# nLab parametrized spectrum

## Theorems

#### Stable Homotopy theory

stable homotopy theory

bundles

cohomology

# Contents

## Idea

A parameterized spectrum is a bundle of spectra (May-Sigurdsson 06). Specifically, for $X$ an ∞-groupoid, then a spectrum parameterized over $X$ is equivalently an (∞,1)-functor $X \longrightarrow Spec$ from $X$ to the stable (∞,1)-category of spectra (Ando-Blumberg-Gepner 11): this assigns to each object of $X$ a spectrum, to each morphism an equivalence of spectra, to each 2-morphism a homotopy between such equivalences, and so forth.

Generally, given an (∞,1)-topos $\mathbf{H}$, then its tangent (∞,1)-topos $T\mathbf{H}$ is the (∞,1)-category of all spectrum objects in $\mathbf{H}$ parameterized over any object of $\mathbf{H}$ (an observation promoted by Joyal).

The intrinsic cohomology of such a tangent (∞,1)-topos of parameterized spectra is twisted generalized cohomology in $\mathbf{H}$, and generally is twisted bivariant cohomology in $\mathbf{H}$.

A comprehensive textbook account on parameterized spectra in ∞Grpd $\simeq$ $L_{whe}$Top is in