Randomized simplicial sets

TL;DR

This paper introduces new geometric realizations of simplicial sets using spaces of random variables, establishing a homotopy equivalence and a new Quillen equivalence with topological spaces.

Contribution

It constructs novel geometric realizations of simplicial sets based on probability measures and random variables, linking them to classical realizations through homotopy.

Findings

The new realizations are homotopy equivalent to classical ones.

The construction provides a new Quillen equivalence between simplicial sets and topological spaces.

The approach uses spaces of probability measures with convergence topology.

Abstract

We construct new geometric realizations of simplicial and pre-simplicial sets where the standard $n$-simplex, viewed as the space of probability measures on $n+1$ elements, is replaced by the space of $(n+1)$-valued random variables, with the topology of probability convergence. We prove that the map which associates to a random variable its probability law is an homotopy equivalence from these new geometric realizations to the classical ones. Finally, we prove that this realization provides a new Quillen equivalence between simplicial sets and topological spaces.