  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 $\mathbb{B}^+_\infty$ 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.
