When I find something interesting and new, I post it here - that's mostly programming, of course, not everything.

Sunday, January 02, 2011

Notes on my relations with CT

Zaki suggested me to write something on this topic...

So, what happened to me... In several stages.

1. I'm 19, a student in Leningrad University (that's USSR, now called Russia), and I attend a special course on homological algebra. Professor Yakovlev starts it with a short intro into category theory, a topic nobody heard of. I was overwhelmed. The thing that generalizes groups and sets and whatever. Diagrams, wow. Commutative diagrams, diagram chasing, wow. The rest was not so interesting; abelian categories, exact diagrams, hom, tensor product, ext and tor... oh whatever. In a couple of years I decided not to go into algebra.

2. I am 24, working as a programmer, trying to pick up some "science". Somewhere (where?) I see a title of a paper, "Applying Category Theory to Computer Science". Wow. My eyes are open. Here it is; database structures, they are just pullbacks. Programs, they are diagrams, are no they? Okay, nobody hears me; and more, there's no way I can get the paper. So just have to guess.

3. I am 27; me and my friends are camping somewhere in the forests of Novgorod region. Vodka, talks... I mention category theory as a good tool for perceiving comp. sci. Andrey responds by inviting me to a category seminar that they have at Technological Institute in Leningrad. Wow! I join them.

4. Several years of studying categories and toposes; MacLane's Categories for the Working Mathematician - I get the microfilm, and print it, samizdat style, in tinest possible print, for the participants. It costs 10 roubles each (like 3 bottles of vodka, if you know what I mean).

5. Andrey's friend is married to a relative of Vonnegut. So this friend, he is an otkaznik, and his American wife stays with him for several years, raising a kid, painting, just being patient. Meanwhile, after one of her trips back to US she brings Johnstone's Topos Theory... and on another trip she brings "Lolita". I devour both books. Have made hand copies of Topos Theory. Twice.

6. Andrey suggests to try to calculate topologies (and, of course, logics), over some specific finite categories; he is specifically interested in Delta3, the initial segment of simplicial category. I write code first in Basic, then figure out that the calculation for delta3 would take 2 weeks. Rewrite it into Fortran; now it's just a couple of days. So I rewrite the core parts (like enumerating subsets, calculating cones and limits) into Assembler. We have the list of 31 Grothendieck topologies over delta3 in just four hours. Meanwhile we calculate topologies for simple categories, like linear segments, commutative squares... a strange thing is discovered, on many occasions the number of topologies is 2^n, where n is the number of objects. That's for posets.

7. I prove it that for finite posets this is the case, and a little bit more. Then I spend months trying to find a typewriter with English letters (KGB is watching); used one in Hungarian trade mission, and one in the Institute of Hydrology (where global warming was then starting). Then I spend some time figuring out how to send the paper, then a KGB colonel suggests a solution, and I use it. It's published in Cah.Topo.Geom.Diff. The hardest part was deciphering French written notes on proofs.

8. I meet Yakovlev; for him, it is all a) "abstract nonsense", and b) it can't be that the result is not known yet.

9. I generalize this stuff to Karoubian categories over Boolean toposes (Booleanness is necessary) and send it to Johnstone. Johnstone, it turned out, was working on a similar problem, from the other end. So I spend the next 15 years trying to get the proof in the opposite direction - that if... then the category is Karoubian. That's how I did not do anything else. At least I tried.

10. The seminar meanwhile had died out - the guy, Stepanov (can I call him professor? something like that... he was not), had married and moved to Novorossiysk, a pretty criminal place; there he died in strange circumstances.

11. Now here in California, the word "monad" rings the bell. So I honesty try to put it in the proper perspective, writing some kind of "tutorials" and "talks", writing again the code that calculates stuff, opening presheaf.com, and ready to talk to anybody willing to listen.

12. And now we have BACAT, a category seminar where we can discuss things. And this is great.


9000 said...
This comment has been removed by the author.
9000 said...

Wow, quite a story. And I wondered how did you manage to dig so deep into category-theory-related things in programming so fast :)

Simon Hawkin said...

Enlightening, indeed. :-)

Still, it would be nice to see something like "Category Theory for a Working Programmer", if you know what I mean.

Vlad Patryshev said...

Simon, this book already exists. Pierce.

eacher said...

It's le'Carre "The Russia House' practically! Wow!


Subscribe To My Podcast