On the topologies of the exponential

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. contents 1 Exponentials 3 1.1 Exponential functors 3 1.2 The exponential functor 6 2 Topologies on the exponential 9 2.1 The topological exponential 9 2.1.1 First steps towards exponentiability 9 2.1.2 The topological exponential is not an exponential 10 2.1.3 The topological exponential is almost an exponential 11 2.1.4 The topological exponential is a restricted exponential 11 2.2 The metric exponential 12 2.3 The minimal exponential 14 2.4 Weak homotopy type of the exponentials 15 3 Interlude: minimal enclosing balls 17 4 Stratification of the exponentials 19 4.1 The exponentials as stratified spaces 19 4.2 Cones and joins 20 4.3 Conical stratification 21 This work was supported by IBS-R003-D1.