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THIS IS GETTING VERY NERDY! FOR THE SAKE OF MOST OF OUR COMPETITORS I SHOULD POINT OUT THAT THE CORRESPONDENTS ARE OLD TIMERS AT BOSS, AND WE DO NOT EXPECT MUCH OF THIS CONVERSATION TO MAKE SENSE TO YOU. AT LEAST NOT YET. IF YOU HANG AROUND LONG ENOUGH MAYBE YOU TOO WILL BE TALKING ABOUT THESE THINGS LIKE A PRO! JODIE
@the_cryptographer_formerly_known_as_madness I’ll need to check that out. I really like MacLane’s exposition in Categories for the Working Mathematician, and I spent a while researching universal algebra and lattice theory (the fields which Birkhoff laid the foundations of). Even though universal algebra has a few nice results – for example Malcev conditions and Birkhoff’s theorem on the equivalence of algebraic varieties and equational classes – I never really came to see what the abstraction gets us.
I don’t really follow the field, but my understanding is that classical universal algebra is quite an unfashionable research area now, and that its unifying approach to algebra has now been taken up by that of category theory, in which one thinks of an algebraic theories as monads over the category of sets (with the perhaps amusing result that compact Hausdorff spaces are viewed as algebras). Category theory people seem to also have the notion of a Lawvere theory, which probably has some relation to the monadic approach which I am too stupid to understand. It’s interesting how picking the right abstraction can be so useful, universal algebra has (maybe) produced some nice theorems but it has been nothing like the revolution that category theory has been for modern mathematics.