  Conjugation spaces are equipped with an involution such that the fixed points
have the same mod 2 cohomology (as a graded vector space, a ring, and even an
unstable algebra) but with all degrees divided by 2, generalizing the classical
examples of complex projective spaces under complex conjugation. Using tools
from stable equivariant homotopy theory we provide a characterization of
conjugation spaces in terms of purity. This conceptual viewpoint, compared to
the more computational original definition, allows us to recover all known
structural properties of conjugation spaces.
