Definition of $A_infty$-module












2












$begingroup$


Let $A$ be a $A_infty$-algebra over a commutative ring $k$, suppose $V$ is a complex of $k$-modules. The "usual" definition of the structure of $A_infty$-module on $V$ the sequence of map
$$
s_n : A^{n-1} otimes V to V,
$$
where $s_0$ is the differential on $V$ and $s_n$ satisfy homotopy relations. Is it true that this definition is equivalent to the existance of $A_infty$-morphism
$$
s: A to operatorname{End}_k(V),
$$
where $operatorname{End}_k(V)$ is the dg algebra of $k$-linear endomorphisms of the graded module $V$?






reference-request homological-algebra







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$endgroup$












  • $begingroup$
    See page 13 here.
    $endgroup$
    – Pedro Tamaroff♦
    Feb 1 at 17:28


















2












$begingroup$


Let $A$ be a $A_infty$-algebra over a commutative ring $k$, suppose $V$ is a complex of $k$-modules. The "usual" definition of the structure of $A_infty$-module on $V$ the sequence of map
$$
s_n : A^{n-1} otimes V to V,
$$
where $s_0$ is the differential on $V$ and $s_n$ satisfy homotopy relations. Is it true that this definition is equivalent to the existance of $A_infty$-morphism
$$
s: A to operatorname{End}_k(V),
$$
where $operatorname{End}_k(V)$ is the dg algebra of $k$-linear endomorphisms of the graded module $V$?






reference-request homological-algebra







share|cite|improve this question









$endgroup$












  • $begingroup$
    See page 13 here.
    $endgroup$
    – Pedro Tamaroff♦
    Feb 1 at 17:28
















2












2








2





$begingroup$


Let $A$ be a $A_infty$-algebra over a commutative ring $k$, suppose $V$ is a complex of $k$-modules. The "usual" definition of the structure of $A_infty$-module on $V$ the sequence of map
$$
s_n : A^{n-1} otimes V to V,
$$
where $s_0$ is the differential on $V$ and $s_n$ satisfy homotopy relations. Is it true that this definition is equivalent to the existance of $A_infty$-morphism
$$
s: A to operatorname{End}_k(V),
$$
where $operatorname{End}_k(V)$ is the dg algebra of $k$-linear endomorphisms of the graded module $V$?






reference-request homological-algebra







share|cite|improve this question









$endgroup$




Let $A$ be a $A_infty$-algebra over a commutative ring $k$, suppose $V$ is a complex of $k$-modules. The "usual" definition of the structure of $A_infty$-module on $V$ the sequence of map
$$
s_n : A^{n-1} otimes V to V,
$$
where $s_0$ is the differential on $V$ and $s_n$ satisfy homotopy relations. Is it true that this definition is equivalent to the existance of $A_infty$-morphism
$$
s: A to operatorname{End}_k(V),
$$
where $operatorname{End}_k(V)$ is the dg algebra of $k$-linear endomorphisms of the graded module $V$?






reference-request homological-algebra




reference-request homological-algebra






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




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asked Feb 1 at 10:12









AlexAlex

2,8991128




2,8991128












  • $begingroup$
    See page 13 here.
    $endgroup$
    – Pedro Tamaroff♦
    Feb 1 at 17:28




















  • $begingroup$
    See page 13 here.
    $endgroup$
    – Pedro Tamaroff♦
    Feb 1 at 17:28


















$begingroup$
See page 13 here.
$endgroup$
– Pedro Tamaroff♦
Feb 1 at 17:28






$begingroup$
See page 13 here.
$endgroup$
– Pedro Tamaroff♦
Feb 1 at 17:28












1 Answer
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$begingroup$

Note that, by adjunction, the family of maps $s_n : A^{otimes (n-1)} otimes Vlongrightarrow V$ are the same as degree $2-n$ maps $s_n : A^{otimes (n-1)} longrightarrow operatorname{End}(V)$. These collect, by desuspending and suspending, into a single degree zero map $s : BA longrightarrow operatorname{End}(V)$, where $BA$ is the bar construction of the $A_infty$-algebra $A$. As you observe, this is the same as giving an $A_infty$-map from $A$ to $operatorname{End}(V)$. See the notes of B. Keller here.






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    $begingroup$

    Note that, by adjunction, the family of maps $s_n : A^{otimes (n-1)} otimes Vlongrightarrow V$ are the same as degree $2-n$ maps $s_n : A^{otimes (n-1)} longrightarrow operatorname{End}(V)$. These collect, by desuspending and suspending, into a single degree zero map $s : BA longrightarrow operatorname{End}(V)$, where $BA$ is the bar construction of the $A_infty$-algebra $A$. As you observe, this is the same as giving an $A_infty$-map from $A$ to $operatorname{End}(V)$. See the notes of B. Keller here.






    share|cite|improve this answer









    $endgroup$


















      2












      $begingroup$

      Note that, by adjunction, the family of maps $s_n : A^{otimes (n-1)} otimes Vlongrightarrow V$ are the same as degree $2-n$ maps $s_n : A^{otimes (n-1)} longrightarrow operatorname{End}(V)$. These collect, by desuspending and suspending, into a single degree zero map $s : BA longrightarrow operatorname{End}(V)$, where $BA$ is the bar construction of the $A_infty$-algebra $A$. As you observe, this is the same as giving an $A_infty$-map from $A$ to $operatorname{End}(V)$. See the notes of B. Keller here.






      share|cite|improve this answer









      $endgroup$
















        2












        2








        2





        $begingroup$

        Note that, by adjunction, the family of maps $s_n : A^{otimes (n-1)} otimes Vlongrightarrow V$ are the same as degree $2-n$ maps $s_n : A^{otimes (n-1)} longrightarrow operatorname{End}(V)$. These collect, by desuspending and suspending, into a single degree zero map $s : BA longrightarrow operatorname{End}(V)$, where $BA$ is the bar construction of the $A_infty$-algebra $A$. As you observe, this is the same as giving an $A_infty$-map from $A$ to $operatorname{End}(V)$. See the notes of B. Keller here.






        share|cite|improve this answer









        $endgroup$



        Note that, by adjunction, the family of maps $s_n : A^{otimes (n-1)} otimes Vlongrightarrow V$ are the same as degree $2-n$ maps $s_n : A^{otimes (n-1)} longrightarrow operatorname{End}(V)$. These collect, by desuspending and suspending, into a single degree zero map $s : BA longrightarrow operatorname{End}(V)$, where $BA$ is the bar construction of the $A_infty$-algebra $A$. As you observe, this is the same as giving an $A_infty$-map from $A$ to $operatorname{End}(V)$. See the notes of B. Keller here.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Feb 1 at 17:35









        Pedro Tamaroff♦Pedro Tamaroff

        97.6k10153299




        97.6k10153299






























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