Simplicial Random Variables

TL;DR

This paper introduces a probabilistic geometric realization of simplicial complexes, linking topology with probability theory by showing the space's homotopy properties and a natural map to probability mass functions.

Contribution

It presents a novel geometric realization inspired by probability theory, establishing its homotopy type and a probabilistic map as a fibration and homotopy equivalence.

Findings

The space has the correct weak homotopy type.

The probability map is a Serre fibration.

The map is a weak homotopy equivalence.

Abstract

We introduce a new `geometric realization' of an (abstract) simplicial complex, inspired by probability theory. This space (and its completion) is a metric space, which has the right (weak) homotopy type, and which can be compared with the usual geometric realization through a natural map, which has probabilistic meaning : it associates to a random variable its probability mass function. This `probability map' function is proved to be a (Serre) fibration and a (weak) homotopy equivalence.