symmetric monoidal (∞,1)-category of spectra
abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
Koszul duality (named after Jean-Louis Koszul) is a duality and phenomenon generalizing the duality between the symmetric and exterior algebra of a vector space to so-called quadratic differential graded algebras (which can be obtained as a free dga module an ideal of relations which live in degree 2). For a pair of Koszul dual algebras, there is a correspondence between certain parts of their derived categories (precise formulation involves some finiteness conditions). In a setup in which one of the algebras is replaced by a cocomplete dg coalgebra, there is a formulation free of finiteness conditions, but involving twisting cochain (see that entry).
Koszul duality is a duality between quadratic operad?s, predicted in
and developed in
There are a lot of algebras whose derived categories are equivalent in surprising ways. Morita equivalences are pretty simple, especially for finite dimensional algebras; essentially the only free parameter is the dimension of the object.
Namely, if $A Mod$ and $B Mod$ are equivalent, then the image of $A$ as a module over itself is a projective generator of $B Mod$, and for a finite-dimensional algebra, essentially the only thing you can do is take several copies of the indecomposible projectives of $B$.
On the other hand, if you take the derived category of dg-modules over $A$ (the dg part of this is not a huge deal; it’s just that they’re very close to, but a bit better behaved than, actual derived/triangulated categories, which are just crude truncations of truly functorial dg/$A_\infty$ versions), this is equivalent to the category of dg-modules over the endomorphism algebra (this is in the dg sense, so it’s a dg-algebra whose cohomology is the $Ext$ algebra) of any generating object. There are a lot more generating objects than projective generators, so there are a lot of derived equivalences.
In particular, let $A$ be a finite dimensional algebra $A$, than an obvious not-very-projective generating object is the sum of all the simple modules, say $L$. As mentioned above, there’s an equivalence $A dg Mod = \mathrm{Ext}(L,L) dg Mod$, just given by taking $\mathrm{Ext}(L,-)$.
In general, $\mathrm{Ext}(L,L)$ is in a very complicated object (for example, often for group algebras over finite fields), but sometimes it turns out to be nice. For example, if $A$ is an exterior algebra, you’ll get a polynomial ring on the dual vector space. Another (closely related) example is that the cohomology of a reductive group (over $\mathbb{C}$) is Koszul dual to the cohomology of its classifying space.
One way to ensure that $\mathrm{Ext}(L,L)$ is nice is if the algebra $A$ is graded. Then $\mathrm{Ext}(L,L)$ inherits an “internal” grading in addition to its homological one. If these coincide, then $A$ is called Koszul.
In this case, $B=\mathrm{Ext}(L,L)$ is forced to be formal (if it had any interesting $A_\infty$ operations, they would break the grading), so you’re dealing with a derived equivalence between actual algebras, though you have to be a bit careful about the dg-issues. Thus the derived category of usual modules over $A$ is equivalent to dg-modules over $B$ (with its unique grading) and vice versa. This can be fixed by taking graded modules on both sides.
The most famous example of Koszul dual algebras are the exterior algebra $Alt^{\bullet}V[-1]$ and the polynomial algebra $Sym^\bullet V^*[-2]$.
A regular block of category O? for any semisimple Lie algebra is a self-Koszul dual.
Braden, Licata, Proudfoot and Webster gave a combinatorial construction of a large family of Koszul dual algebras in Gale duality and Koszul duality.
Other historical references on Koszul duality include
A. A. Beĭlinson, V. A. Ginsburg, V. V. Schechtman, Koszul duality, J. Geom. Phys. 5 (1988), no. 3, 317–350 MR1048505 doi
Alexander Beilinson, Victor Ginzburg, Wolfgang Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473–527.
Koszul duality is also discussed in
The Everything Seminar , koszul-duality-and-lie-algebroids
MathOverflow: What is Koszul duality?, Beilinson-Bernstein and Koszul duality, what-extra-conditions-are-necessary-for-the-following-version-of-koszul-duality
David Eisenbud, Gunnar Fløystad, Frank-Olaf Schreyer, Sheaf cohomology and free resolutions over exterior algebras, Trans. Amer. Math. Soc. 355 (2003), 4397-4426 MR1990756 doi
Tyler Foster, Po Hu, Igor Kriz, D-structures and derived Koszul duality for unital operad algebras, arxiv./1507.07151
A “curved” generalization is discussed in
Bernhard Keller and his student Lefèvre-Hasegawa described rather general framework for Koszul duality using dg-(co)algebras and twisting cochains: