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    • Source: En-ring

      • In mathematics, an






      E



      n




      {\displaystyle {\mathcal {E}}_{n}}

      -algebra in a symmetric monoidal infinity category C consists of the following data:

      An object



      A
      (
      U
      )


      {\displaystyle A(U)}

      for any open subset U of Rn homeomorphic to an n-disk.
      A multiplication map:




      Ī¼
      :
      A
      (

      U

      1


      )
      āŠ—
      ā‹Æ
      āŠ—
      A
      (

      U

      m


      )
      ā†’
      A
      (
      V
      )


      {\displaystyle \mu :A(U_{1})\otimes \cdots \otimes A(U_{m})\to A(V)}


      for any disjoint open disks




      U

      j




      {\displaystyle U_{j}}

      contained in some open disk V
      subject to the requirements that the multiplication maps are compatible with composition, and that



      Ī¼


      {\displaystyle \mu }

      is an equivalence if



      m
      =
      1


      {\displaystyle m=1}

      . An equivalent definition is that A is an algebra in C over the little n-disks operad.


      Examples


      An






      E



      n




      {\displaystyle {\mathcal {E}}_{n}}

      -algebra in vector spaces over a field is a unital associative algebra if n = 1, and a unital commutative associative algebra if n ā‰„ 2.
      An






      E



      n




      {\displaystyle {\mathcal {E}}_{n}}

      -algebra in categories is a monoidal category if n = 1, a braided monoidal category if n = 2, and a symmetric monoidal category if n ā‰„ 3.
      If Ī› is a commutative ring, then



      X
      ā†¦

      C

      āˆ—


      (

      Ī©

      n


      X
      ;
      Ī›
      )


      {\displaystyle X\mapsto C_{*}(\Omega ^{n}X;\Lambda )}

      defines an






      E



      n




      {\displaystyle {\mathcal {E}}_{n}}

      -algebra in the infinity category of chain complexes of



      Ī›


      {\displaystyle \Lambda }

      -modules.


      See also


      Categorical ring
      Highly structured ring spectrum


      References


      http://www.math.harvard.edu/~lurie/282ynotes/LectureXXII-En.pdf
      http://www.math.harvard.edu/~lurie/282ynotes/LectureXXIII-Koszul.pdf


      External links


      "En-algebra", ncatlab.org

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