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\section*{Symmetric spectra}

This page collects material related to the lecture notes

\begin{itemize}%
\item [[Stefan Schwede]],

\emph{Symmetric spectra},

2012

(\href{http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf}{pdf})



\end{itemize}
on [[stable homotopy theory]] in terms of [[symmetric spectra]].

See also the \emph{[[Lectures on Equivariant Stable Homotopy Theory]]} and on \emph{[[Global homotopy theory]]} for the model on [[orthogonal spectra]] and [[equivariant stable homotopy theory]].

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{chapter_i_basics}{Chapter I. Basics}\dotfill \pageref*{chapter_i_basics} \linebreak
\noindent\hyperlink{1_symmetric_spectra}{1. Symmetric spectra}\dotfill \pageref*{1_symmetric_spectra} \linebreak
\noindent\hyperlink{2_properties_of_naive_homotopy_groups}{2. Properties of naive homotopy groups}\dotfill \pageref*{2_properties_of_naive_homotopy_groups} \linebreak
\noindent\hyperlink{3_basic_constructions}{3. Basic constructions}\dotfill \pageref*{3_basic_constructions} \linebreak
\noindent\hyperlink{31_symmetric_spectra_of_simplicial_sets}{3.1 Symmetric spectra of simplicial sets}\dotfill \pageref*{31_symmetric_spectra_of_simplicial_sets} \linebreak
\noindent\hyperlink{32_constructions}{3.2 Constructions}\dotfill \pageref*{32_constructions} \linebreak
\noindent\hyperlink{33_constructions_involving_ring_spectra}{3.3 Constructions involving ring spectra}\dotfill \pageref*{33_constructions_involving_ring_spectra} \linebreak
\noindent\hyperlink{4_stable_equivalences}{4. Stable equivalences}\dotfill \pageref*{4_stable_equivalences} \linebreak
\noindent\hyperlink{5_smash_product}{5. Smash product}\dotfill \pageref*{5_smash_product} \linebreak
\noindent\hyperlink{6_homotopy_groups}{6. Homotopy groups}\dotfill \pageref*{6_homotopy_groups} \linebreak
\noindent\hyperlink{7_relation_to_other_kinds_of_spectra}{7. Relation to other kinds of spectra}\dotfill \pageref*{7_relation_to_other_kinds_of_spectra} \linebreak
\noindent\hyperlink{71_orthogonal_spectra}{7.1 Orthogonal spectra}\dotfill \pageref*{71_orthogonal_spectra} \linebreak
\noindent\hyperlink{72_unitary_spectra}{7.2 Unitary spectra}\dotfill \pageref*{72_unitary_spectra} \linebreak
\noindent\hyperlink{73_continuous_and_simplicial_functors}{7.3 Continuous and simplicial functors}\dotfill \pageref*{73_continuous_and_simplicial_functors} \linebreak
\noindent\hyperlink{74_spaces}{7.4 $\Gamma$-spaces}\dotfill \pageref*{74_spaces} \linebreak
\noindent\hyperlink{75_permutative_categories}{7.5 Permutative categories}\dotfill \pageref*{75_permutative_categories} \linebreak
\noindent\hyperlink{76_spectra_as_enriched_functors}{7.6 Spectra as enriched functors}\dotfill \pageref*{76_spectra_as_enriched_functors} \linebreak
\noindent\hyperlink{8_naive_versus_true_homotopy_groups}{8. Naive versus true homotopy groups}\dotfill \pageref*{8_naive_versus_true_homotopy_groups} \linebreak
\noindent\hyperlink{history_credits_further_reading}{History, credits, further reading}\dotfill \pageref*{history_credits_further_reading} \linebreak
\noindent\hyperlink{chapter_ii_the_stable_homotopy_category}{Chapter II. The stable homotopy category}\dotfill \pageref*{chapter_ii_the_stable_homotopy_category} \linebreak
\noindent\hyperlink{1_the_stable_homotopy_category}{1. The stable homotopy category}\dotfill \pageref*{1_the_stable_homotopy_category} \linebreak
\noindent\hyperlink{2_triangulated_structure}{2. Triangulated structure}\dotfill \pageref*{2_triangulated_structure} \linebreak
\noindent\hyperlink{3_derived_smash_product}{3. Derived smash product}\dotfill \pageref*{3_derived_smash_product} \linebreak
