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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.

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    $\begingroup$ What's wrong with passing to the Deligne-Mumford compactification of the moduli space? $\endgroup$
    – Will Sawin
    Commented Dec 22, 2018 at 13:06
  • $\begingroup$ The homotopy type of the (complex points) of the DM compactification is not equivalent to the space of little disks, so the operad structure is not a model of $E_{n}$. $\endgroup$
    – Patrick Elliott
    Commented Dec 22, 2018 at 13:54
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    $\begingroup$ Sure, but this seems to contradict your earlier claim that there are very few (if any) examples of operads in algebraic geometry. There's an example in your question, and more examples can be constructed from it. $\endgroup$
    – Will Sawin
    Commented Dec 22, 2018 at 16:09

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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.

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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.

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  • $\begingroup$ In what sense is the Fulton Macpherson compactifticaion of configuration space on affine space related to the $E_{d}$ operad? Obviously the spaces are homotopy equivalent, but to my knowledge there is no operad structure (as schemes) on the FM-compactified configuration spaces. $\endgroup$
    – Patrick Elliott
    Commented Dec 22, 2018 at 12:56
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    $\begingroup$ @Patrick By the Fulton Macpherson compactification, I meant the complex version, not the real one. Unless I'm missing something, this is still an operad in schemes, even though it's not cyclic. However the relationship of the complex version to $E_d$ is not as clear, so it's good you asked. $\endgroup$
    – Phil Tosteson
    Commented Dec 22, 2018 at 23:17
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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.)

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    $\begingroup$ Neither the gravity operad nor the paranthesized braid operad come from an operad in schemes. In both cases the operad structure exists only on the level of homology, as in Phil's answer. $\endgroup$
    – Dan Petersen
    Commented Dec 21, 2018 at 16:34
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Shai Haran defines a notion of a Bi-operad which appears to be some kind of a generalized ring. He also defines Bital sets which I don't really understand.

He then proceeds to define generalized schemes as spectra of (distributive) Bi-operads.

The motivation comes from number theory and is geared towards casting spec(Z) as a curve over the mythical F1.

Homotopy and Arithmetic

I wonder if this has connections to spectral algebraic geometry in the sense of Lurie or Toen.

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  • $\begingroup$ I have come across this paper, and I've found it mysterious and hard to read. I also can't ignore the fact that most other literature does not address this paper. I can't help but think it must be flawed. $\endgroup$
    – Patrick Elliott
    Commented Apr 1, 2023 at 17:22
  • $\begingroup$ Quite possible. I am no expert. But the author is no crank, his other papers, especially regarding F1 geometry, have been cited quite a bit by well know researchers in the field. $\endgroup$
    – Shiva
    Commented Apr 15, 2023 at 16:16

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