nLab Parametrized Homotopy Theory

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Stable Homotopy theory

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

This entry collects links related to the book

on parameterized stable homotopy theory, hence on stable homotopy theory in slice (infinity,1)-toposes Top/X/X for given topological base spaces XX: the homotopy theory of ex-spaces and parametrized spectra.

A survey is in the slides

Subtleties

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

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

Applications

One application is twisted cohomology: instead of cocycles given by maps XAX \to A, twisted cocycles are given by sections XPX \to P of a bundle PXP \to X of spectra over XX.

A discussion of some of these issues using tools from (infinity,1)-category theory are in

See also

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

category: reference

Last revised on September 20, 2021 at 09:34:33. See the history of this page for a list of all contributions to it.