  Classification of homotopy n-types has focused on developing algebraic
categories which are equivalent to categories of n-types. We expand this theory
by providing algebraic models of homotopy-theoretic constructions for stable
one-types. These include a model for the Postnikov one-truncation of the sphere
spectrum, and for its action on the model of a stable one-type. We show that a
bicategorical cokernel introduced by Vitale models the cofiber of a map between
stable one-types, and apply this to develop an algebraic model for the
Postnikov data of a stable one-type.
