On the topologies of the exponential

TL;DR

This paper investigates three topologies on the exponential of a space, analyzing their properties and homotopy types, and establishes equivalences between different definitions of factorization algebras.

Contribution

It characterizes the weak homotopy types of exponentials under various topologies and relates different frameworks of factorization algebras.

Findings

Exponential property holds for all three topologies.

Describes the weak homotopy type of Exp(X) for non-connected X.

Shows metric exponential is conically stratified.

Abstract

Factorization algebras have been defined using three different topologies on the Ran space. We study these three different topologies on the exponential, which is the union of the Ran space and the empty configuration, and show that an exponential property is satisfied in each case. As a consequence we describe the weak homotopy type of the exponential Exp(X) for each topology, in the case where X is not connected. We also study these exponentials as stratified spaces and show that the metric exponential is conically stratified for a general class of spaces. As a corollary, we obtain that locally constant factorization algebras defined by Beilinson-Drinfeld are equivalent to locally constant factorization algebras defined by Lurie.