nLab Parametrized Homotopy Theory

This entry collects links related to the book

• Peter May, Johann Sigurdsson,

  Parametrized Homotopy Theory,

  Mathematical Surveys and Monographs 132

  AMS 2006

  ISBN:978-0-8218-3922-5

  arXiv:math/0411656

  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 ex-spaces and parametrized spectra.

A survey is in the slides

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

Subtleties

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

• Michael Cole, Prop. 5.3 in: Many homotopy categories are homotopy categories, Topology and its Applications 153 (2006) 1084–1099 (doi:10.1016/j.topol.2005.02.006)

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

• Tobias Barthel, Emily Riehl, On the construction of functorial factorizations for model categories, Algebr. Geom. Topol. 13 (2013) 1089-1124 (arXiv:1204.5427, doi:10.2140/agt.2013.13.1089, euclid:agt/1513715550)

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

• 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 $\infty$-categories (arXiv:0810.4535)

See also

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

A general abstract context for parameterized spectra are tangent (infinity,1)-toposes.

category: reference