What are Koszul dualities?

I am bewildered by the number of things I've heard referred to as "Koszul duality", and I would like to sort it out. At various different times, I believe I've seen any of the following phenomena referred to as "Koszul duality":

1. Let $$O$$ be an operad (in your favorite symmetric monoidal $$\infty$$-category $$\mathcal V$$), and apply the operadic bar construction to $$O$$ (i.e. regard $$O$$ as a monoid in symmetric sequences equipped with Kelly's convolution monoidal product $$\circ$$, and take the the geometric realization of the simplicial symmetric sequence $$[n] \mapsto O^{\circ n}$$) to obtain $$BO$$. Then (under certain conditions?) $$BO$$ is a cooperad, called the Koszul dual of $$O$$. Dually, if $$C$$ is a cooperad, then applying the operadic cobar construction to $$C$$ yields an operad called the Koszul dual of $$C$$. Under certain conditions, these constructions are adjoint to one another, and under certain further conditions they are inverse to one another.

2. There is a version of (1) called Koszul duality carrying $$O$$-modules to $$BO$$-comodules (and maybe a variant carrying $$O$$-algebras to $$BO$$-coalgebras?).

3. Let $$O$$ be the $$E_k$$ operad, and let $$A$$ be an $$O$$-algebra. Then there is an iterated bar-construction carrying $$A$$ to an $$E_k$$-coalgebra, which is called its Koszul dual.

4. Let $$O$$ be a quadratic operad in the sense of Ginzburg and Kapranov (quadratic-ness only makes sense in certain $$\mathcal V$$ -- basically $$V$$ must be chain complexes). Then there is another quadratic operad Koszul dual to $$O$$, and if $$O$$ is Koszul, then the duality operation is self-inverse at $$O$$.

5. There is also Koszul duality for algebras for the associative operad.

6. I think I've been told that the $$E_k$$-operad is Koszul self-dual. Since the $$E_k$$-operad makes sense in spaces, where quadraticness isn't even defined, I don't know what this means (I only know what it means for an operad to be dual to a coopeard in this generality, following (1)).

I think what's going on is that the fundamental duality is the bar/cobar adjunction between operads $$O$$ and cooperads $$BO$$; duality between $$O$$-modules and $$BO$$-comodules, and between $$O$$-algebras and $$BO$$-coalgebras then comes along for the ride. Then some other form of duality sometimes allows one to relate cooperads back to operads, and we start talking about duality between operads, and between the modules / algebras for an operad and its dual operad from there. But I've never really seen this spelled out in this general a context.

Question: What is the relationship between the above things called Koszul duality (and other things called Koszul duality which I'm missing)?

I hope it's clear that this question has a different focus from this classic MO question.

• 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 21:52
• 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.
– skd
Feb 24, 2021 at 22:00
• 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 22:01
• 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 22:07
• @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 22:14