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]

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.