сладко стянул

сладко стянул

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…

2. (Тут немного пафосно. Идея Гротендика: восстанавливать "объект" по морфизмам из него; следовательно, структуру на объекте — по структуре на морфизмах из него. Так "нетривиальную" локализацию можно запрятать в "тривиальную" — в ослабление изоморфизма категорий до эквивалентности категорий)

If it is inevitable that enhanced categories are only defined up to an equivalence of some sort, let us at least make this equivalence as easy to control as possible.
Then observe that there is another type of controlled localization that is so common and widespread that it usually goes unnoticed by its users: the category Cat of small categories, and the class W of, well, equivalences of categories. In principle, this can be localized by using model category techniques, but this is akin to smelling roses through a gas mask. the answer is actually much simpler, and similar to the homotopy category of chain complexes: objects are small categories, morphisms are isomorphism classes of
functors.

Moreover, we can also consider families of small categories indexed
by some category I. This is conveniently packaged by the Grothendieck construction of [SGA 1, Exposé 6] into a Grothendieck fibration С→I with small fibers, with morphisms given by functors C→C′ cartesian over I. Then again, localizing with respect to equivalences gives the category with the same objects, and isomorphism classes of cartesian functors as morphisms (for precise definitions, see below Subsection 1.4).

Now, an enhanced category C comes equipped with its underlying usual category h(C), but there is more: for any small category I, we also have the enhanced category C^I of functors I→C, and its underlying usual category h(C^I). Thus we actually have a whole family of categories indexed by Cat. This has been described in [Grothendieck, "Pursuing Stacks"] under the name of a derivator; the question was, is it enough to recover C? Our answer is: with some modifications, yes.

(...The main modification compared to [Grothendieck] is that it is not necessary, nor in fact desirable to index our enhanced categories over the whole Cat – it is sufficient to consider the category Pos+ of left-bounded partially ordered sets.)

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1.9K viewsSep 30, 2024 at 22:34

2. (Тут немного пафосно. Идея Гротендика: восстанавливать "объект" по морфизмам из него; следовательно, структуру на объекте — по структуре на морфизмах из него. Так "нетривиальную" локализацию можно запрятать в "тривиальную" — в ослабление изоморфизма категорий до эквивалентности категорий)

If it is inevitable that enhanced categories are only defined up to an equivalence of some sort, let us at least make this equivalence as easy to control as possible.
Then observe that there is another type of controlled localization that is so common and widespread that it usually goes unnoticed by its users: the category Cat of small categories, and the class W of, well, equivalences of categories. In principle, this can be localized by using model category techniques, but this is akin to smelling roses through a gas mask. the answer is actually much simpler, and similar to the homotopy category of chain complexes: objects are small categories, morphisms are isomorphism classes of
functors.

Moreover, we can also consider families of small categories indexed
by some category I. This is conveniently packaged by the Grothendieck construction of [SGA 1, Exposé 6] into a Grothendieck fibration С→I with small fibers, with morphisms given by functors C→C′ cartesian over I. Then again, localizing with respect to equivalences gives the category with the same objects, and isomorphism classes of cartesian functors as morphisms (for precise definitions, see below Subsection 1.4).

Now, an enhanced category C comes equipped with its underlying usual category h(C), but there is more: for any small category I, we also have the enhanced category C^I of functors I→C, and its underlying usual category h(C^I). Thus we actually have a whole family of categories indexed by Cat. This has been described in [Grothendieck, "Pursuing Stacks"] under the name of a derivator; the question was, is it enough to recover C? Our answer is: with some modifications, yes.

(...The main modification compared to [Grothendieck] is that it is not necessary, nor in fact desirable to index our enhanced categories over the whole Cat – it is sufficient to consider the category Pos+ of left-bounded partially ordered sets.)

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