nLab Parametrized Homotopy Theory

Context

Cohomology

This entry collects links related to the book

on parameterized stable homotopy theory, hence on stable homotopy theory in slice (infinity,1)-toposes Top$/X$ for given topological base spaces $X$: the homotopy theory of ex-spaces and parametrized spectra.

A survey is in the slides

Subtleties

Beware that section 4.4 claims a new proof of the Strøm model structure, but relying on

which later was noticed to be false, by Richard Williamson; for details see p. 2 and Rem 5.12 and Sec. 6.1 in:

Applications

One application is twisted cohomology: instead of cocycles given by maps $X \to A$, twisted cocycles are given by sections $X \to P$ of a bundle $P \to X$ of spectra over $X$.

A discussion of some of these issues using tools from (infinity,1)-category theory are in