[Math] $\infty$-operads and $E_\infty$-algebras

Question

I work in algebraic geometry. Lately, the answer to most of my questions seems to be "you should read Lurie's Higher Algebra." I took this advice seriously, however it turned out not to be an easy task.

From what I gather, I should learn how to work with $E_\infty$-rings. I understand the definition of a symmetric monoidal $\infty$-category as a certain coCartesian fibration $p\colon C^\otimes \to N({\rm Fin}_*)$, and that an $E_\infty$-ring in a stable symmetric monoidal $\infty$-category $C$ should be a certain section of $p$.

In Higher Algebra, $E_\infty$-algebras are a special case of algebras over $\infty$-operads, where the operad ${\rm Comm}$ is just the identity $N({\rm Fin}_*)\to N({\rm Fin}_*)$. Lurie treats general $\infty$-operads in detail in two chapters of about 150 pages each, and in later chapters he specializes to the case of the operads $E_k$. Since I seem to be interested in the final operad ${\rm Comm}$ only, it is difficult for me to motivate myself to read through the chapters about the general theory of $\infty$-operads.

Question. As an algebraic geometer interested in making the step from commutative rings to $E_\infty$-rings, should I care about $E_k$-rings for $k<\infty$, or algebras over more general $\infty$-operads? If so, why?

Answer 1

Here are two examples where $E_k$-algebras show up in algebraic geometry, for $1<k<\infty$ (actually, just $k=2$):

• The work of Gaitsgory and Lurie on Weil's conjecture for function fields. See here, here, and here, especially Lectures 21-24. As far as I understand it, the rough idea is that you want to prove a version of the Grothendieck-Lefschetz trace formula, but for a non-constructible sheaf on a certain poorly behaved algebraic prestack (the Ran space). They seem to circumvent this by developing some version of Verdier duality in this setting, that can be viewed as an algebro-geometric version of Koszul duality for $E_2$-algebras. I am not sure if the theory of $E_2$-algebras is strictly necessary for the proof, or just provides motivation.

• The work of Toën and Vezzosi on Bloch's conductor conjecture, see arXiv:1605.08941 and arXiv:1710.05902. A main ingredient in their strategy is also a trace formula, but in the noncommutative setting (smooth proper dg-categories). It turns out that the dg-category $T$ they want to apply this to is not actually proper over the base. However, there exists a certain $E_2$-algebra $B$ over the base and $T$ turns out to be $B$-linear and proper as a $B$-linear dg-category. The trace formula they prove turns out to extend to the $E_2$-setting as well.

Regarding the comment about $E_1$-algebras: The derived category of a variety is equivalent to the derived category of modules over a certain $E_1$-algebra (the endomorphism algebra of the compact generator). A pretty active area of research in algebraic geometry is noncommutative algebraic geometry (via derived/dg-categories) and its relationship to birational geometry. See work of Bondal, Orlov, Kuznetsov, Bridgeland, Huybrechts, Van den Bergh, etc.

Edit: Here's another example.

• Arinkin-Gaitsgory, Singular support on coherent sheaves and the Geometric Langlands conjecture, Selecta 2015. They develop the theory of singular support in the setting of (ind)-coherent sheaves, on pretty general objects (derived Artin stacks), and use this to give a precise formulation of the categorical geometric Langlands conjecture. To that end they revisit some of the work of Benson-Iyengar-Krause on supports in triangulated categories. It seems that the correct setting for this theory turns out to be a dg-category/stable infinity-category equipped with an action of some $E_2$-algebra.