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Дима Каледин, математик (старожилы русского интернета могут знать его имя по старому ЖЖ), опубликовал 600-страничную статью , в которой описывает новый подход к абстрактной теории гомотопии, над которым он работал много лет. Он предлагает этот подход в качестве…

Обзорный текст от Каледина, покороче:
https://arxiv.org/abs/2409.18378
вы туда все равно не полезете, захотелось запостить несколько отрывков из введения

1. (Чем плох "текущий подход" к гомотопическим оснащениям)

...Thus the current thinking goes along more-or-less the following lines.

(i) “Quillen-equivalent model categories have the same homotopy theory”; this is accepted as an article of faith and not discussed.
(ii) One constructs a “category of models” for enhanced small categories; this category of models is equipped with a model structure and produces all the desired data; an “enhanced category” is then simply defined as an object in the corresponding localized category.
(iii) Models are not unique at all, and neither are “categories of models”,
but one checks that they are all Quillen-equivalent, so see (i).

There are two obvious issues with this kind of thinking. Firstly, it is very
set-theoretical in nature and feels like a throwback to 19-th century – a category, something that should be a fundamental notion, is treated as a special type of a simplicial set, or “space”, whatever it is, or something like that. The idea of symmetry so dear to people like Grothendieck is thrown out of the window.
Secondly, a worse problem is the inherent circularity of the argument. Of all the avaliable models, it is best seen in the approach of [BK] based on relative categories.

By definition, a relative category is a small category C equipped with a class of maps W.
Barwick and Kan propose putting a model structure on the category of relative categories, and showing that it is Quillen-equivalent to all the other existing models. Then in this particular model, the result of localizing a category C with
respect to a class of maps W is the relative category ⟨C, W⟩. Effectively, it looks pretty much as if in this approach – and ipso facto in all the others, since they are all Quillen-equivalent – one "solves" the localization problem by declaring it solved.

arXiv.org

How to enhance categories, and why?

This is a companion overview paper to arXiv:2409.17489: we give all the main definitions, constructions and statements, but no proofs.

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www.group-telegram.com/us/sweet_homotopy.com/2029

1.83K viewsedited  Sep 30, 2024 at 21:29

Обзорный текст от Каледина, покороче:
https://arxiv.org/abs/2409.18378
вы туда все равно не полезете, захотелось запостить несколько отрывков из введения

1. (Чем плох "текущий подход" к гомотопическим оснащениям)

...Thus the current thinking goes along more-or-less the following lines.

(i) “Quillen-equivalent model categories have the same homotopy theory”; this is accepted as an article of faith and not discussed.
(ii) One constructs a “category of models” for enhanced small categories; this category of models is equipped with a model structure and produces all the desired data; an “enhanced category” is then simply defined as an object in the corresponding localized category.
(iii) Models are not unique at all, and neither are “categories of models”,
but one checks that they are all Quillen-equivalent, so see (i).

There are two obvious issues with this kind of thinking. Firstly, it is very
set-theoretical in nature and feels like a throwback to 19-th century – a category, something that should be a fundamental notion, is treated as a special type of a simplicial set, or “space”, whatever it is, or something like that. The idea of symmetry so dear to people like Grothendieck is thrown out of the window.
Secondly, a worse problem is the inherent circularity of the argument. Of all the avaliable models, it is best seen in the approach of [BK] based on relative categories.

By definition, a relative category is a small category C equipped with a class of maps W.
Barwick and Kan propose putting a model structure on the category of relative categories, and showing that it is Quillen-equivalent to all the other existing models. Then in this particular model, the result of localizing a category C with
respect to a class of maps W is the relative category ⟨C, W⟩. Effectively, it looks pretty much as if in this approach – and ipso facto in all the others, since they are all Quillen-equivalent – one "solves" the localization problem by declaring it solved.

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