Semi-Cosimplicial Objects and Spreadability

TL;DR

This paper explores semi-cosimplicial objects (SCOs) in categories, linking them to spreadability in probability theory and showing how to construct SCOs from braid monoid actions, with various examples.

Contribution

It introduces a novel connection between semi-cosimplicial objects and spreadability, and provides methods to construct SCOs from braid monoid actions.

Findings

SCOs correspond to spreadable sequences of random variables.

A method to produce SCOs from actions of the infinite braid monoid.

Examples illustrating the construction and application of SCOs.

Abstract

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.