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March 21, 2016

Coalgebraic Geometry

Posted by Qiaochu Yuan

Hi everyone! As some of you may remember, some time ago I was invited to post on the Café, but regrettably I never got around to doing so until now. Mainly I thought that the posts I wanted to write would be old hat to Café veterans, and also I wasn’t used to the interface.

Recently I decided I could at least try occasionally linking to posts I’ve written over at Annoying Precision and seeing how that goes. So, I’ve written two posts on how to start thinking about cocommutative coalgebras as “distributions” on spaces of some sort:

For experts, there’s a background fact I’m dancing around but not stating explicitly, which is that over a field, the category of cocommutative coalgebras is equivalent to the opposite of the category of profinite commutative algebras, which we can interpret as a category of formal schemes. But these posts were already getting too long; I’m trying to say fewer things about each topic I write about so I can write about more topics.

Posted at March 21, 2016 5:00 AM UTC

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6 Comments & 0 Trackbacks

Re: Coalgebraic Geometry

This should be interesting! But it looks like your post was cut off.

A few months ago John and Mike and maybe a few others were discussing coalgebras and distributions, and I had started writing a long comment which I never finished and never posted. I’m wondering if you’re going to save me the trouble of trying to get back to completing it!

Posted by: Todd Trimble on March 21, 2016 11:42 AM | Permalink | Reply to this

Re: Coalgebraic Geometry

Oh, silly me. It wasn’t cut off; you’ve just linked to articles on Annoying Precision.

Anyway, hope to see more from you here!

Posted by: Todd Trimble on March 21, 2016 11:48 AM | Permalink | Reply to this

Re: Coalgebraic Geometry

Qiaochu wrote:

Mainly I thought that the posts I wanted to write would be old hat to Café veterans, and also I wasn’t used to the interface.

I hope you post more here!

I understand how tiresome it is to learn a new interface; if you have any questions just fire me an email. Just for kicks, I rewrote this blog entry of yours after choosing ‘Markdown with itex to MathML’ under ‘Text formatting’ down at the bottom. This means you don’t need stuff like the <br/><br/><br/><br/> that you were using to create line breaks. You’d chosen no text formatting at all, which makes life harder.

I think it’s fine, in fact desirable, to post articles that are “old hat to Café veterans”. As this Café ages, its veterans have become dismayingly erudite, and the posts correspondingly harder to read, discouraging newcomers.

When we started, it was a freewheeling place! Lately I’ve been hearing people say they’re scared to post questions here. When it reaches this point, I think we really have a duty to explain what we’re doing. I spend a lot of time explaining stuff on Azimuth, but not so much pure math, because I’ve gotten caught up in other concerns.

I’d really like you to repost a whole pile of your existing articles over here. I think they’re great, and I don’t think enough people have read them or commented on them. I think they’d get more exposure over here. And a link is not enough: we’re all saturated with links, and a link is like saying “Wanna hear about this? Okay, go out that door! Bye!”

Posted by: John Baez on March 29, 2016 7:43 PM | Permalink | Reply to this

Re: Coalgebraic Geometry

Hear hear.

Posted by: Mike Shulman on March 30, 2016 7:32 AM | Permalink | Reply to this

Re: Coalgebraic Geometry

Thanks for the encouragement! That’s an interesting point about links; I definitely get overwhelmed when a single blog post contains a ton of links, but it hadn’t occurred to me that people might feel like they see too many links in general.

Posted by: Qiaochu Yuan on March 30, 2016 5:58 PM | Permalink | Reply to this

Re: Coalgebraic Geometry

Just a remark : homotopy Lie algebras and rational homotopy theory are just a suspension away from what you describe (suspend the underlying space, go graded and consider a coderivation which squares to zero)

Posted by: yael on April 1, 2016 6:55 AM | Permalink | Reply to this

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