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

[dmm]

This might end up being more readable than famous books by Jacob Lurie...

[dmm]

"Introduction" is super-useful, and actually explains how this is related to and is an alternative for Lurie...

[dmm]

And, actually, hints at some Voevodsky's ideas: page 10 (second page of the Intro):

>However, what the
Galois description strongly suggests is that homotopy theory is something
much more general and widespread than mere algebraic topology. Indeed,
many mathematical objects — starting with sets themselves, actually —
do not naturally form a set, nor a “space”, whatever that is. The problem
is not only the size of things. Even if we only consider finite sets, it is
still completely unnatural, and some would say meaningless, to state that
two sets are “equal”. The natural notion is “isomorphic”, not equal, and
an isomorphism is an additional structure that one needs to preserve. We
have moved beyond Cantor. It is groupoids that appear in nature, not sets,
and “animation” is a crucial part of it.

[dmm]

Problems with Lurie approach, page 27:

>The way Lurie’s
foundations are set up is pretty much all-or-nothing: either you move the
whole of mathematics to an ∞-categorical setting, or you cannot use it
at all. This even extends to some gratuitous rebranding — why on earth
would someone swap the meaning of “final” and “cofinal”, for example? —
but actually goes deeper than such minor quirks. By its very nature, ∞-
categorical formalism does not allow for a clean separation of definitions
(that should be simple) and proofs (that can be as difficult as needed but
can be safely used as black box). Well, the definition of a quasicategory
is simple enough, but already to understand adjunction, you really need
quite a bit of simplicial combinatorics, so that already a definition needs
to be put in a black box. In practice, to write an honest paper using the
formalism, one really needs to study thousands pages of text, and use precise
page references to various relevant facts spread out over all these pages (one
example of such an honest paper that comes to mind now is [NS]). Thus
both the specific model used by Lurie, and the whole foundational paradigm
of [Q] are hard-coded into the formalism as it exists.

[dmm]

And on page 28 he says there are widespread correctness issues in practical day-to-day use of Lurie formalism, which often lead to error...

[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)

[dmm]

The Table of Content is missing the Introduction and its subsections.

Here is that part:

Introduction page 10

Generalities 10

Localization (and its discontents) 11

Categories of models 15

Re-animating the sets 18

A trailer 20

Sociology, or why we do it 26

Safety features presentation 30

Leitfaden, or how we do it 33

Acknowledgements 38

[dmm]

pages 18-20:

people picked a wrong Grothendieck idea. They should have focused on "derivators" and not on infinity-groupoids.

[dmm]

Safety features presentation.

(i) Nothing is ever defined by hand.

(ii) Nothing is ever equal, and nothing commutes “on the nose”.

(iii) All the commutative diagrams have to be enhanced.

>If you
need a commutative diagram in an enhanced category C, you should really
number its vertices and arrows, turn it into a partially ordered set J, or a
category I, or even an enhanced category E if you so wish, and construct
an enhanced functor E → C.

[dmm]

Leitfaden = guide

[dmm]

page 33

>Next, we should say how we deal with the three main problems of category
theory: atrocious notation, idiosyncratic terminology, and inherent
triviality of the arguments.

[dmm]

page 34 :-)

>From time to time, we need an adjective for
something that needs to be named but does not deserve a permanent name;
in these cases, we plagiarize algebraic geometry and assign words such as
“ample”, “proper” or “separated”, more-or-less randomly.