Introduction to Stable homotopy theory -- 1-2 in nLab https://ncatlab.org/nlab/print/Introduction+to+Stable+homotopy+theor...
nLab Introduction to Stable homotopy theory -- 1-2
We give an introduction to the stable homotopy category and to its key computational tool, the Adams spectral sequence. To that end we introduce the modern tools, such as model categories and highly structured ring spectra. In the accompanying seminar we consider applications to cobordism theory and complex oriented cohomology such as to converge in the end to a glimpse of the modern picture of chromatic homotopy theory.
Lecture notes.
Main page: Introduction to Stable homotopy theory.
Previous section: Prelude -- Classical homotopy theory
This section: Part 1 -- Stable homotopy theory
Previous subsection: Part 1.1 -- Stable homotopy theory -- Sequential spectra
This subsection: Part 1.2 - Stable homotopy theory – Structured spectra
Next section: Part 2 -- Adams spectral sequences
Stable homotopy theory – Structured spectra
1. Categorical algebra Monoidal topological categories Algebras and modules Topological ends and coends Topological Day convolution Functors with smash product 2. -Modules Pre-Excisive functors Symmetric and orthogonal spectra As diagram spectra Stable weak homotopy equivalences Free spectra and Suspension spectra 3. The strict model structure on structured spectra Topological enrichment Monoidal model structure Suspension and looping 4. The stable model structure on structured spectra Proof of the model structure Stability of the homotopy theory Monoidal model structure 5. The monoidal stable homotopy category Tensor triangulated structure Homotopy ring spectra 6. Examples Sphere spectrum Eilenberg-MacLane spectra Thom spectra 7. Conclusion 8. References
The key result of part 1.1 was (thm.) the construction of a stable homotopy theory of spectra, embodied by
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a stable model structure on topological sequential spectra SeqSpec(Top ) (thm.) with its corresponding stable homotopy category Ho(Spectra), which stabilizes the canonical looping/suspension adjunction on pointed topological spaces in that it fits into a diagram of (Quillen-)adjunctions of the form