nLab Parametrized Homotopy Theory

Theorems

Stable Homotopy theory

stable homotopy theory

Contents

Cohomology

This entry collects links related to the book

• May, Sigurdsson, Parametrized Homotopy Theory, MAS Mathematical Surveys and Monographs, vol. 132, 2006 (webpage, pdf)

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 parametrized spectra.

A survey is in the slides

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