\noindent\hyperlink{4_grading}{4. Grading}\dotfill \pageref*{4_grading} \linebreak
\noindent\hyperlink{5_generator}{5. Generator}\dotfill \pageref*{5_generator} \linebreak
\noindent\hyperlink{6_homology_and_cohomology}{6. Homology and cohomology}\dotfill \pageref*{6_homology_and_cohomology} \linebreak
\noindent\hyperlink{61_generalized_homology_and_cohomology}{6.1 Generalized homology and cohomology}\dotfill \pageref*{61_generalized_homology_and_cohomology} \linebreak
\noindent\hyperlink{62_ordinary_homology_and_cohomology}{6.2 Ordinary homology and cohomology}\dotfill \pageref*{62_ordinary_homology_and_cohomology} \linebreak
\noindent\hyperlink{63_moore_spectra}{6.3 Moore spectra}\dotfill \pageref*{63_moore_spectra} \linebreak
\noindent\hyperlink{7_finite_spectra}{7. Finite spectra}\dotfill \pageref*{7_finite_spectra} \linebreak
\noindent\hyperlink{8_connective_covers_and_postnikov_sections}{8. Connective covers and Postnikov sections}\dotfill \pageref*{8_connective_covers_and_postnikov_sections} \linebreak
\noindent\hyperlink{9_localization_and_completion}{9. Localization and completion}\dotfill \pageref*{9_localization_and_completion} \linebreak
\noindent\hyperlink{10_the_steenrod_algebra}{10. The Steenrod algebra}\dotfill \pageref*{10_the_steenrod_algebra} \linebreak
\noindent\hyperlink{101_examples_and_applications}{10.1 Examples and applications}\dotfill \pageref*{101_examples_and_applications} \linebreak
\noindent\hyperlink{102_hopf_algebra_structure}{10.2 Hopf algebra structure}\dotfill \pageref*{102_hopf_algebra_structure} \linebreak
\noindent\hyperlink{103_the_adams_spectral_sequence}{10.3 The Adams spectral sequence}\dotfill \pageref*{103_the_adams_spectral_sequence} \linebreak
\noindent\hyperlink{chapter_iii_model_structures}{Chapter III. Model structures}\dotfill \pageref*{chapter_iii_model_structures} \linebreak
\noindent\hyperlink{1_symmetric_spectra_in_a_simplicial_category}{1. Symmetric spectra in a simplicial category}\dotfill \pageref*{1_symmetric_spectra_in_a_simplicial_category} \linebreak
\noindent\hyperlink{2_flat_cofibrations}{2. Flat cofibrations}\dotfill \pageref*{2_flat_cofibrations} \linebreak
\noindent\hyperlink{3_level_model_structures}{3. Level model structures}\dotfill \pageref*{3_level_model_structures} \linebreak
\noindent\hyperlink{4_stable_model_structures}{4. Stable model structures}\dotfill \pageref*{4_stable_model_structures} \linebreak
\noindent\hyperlink{5_operads_and_their_algebras}{5. Operads and their algebras}\dotfill \pageref*{5_operads_and_their_algebras} \linebreak
\noindent\hyperlink{6_model_structures_for_algebras_over_an_operad}{6. Model structures for algebras over an operad}\dotfill \pageref*{6_model_structures_for_algebras_over_an_operad} \linebreak
\noindent\hyperlink{7_connective_covers_of_structured_spectra}{7. Connective covers of structured spectra}\dotfill \pageref*{7_connective_covers_of_structured_spectra} \linebreak
\noindent\hyperlink{chapter_iv_module_spectra}{Chapter IV. Module spectra}\dotfill \pageref*{chapter_iv_module_spectra} \linebreak
\noindent\hyperlink{appendix_a_miscellaneous_tools}{Appendix A. Miscellaneous tools}\dotfill \pageref*{appendix_a_miscellaneous_tools} \linebreak
\noindent\hyperlink{model_category_theory}{Model category theory}\dotfill \pageref*{model_category_theory} \linebreak
\noindent\hyperlink{compactly_generated_spaces}{Compactly generated spaces}\dotfill \pageref*{compactly_generated_spaces} \linebreak
\noindent\hyperlink{simplicial_sets}{Simplicial sets}\dotfill \pageref*{simplicial_sets} \linebreak
\noindent\hyperlink{equivariant_homotopy_theory}{Equivariant homotopy theory}\dotfill \pageref*{equivariant_homotopy_theory} \linebreak
\noindent\hyperlink{errata}{Errata}\dotfill \pageref*{errata} \linebreak


