# A uniqueness theorem for stable homotopy theory

with Stefan Schwede

** ABSTRACT.**
In this paper we study the global structure of the stable homotopy theory
of spectra.
We establish criteria for when the homotopy theory associated to a given stable
model
category agrees with the classical stable homotopy theory of spectra.
One sufficient condition is that the associated homotopy category is equivalent
to the stable homotopy category as a triangulated category with an
action of the ring of stable homotopy groups of spheres, $\pi^s$.
In other words,
the classical stable homotopy theory, with all of its higher order information,
is
determined by the homotopy category as a triangulated category with
an action of $\pi^s$.
Another sufficient condition is the existence of a small generating object
(corresponding to the sphere spectrum) for which a specific `unit map' from
the infinite loop space $QS^0$ to the endomorphism space is a weak equivalence.