On the topologies of the exponential

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 \mathrm{Exp}(X) for each topology, in the case where X is not (necessarily) 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.