nLab Symmetric spectra

This page collects material related to the lecture notes

• Stefan Schwede,

  Symmetric spectra,

  2012

  (pdf)

on stable homotopy theory in terms of symmetric spectra.

See also the Lectures on Equivariant Stable Homotopy Theory and on Global homotopy theory for the model on orthogonal spectra and equivariant stable homotopy theory.

Chapter I. Basics

1. Symmetric spectra

• symmetric spectrum

• ring spectrum

• module spectrum

Basic Examples:

• sphere spectrum

  stable homotopy groups of spheres

  Hopf fibration

• suspension spectrum

• Eilenberg-MacLane spectrumHA

  (the claim about Eilenberg-MacLane spaces here is proven in (Aguilar-Gitler-Prieto 02, corollary 6.4.23) )

• Thom spectrum

  MO, MU

• topological K-theory, KU

2. Properties of naive homotopy groups

• stable homotopy groups

• suspension isomorphism

3. Basic constructions

3.1 Symmetric spectra of simplicial sets

• powering and copowering over sSet

3.2 Constructions

• limits and colimits

• shift spectrum?

• semistable symmetric spectrum

• mapping spectrum

• free symmetric spectrum

3.3 Constructions involving ring spectra

3.4 Symmetric Spectra and $I$-spaces

• I-space

4. Stable equivalences

• spectrification

5. Smash product

• smash product of spectra

6. Homotopy groups

7. Relation to other kinds of spectra

7.1 Orthogonal spectra

• orthogonal spectrum

7.2 Unitary spectra

7.3 Continuous and simplicial functors

• excisive functor, model structure for excisive functors

7.4 $\Gamma$-spaces

• Gamma-space

7.5 Permutative categories

• permutative category

7.6 Spectra as enriched functors

• enriched functor

8. Naive versus true homotopy groups

History, credits, further reading

• highly structured spectra

• symmetric monoidal smash product of spectra

Chapter II. The stable homotopy category

1. The stable homotopy category

• stable homotopy category

2. Triangulated structure

• triangulated category

3. Derived smash product

• smash product of spectra

4. Grading

• ring spectrum

5. Generator

6. Homology and cohomology

6.1 Generalized homology and cohomology

• generalized (Eilenberg-Steenrod) cohomology

• generalized homology

• Milnor exact sequence

• Kronecker pairing

• universal coefficient theorem

6.2 Ordinary homology and cohomology

• ordinary cohomology

• Künneth theorem

• Hurewicz theorem

6.3 Moore spectra

• Moore spectrum

7. Finite spectra

• Spanier-Whitehead category

• finite spectra

8. Connective covers and Postnikov sections

• connective cover

• Postnikov tower

9. Localization and completion

• localization, formal completion

• fracture theorem

10. The Steenrod algebra

• cohomology operation

• Steenrod algebra

10.1 Examples and applications

10.2 Hopf algebra structure

• Hopf algebra

10.3 The Adams spectral sequence

• Adams spectral sequence

Chapter III. Model structures

• model structure on symmetric spectra

1. Symmetric spectra in a simplicial category

2. Flat cofibrations

3. Level model structures

4. Stable model structures

• model structure on symmetric spectra

5. Operads and their algebras

• operad

• algebra over an operad

6. Model structures for algebras over an operad

• model structure on algebras over an operad

7. Connective covers of structured spectra

• connective cover

Chapter IV. Module spectra

Appendix A. Miscellaneous tools

Model category theory

• model category

Compactly generated spaces

• compactly generated topological spaces

• classical model structure on topological spaces

Simplicial sets

• simplicial sets

• classical model structure on simplicial sets

Equivariant homotopy theory

• equivariant homotopy theory

Errata

• p. 273. the definition of Kronecker pairing (construction 6.13) is missign the application of the multiplication map

• p. 331. the second “vertically” should be “horizontally”

• p. 331. “with coordinated”

• p. 333. “stable stemes”

• p. 344. uncompiled source code “eqrefeq-general” etc.

category: reference