Semi-cosimplicial objects and spreadability

To a semi-cosimplicial object (SCO) in a category we associate
a system of partial shifts on the inductive limit. We show how to produce
an SCO from an action of the infinite braid monoid B⁺_∞ and provide examples.
In categories of (noncommutative) probability spaces SCOs correspond
to spreadable sequences of random variables, hence SCOs can be considered
as the algebraic structure underlying spreadability.