\hypertarget{chapter_i_basics}{}\subsection*{{Chapter I. Basics}}\label{chapter_i_basics}

\hypertarget{1_symmetric_spectra}{}\subsubsection*{{1. Symmetric spectra}}\label{1_symmetric_spectra}

\begin{itemize}%
\item [[symmetric spectrum]]


\item [[ring spectrum]]


\item [[module spectrum]]



\end{itemize}
Basic Examples:

\begin{itemize}%
\item [[sphere spectrum]]

[[stable homotopy groups of spheres]]

[[Hopf fibration]]


\item [[suspension spectrum]]


\item [[Eilenberg-MacLane spectrum]] [[HA]]

(the claim about [[Eilenberg-MacLane spaces]] here is proven in (\href{Eilenberg-MacLane+space#AguilarGitlerPrieto02}{Aguilar-Gitler-Prieto 02, corollary 6.4.23}) )


\item [[Thom spectrum]]

[[MO]], [[MU]]


\item [[topological K-theory]], [[KU]]



\end{itemize}
\hypertarget{2_properties_of_naive_homotopy_groups}{}\subsubsection*{{2. Properties of naive homotopy groups}}\label{2_properties_of_naive_homotopy_groups}

\begin{itemize}%
\item [[stable homotopy groups]]


\item \href{homotopy+group+of+a+spectrum#SuspensionIsomorphismOfStableHomotopyGroups}{suspension isomorphism}



\end{itemize}
\hypertarget{3_basic_constructions}{}\subsubsection*{{3. Basic constructions}}\label{3_basic_constructions}

\hypertarget{31_symmetric_spectra_of_simplicial_sets}{}\paragraph*{{3.1 Symmetric spectra of simplicial sets}}\label{31_symmetric_spectra_of_simplicial_sets}

\begin{itemize}%
\item [[powering]] and [[copowering]] over [[sSet]]

\end{itemize}
\hypertarget{32_constructions}{}\paragraph*{{3.2 Constructions}}\label{32_constructions}

\begin{itemize}%
\item [[limits]] and [[colimits]]


\item [[shift spectrum]]


\item [[semistable symmetric spectrum]]


\item [[mapping spectrum]]


\item [[free symmetric spectrum]]



\end{itemize}
\hypertarget{33_constructions_involving_ring_spectra}{}\paragraph*{{3.3 Constructions involving ring spectra}}\label{33_constructions_involving_ring_spectra}

\hypertarget{4_stable_equivalences}{}\subsubsection*{{4. Stable equivalences}}\label{4_stable_equivalences}

\begin{itemize}%
\item [[spectrification]]

\end{itemize}
\hypertarget{5_smash_product}{}\subsubsection*{{5. Smash product}}\label{5_smash_product}

\begin{itemize}%
\item [[smash product of spectra]]

\end{itemize}
\hypertarget{6_homotopy_groups}{}\subsubsection*{{6. Homotopy groups}}\label{6_homotopy_groups}

\hypertarget{7_relation_to_other_kinds_of_spectra}{}\subsubsection*{{7. Relation to other kinds of spectra}}\label{7_relation_to_other_kinds_of_spectra}

\hypertarget{71_orthogonal_spectra}{}\paragraph*{{7.1 Orthogonal spectra}}\label{71_orthogonal_spectra}

\begin{itemize}%
\item [[orthogonal spectrum]]

\end{itemize}
\hypertarget{72_unitary_spectra}{}\paragraph*{{7.2 Unitary spectra}}\label{72_unitary_spectra}

\hypertarget{73_continuous_and_simplicial_functors}{}\paragraph*{{7.3 Continuous and simplicial functors}}\label{73_continuous_and_simplicial_functors}

\begin{itemize}%
\item [[excisive functor]], [[model structure for excisive functors]]

