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June 3, 2019

Why Category Theory Matters

Posted by John Baez

No, I’m not going to tell you why category theory matters. To learn that, you must go here:

It’s interesting to see an outsider’s answer to this subject. He starts with a graph purporting to show the number of times — per year, I guess? — that the phrase “category theory” has been mentioned in books:

I’m curious about the plunge after 1990. I hadn’t noticed that.

I’m amused and flattered, but also a bit unnerved to read that

Generally speaking, there seems to be a cabal of radical category theorists, led by John Baez, who are reinterpreting anything interesting in category theoretic terms.

Of course it’s meant to be humorous. The unnerving part is the idea that I’m “leading” anybody — except perhaps my grad students. It seems to me rather that category theory has an inherent tendency to spread its reach, and I’m just one of a large group of people who have tuned in to this.

Posted at June 3, 2019 12:40 AM UTC

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Re: Why Category Theory Matters

My first guess about the drop-off is rather morbid: it’s the time around which the people who founded category theory started passing away. It took a little while perhaps for the next generation of people to pick up the slack after the intermediate generations were a little jaded by the enthusiasm of category theorists of the 1970s. Or perhaps we got satisfied with existing textbooks. Categories for the Working Mathematician, Topos Theory, Sheaves in Geometry and Logic, and Toposes Triples and Theories, for instance, cast long shadows. Then The Elephant got published at the turn of the century, and slowly people have been writing books again.

Or maybe people really just took up the internet in a serious way, and relied less on the paper medium? TAC got up and running at the time of that peak just before the big dip.

Posted by: David Roberts on June 3, 2019 2:20 AM | Permalink | Reply to this

Re: Why Category Theory Matters

“Category theory is one of, if not the most, abstract fields of mathematics.”

Paradoxically, category theory is also one of the most concrete fields of mathematics.

Posted by: Simon Burton on June 3, 2019 10:50 AM | Permalink | Reply to this

Re: Why Category Theory Matters

I think the drop off in the 90s is an artifact of the corpus. The following links to an ngram showing the same drop off for several similar technical terms: click on me to go to a nice ngram!

This raises some doubt about the reliability of google’s ngram page.

Also, it’s interesting that “category theory” rises again starting in 2000, unlike the other technical terms.

Posted by: James Borger on June 3, 2019 11:25 AM | Permalink | Reply to this

Re: Why Category Theory Matters

Yorkshire terrier vs category theory

Posted by: Tom Leinster on June 3, 2019 10:03 PM | Permalink | Reply to this

Re: Why Category Theory Matters

It looks to me like the 1950-2000 trend is definitely just an artifact of the corpus. The rise in occurrences of “category theory” after 2000, compared to other mathematical terms, is the only really interesting feature of the graph.

Posted by: Mark Meckes on June 4, 2019 1:30 PM | Permalink | Reply to this

Re: Why Category Theory Matters

amused and flattered, but also a bit unnerved

But you failed to note that in the original post, “cabal of radical category theorists” is a link to this blog. Looks like you’re trying to keep all the amusing, flattering, unnerving attention on yourself!

Slightly more seriously, wouldn’t the n-Lab be a more appropriate link for that cabal of radical category theorists?

Actually seriously, to an outsider it’s easy to mistake the most visible public advocate of a movement of any kind for its leader. Leaving aside the question of to what extent you’re a leader of this cabal (or of whether the cabal has a leader, or even exists — and as an outsider myself, I won’t venture an opinion on this), you certainly are the most visible public advocate for the idea that lots of interesting stuff can be reinterpreted in category theoretic terms.

And I’ll also add, again as an outsider, that I don’t see anything wrong with that reinterpretation. It’s not clear to me whether Robb Seaton does, but the quasi-pejorative adjective “radical” and his following parenthetical, “The trend reminds me of reinventing the wheel for the nth time in the nth newest programming language,” suggest that he doesn’t think it’s entirely a good idea. But as far as I’ve seen, nobody around here (including John) has suggested that everything interesting must always be interpreted in category theoretic terms (which really would be radical), just that it is often possible, and often useful to do so.

And after all, something similar goes on with most fields of mathematics. Personally, I try to reinterpret everything in terms of probability if I can. And for that matter, a hundred-some years ago there was a radical reinterpretation of everything interesting in mathematics in set theoretic terms. That may have been taken a bit too far in some ways (as Robb Seaton hints), but it was undeniably useful for continuing progress.

Posted by: Mark Meckes on June 3, 2019 2:05 PM | Permalink | Reply to this

Re: Why Category Theory Matters

I tend to think of category theory as the evolution of a concept: In Euclidean geometry, we draw triangles. In Newtonian mechanics, we put arrows on the edges and call them vectors. Then category theory allows you to curve this edges into paths, and so we get morphisms!

It was actually tackling differential geometry and the tangent bundle that got me hooked onto categories and functors. Plus thinking if category theory could breath new life into such basic ideas as addition and multiplication, then there must be something deep going on in there.

I could swear I once read in the preface of a textbook or another that category theory first took root in algebraic geometry and then they plotted to revolutionise the rest of mathematics, from there! As far as I remember it, the author said it with much more verve than I can put it.

Posted by: Mozibur Rahman Ullah on June 16, 2019 9:54 PM | Permalink | Reply to this

Re: Why Category Theory Matters

It was in Ravi Vakils notes on algebraic geometry where he quotes David Mumford:

“It has now been four decades since David Mumford wrote that algebraic geometry ‘seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics! In one respect this last point is accurate …’ The revolution has now fully come to pass, and has fundamentally changed how we think about many fields of pure mathematics.”

I’m pretty sure that to the mathematical layman the various nuances between abstract and very abstract in mathematics can be a bit oversold, when almost all of it (apart from counting and arithmetic - and that doesn’t count) looks very abstract to him/her!

Posted by: Mozibur Rahman Ullah on June 16, 2019 10:02 PM | Permalink | Reply to this

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