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Parametrized Homotopy Theory

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

• Peter May, What are parameterized spectra good for? (pdf 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

• Matthew Ando, Andrew Blumberg, David Gepner, Michael Hopkins, Charles Rezk, Units of ring spectra and Thom spectra , section 7: Parametrized spectra, units and Thom spectra via $\mathrm{\infty }$-categories (arXiv:0810.4535)

See also

• John Lind, Bundles of spectra and algebraic K-theory (arXiv:1304.5676)

category: reference