55Pxx Homotopy theory
This subtopic introduces the core ideas in homotopy theory, including foundational concepts, standard methods, and the main questions used to organize the area. Typical uses include building mathematical background, framing related research problems, and supporting applications in neighboring fields where these concepts provide useful structure.
Specific topics
55P05 Homotopy extension properties, cofibrations in algebraic topology
Overview
Homotopy extension properties, cofibrations in algebraic topology. This topic develops homotopy theory, including localization methods, model-categorical viewpoints, and foundational equivalence principles.
Related Wikipedia Page
Wikipedia: Homotopy theory
Useful Links
Key Ideas
- homotopy equivalence and deformation
- localization and model structures
- stable and unstable homotopy frameworks
Typical Uses
Used to classify spaces up to deformation and transport algebraic invariants across equivalent models.
Applications
- Algebraic topology foundations
- Higher category theory
- Topological field theory interfaces
References
Recommended Textbooks
55P10 Homotopy equivalences in algebraic topology
Overview
Homotopy equivalences in algebraic topology. This topic develops homotopy theory, including localization methods, model-categorical viewpoints, and foundational equivalence principles.
Related Wikipedia Page
Wikipedia: Homotopy theory
Useful Links
Key Ideas
- homotopy equivalence and deformation
- localization and model structures
- stable and unstable homotopy frameworks
Typical Uses
Used to classify spaces up to deformation and transport algebraic invariants across equivalent models.
Applications
- Algebraic topology foundations
- Higher category theory
- Topological field theory interfaces
References
Recommended Textbooks
55P15 Classification of homotopy type
Overview
Classification of homotopy type. This topic develops homotopy theory, including localization methods, model-categorical viewpoints, and foundational equivalence principles.
Related Wikipedia Page
Wikipedia: Homotopy theory
Useful Links
Key Ideas
- homotopy equivalence and deformation
- localization and model structures
- stable and unstable homotopy frameworks
Typical Uses
Used to classify spaces up to deformation and transport algebraic invariants across equivalent models.
Applications
- Algebraic topology foundations
- Higher category theory
- Topological field theory interfaces
References
Recommended Textbooks
55P20 Eilenberg-MacLane spaces
Overview
Eilenberg-MacLane spaces. This topic develops homotopy theory, including localization methods, model-categorical viewpoints, and foundational equivalence principles.
Related Wikipedia Page
Wikipedia: Homotopy theory
Useful Links
Key Ideas
- homotopy equivalence and deformation
- localization and model structures
- stable and unstable homotopy frameworks
Typical Uses
Used to classify spaces up to deformation and transport algebraic invariants across equivalent models.
Applications
- Algebraic topology foundations
- Higher category theory
- Topological field theory interfaces
References
Recommended Textbooks
55P25 Spanier-Whitehead duality
Overview
Spanier-Whitehead duality. This topic develops homotopy theory, including localization methods, model-categorical viewpoints, and foundational equivalence principles.
Related Wikipedia Page
Wikipedia: Homotopy theory
Useful Links
Key Ideas
- homotopy equivalence and deformation
- localization and model structures
- stable and unstable homotopy frameworks
Typical Uses
Used to classify spaces up to deformation and transport algebraic invariants across equivalent models.
Applications
- Algebraic topology foundations
- Higher category theory
- Topological field theory interfaces
References
Recommended Textbooks
55P30 Eckmann-Hilton duality
Overview
Eckmann-Hilton duality. This topic develops homotopy theory, including localization methods, model-categorical viewpoints, and foundational equivalence principles.
Related Wikipedia Page
Wikipedia: Homotopy theory
Useful Links
Key Ideas
- homotopy equivalence and deformation
- localization and model structures
- stable and unstable homotopy frameworks
Typical Uses
Used to classify spaces up to deformation and transport algebraic invariants across equivalent models.
Applications
- Algebraic topology foundations
- Higher category theory
- Topological field theory interfaces
References
Recommended Textbooks
55P35 Loop spaces in algebraic topology
Overview
Loop spaces in algebraic topology. This topic develops homotopy theory, including localization methods, model-categorical viewpoints, and foundational equivalence principles.
Related Wikipedia Page
Wikipedia: Homotopy theory
Useful Links
Key Ideas
- homotopy equivalence and deformation
- localization and model structures
- stable and unstable homotopy frameworks
Typical Uses
Used to classify spaces up to deformation and transport algebraic invariants across equivalent models.
