nLab
parameterized homotopy theory

Idea

Parameterized (stable) homotopy theory is (stable) homotopy theory of bundles of homotopy types/stable homotopy types over a given base space.

For formalizations see at

• ex-space

• parameterized spectrum, excisive functor

• (infinity,1)-module bundle

• tangent (infinity,1)-topos

• twisted cohomology

Related concepts

• rational parametrized homotopy theory

  • rational fibration lemma

• rational parameterized stable homotopy theory

References

• Ioan Mackenzie James, Introduction to fibrewise homotopy theory, in Ioan Mackenzie James (ed.), Handbook of Algebraic Topology (1995)

A comprehensive textbook account on parameterized spectra in ∞Grpd $\simeq$ $L_{whe}$Top is in

• Peter May, Johann Sigurdsson, Parametrized Homotopy Theory, Mathematical Surveys and Monographs, vol. 132, AMS 2006 (ISBN:978-0-8218-3922-5, arXiv:math/0411656, pdf)

A formulation of aspects of this in (∞,1)-category theory is in

• Matthew Ando, Andrew Blumberg, David Gepner, Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map, Geom. Topol. 22 (2018) 3761-3825 (arXiv:1112.2203)

Discussion of convenient model category presentations:

• Vincent Braunack-Mayer, Combinatorial parametrised spectra, based on the PhD thesis (arXiv:1907.08496)