A new book by Dima Kaledin

[dmm]

"Enhancement for categories and homotopical algebra", arxiv.org/abs/2409.17489

600 pages

"We develop foundations for abstract homotopy theory based on Grothendieck's idea of a "derivator". The theory is model-independent, and does not depend on model categories, nor on simplicial sets. It is designed to accomodate all the usual potential applications, such as e.g. enhancements for derived categories of coherent sheaves, in a way that is as close as possible to usual category theory."

He also released references [K3] and [K4]:

arxiv.org/abs/2409.18380 and arxiv.org/abs/2409.18378

Comments

[juan_gandhi]

Oh, this is cool! Even Karoubian closures got involved. I worked with them for a while. Cool!

[dmm]

Yes. And this might be a "humane" alternative to Lurie's approach!

[dmm]

надо сказать, что Введение совершенно захватывающее, очень интересная история...

да, и ChatGPT оказывается крайне полезен, он, как-то, более по-человечески говорит на эти темы, чем люди (более интуитивно, менее запутанно):

Hi, what's Grothendieck's idea of a "derivator"? https://chatgpt.com/share/66f8cd64-c98c-8010-8ba3-b9aadd8aa249

[dmm]

and what pages 26-33 say, in terms of the analogy with functional programming, is that the traditional approach everyone is practicing after Lurie is "fake functional programming", all the nice words are being said, but in reality one can't avoid using unsafe features, and correctness is very shaky and usually non-existent.

whereas, this new approach is both correct (and simply does not allow to "backtrack into imperative thinking") and also much simpler technically and conceptually.

[dmm]

The way page 29 looks is that Kaledin is actually succeeding where the infinity-categories are failing (at their actual goal, that is, being able to avoid precise equalities)