Applications
- Algebraic topology foundations
- Higher category theory
- Topological field theory interfaces
References
Recommended Textbooks
55P40 Suspensions in algebraic topology
Overview
Suspensions in algebraic topology. This topic develops homotopy theory, including localization methods, model-categorical viewpoints, and foundational equivalence principles.
Related Wikipedia Page
Wikipedia: Homotopy theory
Useful Links
Key Ideas
- homotopy equivalence and deformation
- localization and model structures
- stable and unstable homotopy frameworks
Typical Uses
Used to classify spaces up to deformation and transport algebraic invariants across equivalent models.
Applications
- Algebraic topology foundations
- Higher category theory
- Topological field theory interfaces
References
Recommended Textbooks
55P42 Stable homotopy theory, spectra
Overview
Stable homotopy theory, spectra. This topic develops homotopy theory, including localization methods, model-categorical viewpoints, and foundational equivalence principles.
Related Wikipedia Page
Wikipedia: Homotopy theory
Useful Links
Key Ideas
- homotopy equivalence and deformation
- localization and model structures
- stable and unstable homotopy frameworks
Typical Uses
Used to classify spaces up to deformation and transport algebraic invariants across equivalent models.
Applications
- Algebraic topology foundations
- Higher category theory
- Topological field theory interfaces
References
Recommended Textbooks
55P43 Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.)
Overview
Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.). This topic develops homotopy theory, including localization methods, model-categorical viewpoints, and foundational equivalence principles.
Related Wikipedia Page
Wikipedia: Homotopy theory
Useful Links
Key Ideas
- homotopy equivalence and deformation
- localization and model structures
- stable and unstable homotopy frameworks
Typical Uses
Used to classify spaces up to deformation and transport algebraic invariants across equivalent models.
Applications
- Algebraic topology foundations
- Higher category theory
- Topological field theory interfaces
References
Recommended Textbooks
55P45 $H$-spaces and duals
Overview
$H$-spaces and duals. This topic develops homotopy theory, including localization methods, model-categorical viewpoints, and foundational equivalence principles.
Related Wikipedia Page
Wikipedia: Homotopy theory
Useful Links
Key Ideas
- homotopy equivalence and deformation
- localization and model structures
- stable and unstable homotopy frameworks
Typical Uses
Used to classify spaces up to deformation and transport algebraic invariants across equivalent models.
Applications
- Algebraic topology foundations
- Higher category theory
- Topological field theory interfaces
References
Recommended Textbooks
55P47 Infinite loop spaces
Overview
Infinite loop spaces. This topic develops homotopy theory, including localization methods, model-categorical viewpoints, and foundational equivalence principles.
Related Wikipedia Page
Wikipedia: Homotopy theory
Useful Links
Key Ideas
- homotopy equivalence and deformation
- localization and model structures
- stable and unstable homotopy frameworks
Typical Uses
Used to classify spaces up to deformation and transport algebraic invariants across equivalent models.
Applications
- Algebraic topology foundations
- Higher category theory
- Topological field theory interfaces
References
Recommended Textbooks
55P48 Loop space machines and operads in algebraic topology
Overview
Loop space machines and operads in algebraic topology. This topic develops homotopy theory, including localization methods, model-categorical viewpoints, and foundational equivalence principles.
Related Wikipedia Page
Wikipedia: Homotopy theory
Useful Links
Key Ideas
- homotopy equivalence and deformation
- localization and model structures
- stable and unstable homotopy frameworks
Typical Uses
Used to classify spaces up to deformation and transport algebraic invariants across equivalent models.
Applications
- Algebraic topology foundations
- Higher category theory
- Topological field theory interfaces
References
Recommended Textbooks
55P50 String topology
Overview
String topology. This topic develops homotopy theory, including localization methods, model-categorical viewpoints, and foundational equivalence principles.
Related Wikipedia Page
Wikipedia: Homotopy theory
Useful Links
Key Ideas
- homotopy equivalence and deformation
- localization and model structures
- stable and unstable homotopy frameworks
Typical Uses
Used to classify spaces up to deformation and transport algebraic invariants across equivalent models.
Applications
- Algebraic topology foundations
- Higher category theory
- Topological field theory interfaces
References
Recommended Textbooks
55P55 Shape theory in algebraic topology
Overview
Shape theory in algebraic topology. This topic develops homotopy theory, including localization methods, model-categorical viewpoints, and foundational equivalence principles.
