  We define a commutative monoid structure on the poset of left-exact
localizations of a higher topos, that we call the acyclic product. Our approach
is anchored in a structural analogy between the poset of left-exact
localizations of a topos and the poset of ideals of a commutative ring. The
acyclic product is analogous to the product of ideals. The sequence of powers
of a given left-exact localization defines a tower of localizations. We show
how this recovers the towers of Goodwillie calculus in the unstable homotopical
setting. We use this to describe the topoi of $n$-excisive functors as
classifying $n$-nilpotent objects.
