# nLab Parametrized Homotopy Theory

## Theorems

cohomology

### Theorems

This entry is about the book

• May, Sigurdsson, Parametrized Homotopy Theory (pdf)

which discusses homotopy theory and cohomology over base spaces, hence in slice categories 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