\end{itemize}
\hypertarget{74_spaces}{}\paragraph*{{7.4 $\Gamma$-spaces}}\label{74_spaces}

\begin{itemize}%
\item [[Gamma-space]]

\end{itemize}
\hypertarget{75_permutative_categories}{}\paragraph*{{7.5 Permutative categories}}\label{75_permutative_categories}

\begin{itemize}%
\item [[permutative category]]

\end{itemize}
\hypertarget{76_spectra_as_enriched_functors}{}\paragraph*{{7.6 Spectra as enriched functors}}\label{76_spectra_as_enriched_functors}

\begin{itemize}%
\item [[enriched functor]]

\end{itemize}
\hypertarget{8_naive_versus_true_homotopy_groups}{}\subsubsection*{{8. Naive versus true homotopy groups}}\label{8_naive_versus_true_homotopy_groups}

\hypertarget{history_credits_further_reading}{}\subsubsection*{{History, credits, further reading}}\label{history_credits_further_reading}

\begin{itemize}%
\item [[highly structured spectra]]


\item [[symmetric monoidal smash product of spectra]]



\end{itemize}
\hypertarget{chapter_ii_the_stable_homotopy_category}{}\subsection*{{Chapter II. The stable homotopy category}}\label{chapter_ii_the_stable_homotopy_category}

\hypertarget{1_the_stable_homotopy_category}{}\subsubsection*{{1. The stable homotopy category}}\label{1_the_stable_homotopy_category}

\begin{itemize}%
\item [[stable homotopy category]]

\end{itemize}
\hypertarget{2_triangulated_structure}{}\subsubsection*{{2. Triangulated structure}}\label{2_triangulated_structure}

\begin{itemize}%
\item [[triangulated category]]

\end{itemize}
\hypertarget{3_derived_smash_product}{}\subsubsection*{{3. Derived smash product}}\label{3_derived_smash_product}

\begin{itemize}%
\item [[smash product of spectra]]

\end{itemize}
\hypertarget{4_grading}{}\subsubsection*{{4. Grading}}\label{4_grading}

\begin{itemize}%
\item [[ring spectrum]]

\end{itemize}
\hypertarget{5_generator}{}\subsubsection*{{5. Generator}}\label{5_generator}

\hypertarget{6_homology_and_cohomology}{}\subsubsection*{{6. Homology and cohomology}}\label{6_homology_and_cohomology}

\hypertarget{61_generalized_homology_and_cohomology}{}\paragraph*{{6.1 Generalized homology and cohomology}}\label{61_generalized_homology_and_cohomology}

\begin{itemize}%
\item [[generalized (Eilenberg-Steenrod) cohomology]]


\item [[generalized homology]]


\item [[Milnor exact sequence]]


\item [[Kronecker pairing]]


\item [[universal coefficient theorem]]



\end{itemize}
\hypertarget{62_ordinary_homology_and_cohomology}{}\paragraph*{{6.2 Ordinary homology and cohomology}}\label{62_ordinary_homology_and_cohomology}

\begin{itemize}%
\item [[ordinary cohomology]]


\item [[Künneth theorem]]


\item [[Hurewicz theorem]]



\end{itemize}
\hypertarget{63_moore_spectra}{}\paragraph*{{6.3 Moore spectra}}\label{63_moore_spectra}

\begin{itemize}%
\item [[Moore spectrum]]

\end{itemize}
\hypertarget{7_finite_spectra}{}\subsubsection*{{7. Finite spectra}}\label{7_finite_spectra}

\begin{itemize}%
\item [[Spanier-Whitehead category]]


\item [[finite spectra]]



\end{itemize}
\hypertarget{8_connective_covers_and_postnikov_sections}{}\subsubsection*{{8. Connective covers and Postnikov sections}}\label{8_connective_covers_and_postnikov_sections}

\begin{itemize}%
\item [[connective cover]]


\item [[Postnikov tower]]



\end{itemize}
\hypertarget{9_localization_and_completion}{}\subsubsection*{{9. Localization and completion}}\label{9_localization_and_completion}

\begin{itemize}%
\item [[localization]], [[formal completion]]


\item [[fracture theorem]]



\end{itemize}
\hypertarget{10_the_steenrod_algebra}{}\subsubsection*{{10. The Steenrod algebra}}\label{10_the_steenrod_algebra}

\begin{itemize}%
\item [[cohomology operation]]


