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posicНа улицах Праги лежит немного снега, выпавшего ночью. Прогноз погоды обещает потепление и дождь.
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Leonid Positselski
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Date: 2018-12-01 01:19 pm (UTC)no subject
Date: 2018-12-01 04:03 pm (UTC)no subject
Date: 2018-12-01 04:11 pm (UTC)Сейчас выйду на улицу, посмотрю. Нужно распечатать на принтере параграф из книжки, почитать. Потом, может быть, зайду в кофейню выпить чаю.
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Date: 2018-12-01 08:23 pm (UTC)no subject
Date: 2018-12-01 08:33 pm (UTC)no subject
Date: 2018-12-01 09:03 pm (UTC)no subject
Date: 2018-12-01 09:18 pm (UTC)Но на улице Яна Масарика я, кажется, никогда не был (хотя несколько раз проходил мимо нее этой осенью, направляясь по от площади Мира по Французской улице в сторону парка). Надо бы, раз такое дело, сходить туда посмотреть на липы.
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Date: 2018-12-01 09:29 pm (UTC)no subject
Date: 2018-12-01 09:35 pm (UTC)no subject
Date: 2018-12-01 09:47 pm (UTC)no subject
Date: 2018-12-01 09:57 pm (UTC)no subject
Date: 2018-12-01 10:04 pm (UTC)no subject
Date: 2018-12-01 08:47 pm (UTC)localization of an adjoint pair yields an equivalence (like the passage from Quillen equivalence to
the equivalence of the homotopy categoires).
VH
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Date: 2018-12-01 08:55 pm (UTC)One might even try to do it in a canonical way: invert all the adjunction units as morphisms in one of the categories, and all the adjunction counits as morphisms in the other category. Then the functors are likely to remain adjoint, and consequently become adjoint equivalences.
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Date: 2018-12-01 09:22 pm (UTC)On my part (as it often happens with me on MathOverflow nowadays) it is just an example of "у кого о чем болит, тот о том и говорит". I am using this "generalized Foxby equivalence" in a recent preprint of mine, on which I am still working now (will be speaking about related stuff at the Algebra seminar in Prague on Monday -- https://ka.karlin.mff.cuni.cz//seminar ). So it comes to my mind faster than the much more commonly used example with Quillen equivalences.
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Date: 2018-12-02 02:41 pm (UTC)"correctly" derive functors. The setup is as follows: you have a functor f:C->D and a localization (may be in oo-categorical sense) q:C->C'. My recipe: convert f into a cocartesian fibration E--->[1] (with fibers C and D over 0 and 1).
Then localize at weak equivalences in C. You get E'--->[1]. If it remains a cocartesian fibration, it gives you left derived functor Lf:C'--->D. The definition is less general than the one via right Kan extension. But, of course, in all meaingful cases it works ad gives correct answer. And, in cases you expect to get an adjunction, you get it automatically.
This is to explain that you suggestion of localizing with units and counits is not the correct answer. I am not sure there is a universal (minimal) localization converting an adjoint pair into equivalence even though I thought precisely about this (in the context of Koszul duality, of course).
VH
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Date: 2018-12-02 03:25 pm (UTC)In the context of Koszul duality, though -- e.g, if you are interested in homogeneous Koszul duality over a field, or even in nonhomogeneous Koszul duality over a field -- the bar- and cobar-constructions are exact functors (if you have chosen the equivalence relation on the complexes/DG-modules/DG-comodules/... on the left- and the right-hand side of your intended category equivalence properly).
In other words, one does not have to resolve a complex or a DG-module (i.e., replace it by a more adjusted one) before applying the bar-construction to it, and similarly with the cobar-construction. So I would expect the idea of inverting the units and counits of the adjunction to be applicable in the case of Koszul duality (at least, over a field).
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Date: 2018-12-01 11:19 pm (UTC)