Cosheaves and distributions on toposes

Distributions on a Grothendieck topos were introduced by Lawvere [12] (cf. also [13]) as a generalization of the classical notion (cf. [20]) of real-valued distributions on a topological space. The cosheaves approach to distributions which is implicit in work of Pitis [19] is used here first, in order to answer affirmatively a question posed in [12] concerning the existence of the ''symmetric topos'' and next, in order to prove a structure theorem for categories of distributions on Grothendieck toposes that is similar in spirit to the Joyal-Tierney [11] structure theorem for Grothendieck toposes.