Homotopy operator

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In the article "Lie theory and the Chern-Weil homomorphism" of A. Alekseev and Meinrenken on page 8 you have the following definition:



2.8.Homotopy operators. The space $Lleft ( E,E' right )$ of linear maps $phi :Erightarrow E'$ between differential spaces is itself a differential space, with differential $dleft ( phi right )=dcirc phi -left ( -1 right )^left phi circ d$. Chain maps correspond to cocycles in $Lleft ( E,E' right )^overline0$.



A homotopy operator between two chains maps $phi _0,phi _1:Erightarrow E'$ is an odd linear map $hin Lleft ( E,E' right )^overline1$ such that $dleft ( h right )=phi _0-phi _1$.



Taking into account the definition given on page 6 of the same article:



2.5. A differential space (ds) is a super vector space $E$, together with a differential,i.e. an odd endomorphism $din Endleft ( E right )^overline1$ satisfying $dcirc d=0$.



So $Lleft ( E,E' right )$ is a differential space, together with a odd endomorphism $din End(Lleft ( E,E' right )^overline1)$ satisfying $dcirc d=0$.



I want to know in which part of the definition 2.5 works



$dleft ( phi right )=dcirc phi -left ( -1 right )^left phi circ d$. Chain maps correspond to cocycles in $Lleft ( E,E' right )^overline0$.



Thanks for the help




algebraic-topology




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    In the article "Lie theory and the Chern-Weil homomorphism" of A. Alekseev and Meinrenken on page 8 you have the following definition:



    2.8.Homotopy operators. The space $Lleft ( E,E' right )$ of linear maps $phi :Erightarrow E'$ between differential spaces is itself a differential space, with differential $dleft ( phi right )=dcirc phi -left ( -1 right )^left phi circ d$. Chain maps correspond to cocycles in $Lleft ( E,E' right )^overline0$.



    A homotopy operator between two chains maps $phi _0,phi _1:Erightarrow E'$ is an odd linear map $hin Lleft ( E,E' right )^overline1$ such that $dleft ( h right )=phi _0-phi _1$.



    Taking into account the definition given on page 6 of the same article:



    2.5. A differential space (ds) is a super vector space $E$, together with a differential,i.e. an odd endomorphism $din Endleft ( E right )^overline1$ satisfying $dcirc d=0$.



    So $Lleft ( E,E' right )$ is a differential space, together with a odd endomorphism $din End(Lleft ( E,E' right )^overline1)$ satisfying $dcirc d=0$.



    I want to know in which part of the definition 2.5 works



    $dleft ( phi right )=dcirc phi -left ( -1 right )^left phi circ d$. Chain maps correspond to cocycles in $Lleft ( E,E' right )^overline0$.



    Thanks for the help




    algebraic-topology




    share|cite|improve this question





















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      In the article "Lie theory and the Chern-Weil homomorphism" of A. Alekseev and Meinrenken on page 8 you have the following definition:



      2.8.Homotopy operators. The space $Lleft ( E,E' right )$ of linear maps $phi :Erightarrow E'$ between differential spaces is itself a differential space, with differential $dleft ( phi right )=dcirc phi -left ( -1 right )^left phi circ d$. Chain maps correspond to cocycles in $Lleft ( E,E' right )^overline0$.



      A homotopy operator between two chains maps $phi _0,phi _1:Erightarrow E'$ is an odd linear map $hin Lleft ( E,E' right )^overline1$ such that $dleft ( h right )=phi _0-phi _1$.



      Taking into account the definition given on page 6 of the same article:



      2.5. A differential space (ds) is a super vector space $E$, together with a differential,i.e. an odd endomorphism $din Endleft ( E right )^overline1$ satisfying $dcirc d=0$.



      So $Lleft ( E,E' right )$ is a differential space, together with a odd endomorphism $din End(Lleft ( E,E' right )^overline1)$ satisfying $dcirc d=0$.



      I want to know in which part of the definition 2.5 works



      $dleft ( phi right )=dcirc phi -left ( -1 right )^left phi circ d$. Chain maps correspond to cocycles in $Lleft ( E,E' right )^overline0$.



      Thanks for the help




      algebraic-topology




      share|cite|improve this question











      In the article "Lie theory and the Chern-Weil homomorphism" of A. Alekseev and Meinrenken on page 8 you have the following definition:



      2.8.Homotopy operators. The space $Lleft ( E,E' right )$ of linear maps $phi :Erightarrow E'$ between differential spaces is itself a differential space, with differential $dleft ( phi right )=dcirc phi -left ( -1 right )^left phi circ d$. Chain maps correspond to cocycles in $Lleft ( E,E' right )^overline0$.



      A homotopy operator between two chains maps $phi _0,phi _1:Erightarrow E'$ is an odd linear map $hin Lleft ( E,E' right )^overline1$ such that $dleft ( h right )=phi _0-phi _1$.



      Taking into account the definition given on page 6 of the same article:



      2.5. A differential space (ds) is a super vector space $E$, together with a differential,i.e. an odd endomorphism $din Endleft ( E right )^overline1$ satisfying $dcirc d=0$.



      So $Lleft ( E,E' right )$ is a differential space, together with a odd endomorphism $din End(Lleft ( E,E' right )^overline1)$ satisfying $dcirc d=0$.



      I want to know in which part of the definition 2.5 works



      $dleft ( phi right )=dcirc phi -left ( -1 right )^left phi circ d$. Chain maps correspond to cocycles in $Lleft ( E,E' right )^overline0$.



      Thanks for the help





      algebraic-topology





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