Are there prominent examples of operads in schemes?
There is an abundance of examples of operads in topological spaces, chain complexes, and simplicity sets. However, there are very few (if any) examples of operads in algebraic geometric objects, even in $A^{1}$-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 operatic composition without passing to the Deligne–Mumford compactification of the moduli space.
ag.algebraic-geometry homotopy-theory operads algebraic-stacks
edited 57 mins ago
LSpice
2,82322627
asked 1 hour ago
Patrick Elliott
1032
add a comment |
There is an abundance of examples of operads in topological spaces, chain complexes, and simplicity sets. However, there are very few (if any) examples of operads in algebraic geometric objects, even in $A^{1}$-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 operatic composition without passing to the Deligne–Mumford compactification of the moduli space.
ag.algebraic-geometry homotopy-theory operads algebraic-stacks
edited 57 mins ago
LSpice
2,82322627
asked 1 hour ago
Patrick Elliott
1032
add a comment |
There is an abundance of examples of operads in topological spaces, chain complexes, and simplicity sets. However, there are very few (if any) examples of operads in algebraic geometric objects, even in $A^{1}$-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 operatic composition without passing to the Deligne–Mumford compactification of the moduli space.
ag.algebraic-geometry homotopy-theory operads algebraic-stacks
edited 57 mins ago
LSpice
2,82322627
asked 1 hour ago
Patrick Elliott
1032
There is an abundance of examples of operads in topological spaces, chain complexes, and simplicity sets. However, there are very few (if any) examples of operads in algebraic geometric objects, even in $A^{1}$-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 operatic composition without passing to the Deligne–Mumford compactification of the moduli space.
ag.algebraic-geometry homotopy-theory operads algebraic-stacks
ag.algebraic-geometry homotopy-theory operads algebraic-stacks
edited 57 mins ago
LSpice
2,82322627
asked 1 hour ago
Patrick Elliott
1032
edited 57 mins ago
LSpice
2,82322627
asked 1 hour ago
Patrick Elliott
1032
edited 57 mins ago
LSpice
2,82322627
edited 57 mins ago
LSpice
2,82322627
edited 57 mins ago
LSpice
2,82322627
2,82322627
asked 1 hour ago
Patrick Elliott
1032
asked 1 hour ago
Patrick Elliott
1032
asked 1 hour ago
Patrick Elliott
1032
1032
add a comment |
add a comment |
1 Answer
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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.
answered 34 mins ago
Phil Tosteson
773147
add a comment |
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1 Answer
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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.
answered 34 mins ago
Phil Tosteson
773147
add a comment |
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.
answered 34 mins ago
Phil Tosteson
773147
add a comment |
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.
answered 34 mins ago
Phil Tosteson
773147
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.
answered 34 mins ago
Phil Tosteson
773147
edited 25 mins ago
answered 34 mins ago
Phil Tosteson
773147
answered 34 mins ago
Phil Tosteson
773147
answered 34 mins ago
Phil Tosteson
773147
773147
add a comment |
add a comment |
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