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This semester I have been asked to join Jaap with overlooking a handful of students who run a seminar on basic category theory.
In the course of that I will be re-looking at some nLab entries on basic stuff. Today I started looking at the cornerstone entry of the whole nLab: category theory.
I was very unhappy with that entry. Until a few minutes back. Now I am feeling a little better. That entry had consisted to a large extent (and still somewhat does) of lengthy lists of statements, all not exactly to the point, interspersed with lots of discussion with people like Todd and Toby continuously disagreeing with what somebody had written.
I think it is not sufficient to try to steer that somebody (who seems to have left us anyway). We need to rewrite this entry. If we can’t get a decent entry on category theory on the nLab, then we have no business making any claims about having a useful wiki focused on category theory.
So, I started reworking the entry:
I moved the historical remark from the very beginning to a dedicated section. An entry should start with explaining something, not with recounting how other people eventually understood that something.
After editing further the Idea section a bit, I inserted two new sections, in order to get to the main point of it all, and not bury that beneath various secondary aspects:
A section: “Basic constructions” namely universal constructions. That’s what category theory is all about, after all. There is not much to be said about the concept of category itself, that’s pretty trivial. The magic is in the fact that categories support universal constructions.
A section “Basic theorems”: a list of the half-dozen or so cornerstone theorems that rule category theory and mathematics as a whole. I want that nobody who glances at the entry can get away with the impression that its “just language”.
I haven’t edited much more beyond that, except that I did remove large chunks of old discussion that looked to me like mostly resolved, mostly about content that I didn’t find too exciting anyway. Should I have accidentally removed something of value, those who remember it will be able to find it in the entry’s history.
I am still not happy with the entry, but at least now I am feeling a bit better about its first third or so. I would wish a genuine category theory guru – you know who you are – would take an hour and set himself the task: here I have the chance to expose the beautiul power of category theory to the world.
Oh, and one thing: Andrew: if you have a minute, I think it would be nice if you added your little piece on “Why teach category theory?” to a section of that or similar title to this entry.
Added to “central constructions” a section “Presheaves” that leads over to topos theory.
True. I added a link in the References-section (wasn’t sure how to call the corresponding Sub-section…)
Also added a link to Jaap’s lecture notes.
I have now decided to also remove the long lists of keywords of “related structures”, that ran randomly over items like operads, computads, weak n-category, doctrines, and so forth.
I started trying to bring these into some logical order. But I think for the moment I just leave it at the following:
First of all I moved the “Applications” section up, so that the entry logic now flows
Idea
Basic constructions
Basic Theorems
Applications
which I think is a good structure.
Then I thought how to organize the main applications. This is stubby, bur for the moment I divided into
In pure mathematics
Outside of mathematics
and in the first I put two big headline-links
This should now be fleshed out, and most of the items of the lists that I removed would find there place within somewhere,
Then under “Applications outside math” I put
Here, too, one could go into more detail indefinitely. But at least the links will lead the reader to further links (well, I am also still very unhappy with the more-than-stubby “higher category theory and physics”, but this is a task for another day).
I removed and would like to not see again links (to non-existent pages) titled “psychology” and the like.
Added one more “headline link” for the applications in pure math, now that list is
logic
geometry
algebra.
That could serve as a good top-level organization. Now we could go into finer sub-topics.
I’ve suggested a slight rewording of one of the early paragraphs, as I felt that the way the cliff face of higher category theory loomed over the foothills of standard category theory was a bit daunting. What do you think?
I have now decided to also remove the long lists of keywords of "related structures", that ran randomly over items like operads, computads, weak n-category, doctrines, and so forth.
We could have an entry higher structure or something of the sort with intro few words about higher categories and then the list of some related notions like the mentioned.
I removed redirect applied category theory. First because the link applied category theory is called from within the category theory entry! Second we should soon have a genuine separate page for that while keeping the section on applied category about as it is now.
Feels a bit lonely sitting there all by myself in the “Teaching Category Theory” category. If nothing else, I ought to polish that essay a little.
(I completely agree that the essay itself does not belong on that page, indeed I wouldn’t put it on the nLab as it is very much my personal opinion and whilst others might agree with it, I wouldn’t want them editing it as then it might not be my personal opinion. That said, I’d welcome any comments on it and suggestions for improvement, of course.)
Feels a bit lonely sitting there all by myself in the “Teaching Category Theory” category.
Yes, as I said. I wasn’t sure where to put it. But I want to have links to this kind of stuff on the page – explanations of why it is nonsense to disregard category theory. As you nicely put it: as nonsensical as disregarding logic in math.
This is not about the old folks who won’t change their mind. This is about young students. They need to be provided with the proper information. The page category theory must eventually be such that a math senior student (or whatever it’s called in anglosaxony) with a good background in general math gets away with an accurate modern impression of what category theory is and does.
I’ve suggested a slight rewording of one of the early paragraphs, as I felt that the way the cliff face of higher category theory loomed over the foothills of standard category theory was a bit daunting. What do you think?
Thanks, Tim, I like that! I added in some more hyperlinks.
But I think this paragraph is important, and important at this point of the introduction. And let’s be frank: at the level of precision of the Idea-section at this point, it is really just as easy to say “morphism between objects” as it is to say “morphism between morphisms”. Every child can say that and have that thought. The thought is simple and god given. All the subtlety is “only” in making it fully precise.
I removed redirect applied category theory. First because the link applied category theory is called from within the category theory entry! Second we should soon have a genuine separate page for that while keeping the section on applied category about as it is now.
To some extent the page nPOV is nothing but a list of applications. I would enjoy it if we could incrementally further brush up that nPOV page, by bringing more structure into its contents, adding more aspects and so on. This nPOV page is a lot like the Conclusion-section of the whole nLab. Q: “What is it all good for?” A. “See the page nPOV”.
Maybe we can rename it to something less whimsical and keep nPOV just as a redirect.
Okay, following what I just said I
split off applications of (higher) category theory from nPOV and linked to it from the relevant pages;
started to bring the structure of the entry higher category theory into roughly the same form as we have now for category theory. But partly the sections contain just link.lists for the moment.
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