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I edited Trimble n-category:
added table of contents
added hyperlinks
moved the query boxes that seemed to contain closed discussion to the bottom. I kept the query box where I ask for a section about category theory for Trimble n-categories, but maybe we want to remove that, too. Todd has more on this on his personal web.
removed query boxes of discussions:
+–{.query} John has just substituted “topological” for my (Todd’s) original “tautological”, which I don’t mind at all – “topological” is correct and arguably more informative – but maybe this is a good place to explain that “tautological” was not a typo, but rather an expansion of a technical usage:
One of the first examples of an operad that people are taught is the one where the n-th component consists of all n-ary operations $X^n \to X$. (Here $X^n$ denotes an n-fold tensor power in a monoidal category.) Because it’s the simplest or most obvious example of an operad, and because it’s canonically there no matter what $X$ is, it’s often in the literature called a “tautological operad”. What is less often noted is that by the same logic, $hom(X, X^n)$ is an equally tautological operad, except that it consists of all “co-operations” instead of operations. Then, I am applying this observation to the example of the monoidal category of topological cospans from a point to itself, where $X$ is the interval.
Toby: Considering that the original was ’tautological topological’, which is more informative, I've changed it back. But perhaps this explanation has a place in the main text?
Todd: Sounds like a good idea. If you can figure out a smooth way to do that, I’d be grateful.
Toby: Actually, I just realised that most of it is already in the next paragraph below. I'll just add a sentence noting that this explains the name. =–
+–{.query} Urs: Thanks, Todd, this is great. If I may, let me try, also to check if I am following, to see how much of this we can do while abstracting away from Top and the concrete interval object in there.
It seems that we need:
a closed monoidal homotopical category $V$ (not your $V$ above, but I’ll call it $V$ nevertheless);
an object $I \in V$ fitting in an internal co-graph $pt \stackrel{\sigma, \tau}{\to} I$ such that
all the end-to-end pushouts $I^{\vee n}$ exist in $V$;
all the $V$-internal hom-objects $[I, I^{\vee n}]$ are weakly equivalent to the point (the terminal object) $[I, I^{\vee n}] \stackrel{\sim}{\to} pt$;
equipped with the structure of a co-operad internal to $V$ on $\{[I,I^{\vee n}]\}$, $n \in \mathbb{N}$.
Then
Hm, let me just let it stand this way for you to either erase all of this or maybe comment on it for the moment…
Todd: Yes, that’s more or less right. But there is a god-given nonpermutative operad (not co-operad) structure on the graded $V$-object $\{[I, I^{\vee n}]\}$, and that’s what I’m using here. If you know how the “endomorphism operad” works (as in the entry operad), then the operad here works the same way, mutatis mutandis.
As for the last part: in the original definition, I didn’t know how to pass to the full-fledged $\omega$-categorical definition; I just had $n$-categories, one for each $n$. But Tom and Eugenia figured out how to make “Trimble-like” $\omega$-categories work by using coalgebraic methods. I haven’t thought about whether that allows a $\Pi_\omega$… But aside from these technicalities: yes, you seem to understand what this is all about.
There were some comments to this effect after I gave that lecture; someone asked what you really need to do this definition very generally. Martin Hyland cried, “Not much!”
Urs Schreiber: Okay, thanks. I tried now to formalize this at interval object.
=–
John Wick
Thanks for looking into cleaning up this old entry, there is much room left to do so. (This entry dates from the early days of the nLab in 2009 and had been hardly touched since.)
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