stable homotopy theory
loop space object
stable (∞,1)-category of spectra
stable homotopy category
smash product of spectra
symmetric smash product of spectra
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A bundle of spectra.
Specifically, for X an ∞-groupoid a spectrum parameterized over X is an (∞,1)-functor X→Spec from X to the stable (∞,1)-category of spectra.
The collection of all parameterized spectra for all ∞-groupoids X should form the tangent (∞,1)-category of ∞Grpd.
In twisted cohomology.
The comprehensive textbook on the matter is
A formulation of aspects of this in (∞,1)-category theory is in
See also the references at (∞,1)-vector bundle.