Suppose we have a non symmetric operad $\mathcal{O}$, a collection of sets
$\{P(n)\}\_{n\geq 0}$ and maps


$$P(n)\otimes \mathcal{O}(k\_1)\otimes\cdots \otimes \mathcal{O}(k\_n)\to P(k\_1+\cdots + k\_n)$$


satisfying some conditions on associativity when composed with the operadic composition, etc.



> 
> **Question.** Is there a name for the collection $P$ equipped with this structure?
> 
> 
> 


It is not an algebra over the operad, nor does it seem to fit into any definition of a module I have come across. We assume no further structure on $P$.



Right action by an operad on a non symmetric collection
This structure is called a right module over the operad. You can have a look at the book "Modules over operads and functors" by Benoit Fresse for a thorough study of these objects.


