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I'm asking here this question I asked on MSE that got no answers.

Let $V$ be a dg-module and $P$ an operad. The free $P$-algebra on $V$ is defined by $P(V)=\bigoplus_{r=0}^\infty (P(r)\otimes V^{\otimes r})_{\Sigma_r}$, where the $Σ_r$-quotient identifies tensor permutations with the action of permutations on $P(r)$.

On the other hand, $V$ is said to be a $P$-algebra if there is a morphism of operads $P\to End_V$, where $End_V$ is the endomorphism operad of $V$. Equivalently, $V$ is a $P$-algebra if there is a collection of maps $P(r)\otimes V^{\otimes r}\to V$ satisfying certain conditions.

How do these two notions reconcile?

An element of $p\otimes x_1\otimes\cdots \otimes x_r\in P(V)$ can be written as $p(x_1\otimes\cdots\otimes x_r)$ and therefore $p$ is interpreted as a map $V^{\otimes r}\to V$. But how can we realize $p$ as an element of $End_V(r)$ so that we do have the map of operads $P\to End_V$? Or equivalently, how can identify $p(x_1\otimes\cdots\otimes x_r)$ with an element of $V$ so that we have the maps $P(r)\otimes V^{\otimes r}\to V$?

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    $\begingroup$ Is it possible you are conflating the two different "V"s? If V is a module, then the free thing P(V) will be a P-algebra so there is a map $P \to \mathrm{End}_{P(V)}$ not a map to $\mathrm{End}_{V}$. Maybe an additional confusion is that, if $V$ has the structure of a P-algebra, then there is a map of $P$-algebras $P(V) \to V$. $\endgroup$
    – Dylan Wilson
    Commented Jul 16, 2020 at 17:04
  • $\begingroup$ It's also useful to remember that you have the operad structure of $P$ available to work with. When you want to apply some $p\in P(r)$ to $y_1\otimes\cdots\otimes y_r$ where each $y_i$ has the form $[q_i\otimes$ some product of elements of $V]$, you'll want to combine $p$ with the $q_i$'s using the operad structure of $P$. $\endgroup$
    – Andreas Blass
    Commented Jul 16, 2020 at 17:12
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    $\begingroup$ Take any book about operads. For instance, in Fresse's "Homotopy of Operads...", you can find how the free operad algebra is an algebra over that operad on page 39. In Markl-Shnider-Stasheff's "Operads in Algebra...", it's on page 48. $\endgroup$
    – Agustí Roig
    Commented Jul 16, 2020 at 18:21
  • $\begingroup$ @DylanWilson you're right I was confusing the role of $V$ on each case, thanks for pointing out. $\endgroup$
    – Javi
    Commented Jul 16, 2020 at 18:41
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    $\begingroup$ @AgustíRoig thank you for the references $\endgroup$
    – Javi
    Commented Jul 16, 2020 at 18:41

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