### 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.

## 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!