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I hope to be adding bits and pieces to an article real coalgebra, which I’ve started. (In some sense it might fit better on my web, but for some reason I’m placing it on the main nLab.)
Okay, thanks for the feedback! I’ll consider that.
I’d second that. It would be good if the word “interval” appeared in the title.
It does now.
Or, at least it would if it weren’t for the !@#$%^ cache bug. Was supposed to redirect to coalgebra of the real interval.
I have cleared the cache of the entry. Should work now.
It does redirect. Only the cache bug makes you think that it doesn’t.
Hi Todd, is there some connection between Freyd’s real interval as a coalgebra, and co-categories? Why do we represent morphisms by interval-like notation? But then if categories are defined by composition, maybe we’d expect the initial algebra to appear, i.e., dyadic rationals, as here. I guess at least its completion is the real interval.
Oh, I thought this rang a bell with something to do with you said on the Cafe. Here’s John Baez over there
Todd was too modest to emphasize it here, but the observation that the closed unit interval is an A∞-cocategory is key to his definition of ∞-categories.
Here are some of the basic ideas, too basic too be explained so far in the nLab entry.
An arrow looks like an interval. So, the theory of categories and even n-categories should have a lot to do with the interval — especially when it comes to applications to topology!
In a category we can glue arrows together, ‘composing’ them. But an interval can be chopped apart or ‘decomposed’ into a bunch of intervals. So, there should be a cocategory or something like that lurking around here.
In fact the closed unit interval gives an A∞-cocategory: a cocategory where the laws hold up to homotopy, where the homotopies satisfy nice laws up to homotopy, ad infinitum.
The space of maps out of an A∞-cocategory into something should form an A∞-category. So, the space of maps out of an interval into a space forms an A∞-category. And this is an important first step in how Todd constructs the fundamental n-groupoid of a space!
But the really cool part is how this construction goes hand in hand with his definition of n-categories, in an inductive kind of way.):
Hi David – yes, this is very much what I had in mind when I felt the urge to write things down. Basically I want to derive all the ${\operatorname{Spec}}{\mathbb ZA_\infty$ cocategory stuff by means of coinduction. As part of this, I want to be able to first of all derive the convexity structure of $I$ (the ternary operation $(t, x, y) \mapsto t x + (1-t)y$) from the terminal coalgebra structure. But the road to that seems to be a little longer than I was expecting!
Just deriving the midpoint operation alone takes a lot more work than Freyd was letting on to in his first real coalgebra postings at the categories list. But maybe it gets easier from there, once you have it.
(Side note: all of this is also to be related to the Thompson group in an appropriate way.)
Actually, ultimately, I want to bring “my” $n$-categories more into the fold of $(\infty, n)$-category theory in its geometric forms, as Urs has long been urging me to do.
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