Deligne's conjecture states that the Hochschild cochain complex of an A-infinity algebra is an algebra over the operad of chains on the topological little discs operad.


Of course the conjecture has been proven, but proofs aside -- what are some moral reasons why we should believe in Deligne's conjecture?



Deligne's conjecture (the little discs operad one)
Here is a super simple way to think about it: The Hochshcild cochains on $A$ is derived $Hom$ over $A \otimes A^{op}$ of $A$ into$ A$. This has two multiplications, given by composition and multiplication in the target respectively.


These commute, or intertwine, so the result has an action of the tensor product of two $A$-infinity operads, which is (at least if you chose the right $A$-infinity operads) an $E\_2$ operad. 