Related Wikipedia Page
Wikipedia: Homotopy theory
Useful Links
Key Ideas
- homotopy equivalence and deformation
- localization and model structures
- stable and unstable homotopy frameworks
Typical Uses
Used to classify spaces up to deformation and transport algebraic invariants across equivalent models.
Applications
- Algebraic topology foundations
- Higher category theory
- Topological field theory interfaces
References
Recommended Textbooks
55P57 Rational homotopy theory
Overview
Rational homotopy theory. This topic develops homotopy theory, including localization methods, model-categorical viewpoints, and foundational equivalence principles.
Related Wikipedia Page
Wikipedia: Homotopy theory
Useful Links
Key Ideas
- homotopy equivalence and deformation
- localization and model structures
- stable and unstable homotopy frameworks
Typical Uses
Used to classify spaces up to deformation and transport algebraic invariants across equivalent models.
Applications
- Algebraic topology foundations
- Higher category theory
- Topological field theory interfaces
References
Recommended Textbooks
55P60 Localization and completion in homotopy theory
Overview
Localization and completion in homotopy theory. This topic develops homotopy theory, including localization methods, model-categorical viewpoints, and foundational equivalence principles.
Related Wikipedia Page
Wikipedia: Homotopy theory
Useful Links
Key Ideas
- homotopy equivalence and deformation
- localization and model structures
- stable and unstable homotopy frameworks
Typical Uses
Used to classify spaces up to deformation and transport algebraic invariants across equivalent models.
Applications
- Algebraic topology foundations
- Higher category theory
- Topological field theory interfaces
References
Recommended Textbooks
55P62 Rational homotopy theory
Overview
Rational homotopy theory. This topic develops homotopy theory, including localization methods, model-categorical viewpoints, and foundational equivalence principles.
Related Wikipedia Page
Wikipedia: Homotopy theory
Useful Links
Key Ideas
- homotopy equivalence and deformation
- localization and model structures
- stable and unstable homotopy frameworks
Typical Uses
Used to classify spaces up to deformation and transport algebraic invariants across equivalent models.
Applications
- Algebraic topology foundations
- Higher category theory
- Topological field theory interfaces
References
Recommended Textbooks
55P65 Homotopy functors in algebraic topology
Overview
Homotopy functors in algebraic topology. This topic develops homotopy theory, including localization methods, model-categorical viewpoints, and foundational equivalence principles.
Related Wikipedia Page
Wikipedia: Homotopy theory
Useful Links
Key Ideas
- homotopy equivalence and deformation
- localization and model structures
- stable and unstable homotopy frameworks
Typical Uses
Used to classify spaces up to deformation and transport algebraic invariants across equivalent models.
Applications
- Algebraic topology foundations
- Higher category theory
- Topological field theory interfaces
References
Recommended Textbooks
55P91 Equivariant homotopy theory in algebraic topology
Overview
Equivariant homotopy theory in algebraic topology. This topic develops homotopy theory, including localization methods, model-categorical viewpoints, and foundational equivalence principles.
Related Wikipedia Page
Wikipedia: Homotopy theory
Useful Links
Key Ideas
- homotopy equivalence and deformation
- localization and model structures
- stable and unstable homotopy frameworks
Typical Uses
Used to classify spaces up to deformation and transport algebraic invariants across equivalent models.
Applications
- Algebraic topology foundations
- Higher category theory
- Topological field theory interfaces
References
Recommended Textbooks
55P92 Relations between equivariant and nonequivariant homotopy theory
Overview
Relations between equivariant and nonequivariant homotopy theory. This topic develops homotopy theory, including localization methods, model-categorical viewpoints, and foundational equivalence principles.
Related Wikipedia Page
Wikipedia: Homotopy theory
Useful Links
Key Ideas
- homotopy equivalence and deformation
- localization and model structures
- stable and unstable homotopy frameworks
Typical Uses
Used to classify spaces up to deformation and transport algebraic invariants across equivalent models.
Applications
- Algebraic topology foundations
- Higher category theory
- Topological field theory interfaces
References
Recommended Textbooks
55P99 None of the above
Overview
None of the above. This topic develops homotopy theory, including localization methods, model-categorical viewpoints, and foundational equivalence principles.
Related Wikipedia Page
Wikipedia: Homotopy theory
Useful Links
Key Ideas
- homotopy equivalence and deformation
- localization and model structures
- stable and unstable homotopy frameworks
Typical Uses
Used to classify spaces up to deformation and transport algebraic invariants across equivalent models.
Applications
- Algebraic topology foundations
- Higher category theory
- Topological field theory interfaces
References
Recommended Textbooks