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Stream: event: Online CT seminar

Topic: Aug 1 0100 GMT - David Roberts - Grothendieck "canonical"

Eric M Downes (Jul 25 2024 at 15:57):

Different Time! 2024-08-01T01:00:00+00:00 (David confirmed!)
For the Aussies, European insomniacs, and any North/South Americans desiring an after-dinner think about Grothendieck's use of "canonical"!

From DMR himself

Title: Tracing the origin of 'canonical' in Grothendieck's work.

Abstract: Unlike naturality, the concept of 'canonicality' doesn't have an accepted rigorous definition. Yet we find mathematicians use the word with a 'you know it when you see it' approach, and are even happy to 'canonically identify' objects when convenient. A prominent example of this is Grothendieck and Dieudonné's seminal work EGA. Kevin Buzzard has recently popularised the potential issues with such an attitude, most importantly because in computer formalisation one cannot sweep such things under the rug. I will trace the usage of 'canonical' in this context backwards from EGA1, and offer some speculative ideas on how the choice of word might have been prompted at the moment it was introduced.

In a separate thread David posed a challenge to me, which I will extend to anyone else wanting to engage concrete canonicity creatively:

What I'm after is a nontrivial functor $F$ with codomain Set (for now, the category of ZFC sets) that takes the value $X$ on some object $A$, another functor $G$ that takes the value $S$ on that same object, and also an isomorphism natural transformation from the second functor to the first. This means that if I have $f\colon A\stackrel{\simeq}{\to} B$, then $F(B)$ contains $G(B)$ as a literal subset, and then we have $F(f)\big|_S = G(f)\colon S \stackrel{\simeq}{\to} G(B)$ as functions.

One can get this in a stupid way by saying the domain of $F$ and $G$ is the trivial, single object, single arrow category. But then this would imply that every single morphism in any category is 'canonical'.

Alexander Campbell (Jul 25 2024 at 22:03):

The time of the talk in your message displays to me as "Thu, Aug 1, 2024, 11:00 AM" according to my Sydney time zone (UTC+10), which is half an hour ahead of David's time zone.

Eric M Downes (Jul 25 2024 at 23:03):

Thanks, Alexander. Yes, I did actually check the time with David previously, but it doesn't seem like we confirmed the day. Hurray for half-hour time-zones. :)

David Michael Roberts (Jul 26 2024 at 00:29):

I like your way of putting it, that one has a chance of engaging creatively.

David Michael Roberts (Jul 26 2024 at 00:32):

I'm not convinced that the function $F$ in that challenge exists, but I'm willing to be proved wrong.

Todd Trimble (Jul 26 2024 at 00:50):

I confess that I'm having trouble even understanding that question. Is the separate thread here in this Zulip?

David Michael Roberts (Jul 26 2024 at 01:14):

https://categorytheory.zulipchat.com/#narrow/stream/229199-learning.3A-questions/topic/Aut.28Torsor.29.20.3D.20Inn.28Torsor.29.3F/near/447612243

David Michael Roberts (Jul 26 2024 at 01:15):

@Todd Trimble Bourbaki call a subset inclusion (in a material set theory) a canonical map. My question was to people who think that 'canonical' should mean 'core-natural transformation': what are the functors and what is the transformation for such an inclusion map, in that case?

Todd Trimble (Jul 26 2024 at 02:09):

Okay, it took a while even to get a sense of what the question was driving at, but if I had to rephrase what I'm guessing is the intent, I'd try "what's so canonical about a subset inclusion?" Nothing? Then why does Bourbaki call it canonical?

David Michael Roberts (Jul 26 2024 at 02:32):

I mean, it seems like the type of function you'd call canonical...

I have Thoughts about why it might be like that, but it's a bit of a guess.

Todd Trimble (Jul 26 2024 at 03:10):

Well, obviously I was interpreting it in the sense of "what's so core-natural canonical about a subset inclusion"?, but I guess you're now trying to imagine the mindset of Bourbaki in your reply.

David Michael Roberts (Jul 26 2024 at 05:27):

Well the mindset of one particular Bourbaki member, yes ;-)

David Michael Roberts (Jul 26 2024 at 05:29):

Core-natural tries to capture what "canonicity" is in practice, but it doesn't quite get there, imo. Bourbaki also had a definition and I presume subset inclusion satisfied it.

Federica Pasqualone (Jul 26 2024 at 08:53):

Alexander Campbell said:

The time of the talk in your message displays to me as "Thu, Aug 1, 2024, 11:00 AM" according to my Sydney time zone (UTC+10), which is half an hour ahead of David's time zone.

... and it is 3AM Berlin time ! :grinning: