A shuffle algebra point of view on operator-valued probability theory

We extend the shuffle algebra perspective on scalar-valued non-commutative probability theory to operator-valued case. Given an space with B acting it (on left and right), we associate operators in operad of multilinear maps distribution free cumulants a random variable. These define representation PRO non-crossing partitions. Using concepts from higher category theory, specifically 2-monoidal categories, notion unshuffle Hopf underlying PRO. introduce words insertions show that both latter partitions are algebras. The two relate by mean map bialgebra (in sense) which call splitting map. Ultimately, obtain half-shuffle fixed point equation corresponding moment-cumulant relations bicollection homomorphisms insertions. Right laws interpreted framework boolean monotone respectively.