Argthtjtr

There is an abundance of examples of operads in topological spaces, chain complexes, and simplicial sets. However, there are very few (if any) examples of operads in algebraic geometric objects, even in $mathbb A^{1}$-homotopy theory.

My question is twofold:

• Are there useful examples of operads in algebraic geometry?




• Is there some intrinsic property of schemes which makes them incompatible with operatic structures? What about (higher) stacks?

For context, my main interest is in the $E_{n}$ operads. For instance, one would naïvely imagine that the moduli space $M_{0,n}$ of genus-zero marked curves could be an algebraic geometric counterpart of the space of little disks, but there seems to be no way to define the operadic composition without passing to the Deligne–Mumford compactification of the moduli space.

Mikhail Kapranov's invited lecture at the 1998 ICM was about exactly this:

Mikhail Kapranov, Operads and algebraic geometry. Documenta Mathematica Extra Volume ICM II (1998), 277-286. (PDF of the whole volume here.)

Abstract:

The study (motivated by mathematical physics) of algebraic varieties related to the moduli spaces of curves, helped to uncover important connections with the abstract algebraic theory of operads. This interaction led to new developments in both theories, and the purpose of the talk is to discuss some of them.

The homology of $M_{0,n}$ is an operad in a natural way that is compatible with AG structure (like weights). The construction uses the operad structure on the compactification plus Gysin maps-- it is general enough that one would expect it to work in the stable $mathbb A^1$ homotopy category. In fact, there you can likely define it as the Koszul dual operad to the homology of the compactification. I'm not well-informed enough to know whether/where this has been done, or whether the construction should work unstably as well.

The same goes for the Fulton Macpherson compactification of configuration space in affine space which is closer to the $E_d$ operad.

I recommend reading the paper Mixed Hodge structures and formality of symmetric monoidal functors by Cirici and Horel. They give several examples, and the paper is a great motivation for why one would be interested in getting operads in schemes (because under good circumstances, they are formal).

Among the examples they give, you have: the noncommutative associative operad; self maps of the projective line; (actually, a variant of the little disks operad given by parenthesized braids and the gravity operad aren't examples, thanks to Dan Petersen for the clarification).

(I am not sure that you will be able to view $E_n$ for $n > 2$ as an operad in schemes, but I would be happy if it were the case.)