elephant there is a remark that one can obtain the topos of presheaves on a discrete (meaning internal to the base topos) category as a "lax descent topos" which can be proven to be a topos "in the same way as for coinserters". Then in B3.4.14(b) the topos of sheaves on a localic groupoid is defined using a descent topos over its nerve. It seems clear that mimicking the construction of (b) using a localic groupoid and the lax descent topos referred to in (a) would yield a topos of sheaves on any localic category, but this doesn't seem to be stated explicitly there.

I wonder if it is in Moerdijk's monograph Classifying spaces and classifying topoi -- I don't have that with me right now to check.

Joshua Wrigley (Feb 23 2024 at 09:35):

Mike Shulman said:

Kevin Arlin said:

I guess you can define sheaves on a topological category as well--do you know whether you get a topos in the usual way

I believe this is true, even for localic categories, but I don't know a reference offhand.

Given a localic/topological category $\mathbb{C} = (C_1 \rightrightarrows C_0)$, its category of sheaves ${\bf Sh}(\mathbb{C})$ inherits the necessary exactness properties from the topos ${\bf Sh}(C_0)$, so one just needs to show that ${\bf Sh}(\mathbb{C})$ has a small set of generators. Much like Mike, I can't recall a reference for this off the top of my head (in the case of localic/topological groupoids, this is found in Moerdijk).

Joshua Wrigley (Feb 23 2024 at 09:43):

Kevin Arlin said:

Looks very cool, Josh! I've been trying to understand a bit about toposes with enough points lately and I've been wondering why it's always about sheaves on topological groupoids.

As regards for why I focus on open topological groupoids: there is a technical lemma (results VII.23-24 in the thesis) lifted straight from Henrik Forssell's work that resists an obvious generalisation. This result is required to apply the theory of internal locales and deduce the classification theorem.

Joshua Wrigley (Feb 23 2024 at 10:09):

@John Baez said:

I think most topos theorists would argue locales are better behaved than topological spaces.

This is precisely what Johnstone said when I presented at CT2023! I will say something vague in reply: locales are syntactic constructions, but for topological spaces points need to exist. This parallels the idea that straight logic is the syntactic manipulation of formulae, but model theory concerns what those formulae describe. So, with much hand-waving, to get a model-theoretic flavour, one needs to consider topological representing data.

This is most clearly seen in the distinction between the Joyal-Tierney/Dubuc (i.e. localic) and Caramello (i.e. topological) representations of a connected atomic topos. Let $\mathcal{E}$ be a connected atomic topos with a point ${\bf Set} \xrightarrow{M} \mathcal{E}$, then:

1. $\mathcal{E}$ is always equivalent to ${\bf B} G$ for $G$ the localic automorphism group of $M$.
2. but $\mathcal{E} \simeq {\bf B} G$ for $G$ the topological automorphism group of $M$ if and only if $M$ is ultrahomogeneous in the model-theoretic sense.