We give an explicit description of the quadratic approximation of theidentity functor on the category of unpointed spaces. To do this, we construct a version of the stable James-Hopf map that is valid for unpointedspaces. While previous constructions of the James-Hopf map relied onconfiguration space models of Ω∞Σ∞X, our approach uses equivariantstable homotopy theory.By way of “application” we use the quadratic approximation to showan example of two fiberwise spaces that are stably fiber homotopy equivalent, but are not unstably fiber homotopy equivalent. The difference isdetected on the level of quadratic approximations. We use a simplifiedversion of an example proposed by Henry Adams in the context of thepath evasion problem.