Are there prominent examples of operads in schemes?












6














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







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    6














    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







    share|cite|improve this question



























      6












      6








      6


      1





      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







      share|cite|improve this question















      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






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 57 mins ago









      LSpice

      2,82322627




      2,82322627










      asked 1 hour ago









      Patrick Elliott

      1032




      1032






















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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.






          share|cite|improve this answer























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            2














            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.






            share|cite|improve this answer




























              2














              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.






              share|cite|improve this answer


























                2












                2








                2






                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.






                share|cite|improve this answer














                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.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited 25 mins ago

























                answered 34 mins ago









                Phil Tosteson

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                773147






























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