Timeline for What are Koszul dualities?
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11 events
| when toggle format | what | by | license | comment | |
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| Apr 27, 2022 at 12:48 | answer | added | Tim Campion | timeline score: 5 | |
| Mar 12, 2021 at 4:11 | comment | added | Bbb | Of course one can imagine turning this on its head and take operadic KD to be more fundamental, prove E_n self duality somehow, then deduce E_1 KD - maybe there exists a language where this path is convenient, my perspective is coming from someone more comfortable at thinking about E_1 algebras than at operadic homological algebra | |
| Mar 12, 2021 at 3:58 | comment | added | Bbb | Also potentially confusing is that for an algebra you can talk about a KD algebra OR a KD coalgebra (regarding your comment about E_n self-duality), they’re variants of the same construction, at the level of underlying objects they’re related by linear duality. These are explained respectively in 5.2.2 and 5.2.5 of HA. | |
| Mar 12, 2021 at 3:55 | comment | added | Bbb | One way to think about this is that everything is a special case of 5. One model for Infinity operads are E_1 algebras in symmetric sequences, E_1 KD in that case gives you operad KD. E_n KD is an iterated version of the E_1 case as explained in HA, but can also be gotten from the E_n self-duality. Statements about modules tend to be some Morita theory using that the underlying module of the koszul dual looks like an endomorphism algebra | |
| Feb 24, 2021 at 22:26 | comment | added | Phil Tosteson | Ah, yes you're right. I take back that you need stability to talk about duality between the $E_n$'s. I'm not really sure what the right level of generality is-- or really if there is one. As you make more assumptions you probably get more features/simplifications. | |
| Feb 24, 2021 at 22:14 | comment | added | Tim Campion | @PhilTosteson Thanks, this is enlightening. One thing that still confuses me is that in Higher Algebra Ch 5.2 Lurie discusses a notion of Koszul duality for E_k algebras which seems to be the composite of the bar/cobar duality and some other form of duality, but this second form is not simply dualization in a monoidal category as far as I can see, and I think this is related to some nonabelian sense in which the E_k operad is "Koszul self-dual". The point about model-dependency is well-taken. | |
| Feb 24, 2021 at 22:07 | comment | added | Phil Tosteson | From this point of view, Koszul duality is concerned with constructing small models for Bar constructions. As a special case, this theory contains the fact that you can use the Koszul complex to compute Tor groups for modules over a polynomial ring. I'd recommend Loday and Valette's book Algebraic Operads for more on this | |
| Feb 24, 2021 at 22:01 | comment | added | Phil Tosteson | I should add that much of the power Koszul duality as practiced by representation theorists (and originally by Priddy) comes from model dependent formulations. From an general perspective, every operad with operations of arity $\geq 2$ has a "Koszul dual"--given by a bar construction. But for quadratic operads there is a much smaller co-operad, defined in terms of the presentation, which is often quasi-isomorphic to the one you get from the bar construction. A representation theorist would say that the original operad is Koszul only when this small model agrees with the bar construction. | |
| Feb 24, 2021 at 22:00 | comment | added | skd | I like think about Koszul duality as "exchanging the forget/free adjunction for O with the square-zero/cotangent complex adjunction for the Koszul dual O^". For instance, if k = char 0, then the cotangent complex/bar construction of a free commutative algebra on a (perfect, say) k-module V is the dual V* with the "trivial" Lie structure. You can similarly calculate the cotangent complex of a square-zero extension k (+) V to be the free Lie algebra on V*. In a precise sense, establishing such equivalences is the main technical content of Koszul duality. | |
| Feb 24, 2021 at 21:52 | comment | added | Phil Tosteson | Your broad picture is essentially correct. It's tough to write a full answer, because there are technicalities that make it hard to say what the "correct" general setting is. To say something like $E_k$ is self dual, you $\mathcal V$ to be stable and have duals so that you can dualize a co-operad and make it into an operad. | |
| Feb 24, 2021 at 20:45 | history | asked | Tim Campion | CC BY-SA 4.0 |