  We describe a point-set category of parametrized orthogonal spectra, a model
structure on this category, and a separate, more geometric class of
cofibrant-and-fibrant objects. The structures we describe are "convenient" in
that they are preserved by the most common operations. They allow us to reduce
sophisticated statements about the homotopy category to straightforward claims
at the point-set level. We use this framework to give a construction of the
bicategory of parametrized spectra, one that is far more direct than earlier
approaches. This gives a clean bridge between the concrete index theory
pioneered by Dold, and the formal bicategorical theory developed by May and
Sigurdsson, Ponto, and Shulman.
