nLab Symmetric spectra


This page collects material related to the lecture notes

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

Basic Examples:

2. Properties of naive homotopy groups

3. Basic constructions

3.1 Symmetric spectra of simplicial sets

3.2 Constructions

3.3 Constructions involving ring spectra

3.4 Symmetric Spectra and II-spaces

4. Stable equivalences

5. Smash product

6. Homotopy groups

7. Relation to other kinds of spectra

7.1 Orthogonal spectra

7.2 Unitary spectra

7.3 Continuous and simplicial functors

7.4 Γ\Gamma-spaces

7.5 Permutative categories

7.6 Spectra as enriched functors

8. Naive versus true homotopy groups

History, credits, further reading

Chapter II. The stable homotopy category

1. The stable homotopy category

2. Triangulated structure

3. Derived smash product

4. Grading

5. Generator

6. Homology and cohomology

6.1 Generalized homology and cohomology

6.2 Ordinary homology and cohomology

6.3 Moore spectra

7. Finite spectra

8. Connective covers and Postnikov sections

9. Localization and completion

10. The Steenrod algebra

10.1 Examples and applications

10.2 Hopf algebra structure

10.3 The Adams spectral sequence

Chapter III. Model structures

1. Symmetric spectra in a simplicial category

2. Flat cofibrations

3. Level model structures

4. Stable model structures

5. Operads and their algebras

6. Model structures for algebras over an operad

7. Connective covers of structured spectra

Chapter IV. Module spectra

Appendix A. Miscellaneous tools

Model category theory

Compactly generated spaces

Simplicial sets

Equivariant homotopy theory


  • 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

Last revised on April 1, 2023 at 20:51:57. See the history of this page for a list of all contributions to it.