Simplicial random variables

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 law' map is proved to be a Serre fibration and an homotopy equivalence.