\item [[Steenrod algebra]]



\end{itemize}
\hypertarget{101_examples_and_applications}{}\paragraph*{{10.1 Examples and applications}}\label{101_examples_and_applications}

\hypertarget{102_hopf_algebra_structure}{}\paragraph*{{10.2 Hopf algebra structure}}\label{102_hopf_algebra_structure}

\begin{itemize}%
\item [[Hopf algebra]]

\end{itemize}
\hypertarget{103_the_adams_spectral_sequence}{}\paragraph*{{10.3 The Adams spectral sequence}}\label{103_the_adams_spectral_sequence}

\begin{itemize}%
\item [[Adams spectral sequence]]

\end{itemize}
\hypertarget{chapter_iii_model_structures}{}\subsection*{{Chapter III. Model structures}}\label{chapter_iii_model_structures}

\begin{itemize}%
\item [[model structure on symmetric spectra]]

\end{itemize}
\hypertarget{1_symmetric_spectra_in_a_simplicial_category}{}\subsubsection*{{1. Symmetric spectra in a simplicial category}}\label{1_symmetric_spectra_in_a_simplicial_category}

\hypertarget{2_flat_cofibrations}{}\subsubsection*{{2. Flat cofibrations}}\label{2_flat_cofibrations}

\hypertarget{3_level_model_structures}{}\subsubsection*{{3. Level model structures}}\label{3_level_model_structures}

\hypertarget{4_stable_model_structures}{}\subsubsection*{{4. Stable model structures}}\label{4_stable_model_structures}

\begin{itemize}%
\item [[model structure on symmetric spectra]]

\end{itemize}
\hypertarget{5_operads_and_their_algebras}{}\subsubsection*{{5. Operads and their algebras}}\label{5_operads_and_their_algebras}

\begin{itemize}%
\item [[operad]]


\item [[algebra over an operad]]



\end{itemize}
\hypertarget{6_model_structures_for_algebras_over_an_operad}{}\subsubsection*{{6. Model structures for algebras over an operad}}\label{6_model_structures_for_algebras_over_an_operad}

\begin{itemize}%
\item [[model structure on algebras over an operad]]

\end{itemize}
\hypertarget{7_connective_covers_of_structured_spectra}{}\subsubsection*{{7. Connective covers of structured spectra}}\label{7_connective_covers_of_structured_spectra}

\begin{itemize}%
\item [[connective cover]]

\end{itemize}
\hypertarget{chapter_iv_module_spectra}{}\subsection*{{Chapter IV. Module spectra}}\label{chapter_iv_module_spectra}

\hypertarget{appendix_a_miscellaneous_tools}{}\subsection*{{Appendix A. Miscellaneous tools}}\label{appendix_a_miscellaneous_tools}

\hypertarget{model_category_theory}{}\subsubsection*{{Model category theory}}\label{model_category_theory}

\begin{itemize}%
\item [[model category]]

\end{itemize}
\hypertarget{compactly_generated_spaces}{}\subsubsection*{{Compactly generated spaces}}\label{compactly_generated_spaces}

\begin{itemize}%
\item [[compactly generated topological spaces]]


\item [[classical model structure on topological spaces]]



\end{itemize}
\hypertarget{simplicial_sets}{}\subsubsection*{{Simplicial sets}}\label{simplicial_sets}

\begin{itemize}%
\item [[simplicial sets]]


\item [[classical model structure on simplicial sets]]



\end{itemize}
\hypertarget{equivariant_homotopy_theory}{}\subsubsection*{{Equivariant homotopy theory}}\label{equivariant_homotopy_theory}

\begin{itemize}%
\item [[equivariant homotopy theory]]

\end{itemize}
\hypertarget{errata}{}\subsection*{{Errata}}\label{errata}

\begin{itemize}%
\item p. 273. the definition of [[Kronecker pairing]] (construction 6.13) is missign the application of the multiplication map


\item p. 331. the second ``vertically'' should be ``horizontally''


\item p. 331. ``with coordinated''


\item p. 333. ``stable stemes''


\item p. 344. uncompiled source code ``eqrefeq-general'' etc.



\end{itemize}
category: reference



\end{document}