18Nxx Higher categories
This subtopic studies higher categories, where morphisms between morphisms and multi-level coherence conditions refine categorical foundations.
Specific topics
18N10 $2$-categories, bicategories, double categories
Overview
18N10 studies 2-categories and bicategories in higher categories. It generalizes category theory to morphisms between morphisms and beyond, capturing coherence and higher-dimensional algebraic structure.
Related Wikipedia Page
2-categories and bicategories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for 2-categories and bicategories
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in topology, algebraic geometry, and modern homotopy theory to encode higher-order equivalences.
Applications
- Higher-dimensional category theory
- Coherence and strictification
- Applications to homotopy and topology
References
Recommended Textbooks
18N20 Tricategories, weak $3$-categories
Overview
18N20 studies n-categories and weak n-categories in higher categories. It generalizes category theory to morphisms between morphisms and beyond, capturing coherence and higher-dimensional algebraic structure.
Related Wikipedia Page
n-categories and weak n-categories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for n-categories and weak n-categories
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in topology, algebraic geometry, and modern homotopy theory to encode higher-order equivalences.
Applications
- Higher-dimensional category theory
- Coherence and strictification
- Applications to homotopy and topology
References
Recommended Textbooks
18N25 Simplicial categories, quasi-categories
Overview
18N25 studies infinity-categories and quasi-categories in higher categories. It generalizes category theory to morphisms between morphisms and beyond, capturing coherence and higher-dimensional algebraic structure.
Related Wikipedia Page
Infinity-categories and quasi-categories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for infinity-categories and quasi-categories
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in topology, algebraic geometry, and modern homotopy theory to encode higher-order equivalences.
Applications
- Higher-dimensional category theory
- Coherence and strictification
- Applications to homotopy and topology
References
Recommended Textbooks
18N30 $(\infty,1)$-categories
Overview
18N30 studies higher operads and higher monads in higher categories. It generalizes category theory to morphisms between morphisms and beyond, capturing coherence and higher-dimensional algebraic structure.
Related Wikipedia Page
Higher operads and higher monads (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for higher operads and higher monads
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in topology, algebraic geometry, and modern homotopy theory to encode higher-order equivalences.
Applications
- Higher-dimensional category theory
- Coherence and strictification
- Applications to homotopy and topology
References
Recommended Textbooks
18N40 $\infty$-groupoids, homotopy types
Overview
18N40 studies higher-category foundations in higher categories. It generalizes category theory to morphisms between morphisms and beyond, capturing coherence and higher-dimensional algebraic structure.
Related Wikipedia Page
Higher-category foundations (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for higher-category foundations
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in topology, algebraic geometry, and modern homotopy theory to encode higher-order equivalences.
Applications
- Higher-dimensional category theory
- Coherence and strictification
- Applications to homotopy and topology
References
Recommended Textbooks
18N45 $n$-categories for $n \geq 3$
Overview
18N45 studies coherence and strictification results in higher categories. It generalizes category theory to morphisms between morphisms and beyond, capturing coherence and higher-dimensional algebraic structure.
Related Wikipedia Page
Coherence and strictification results (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for coherence and strictification results
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in topology, algebraic geometry, and modern homotopy theory to encode higher-order equivalences.
Applications
- Higher-dimensional category theory
- Coherence and strictification
- Applications to homotopy and topology
References
Recommended Textbooks
18N50 Homotopy type theory, univalent foundations
Overview
18N50 studies higher-categorical algebra in higher categories. It generalizes category theory to morphisms between morphisms and beyond, capturing coherence and higher-dimensional algebraic structure.
Related Wikipedia Page
Higher-categorical algebra (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for higher-categorical algebra
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in topology, algebraic geometry, and modern homotopy theory to encode higher-order equivalences.
Applications
- Higher-dimensional category theory
- Coherence and strictification
- Applications to homotopy and topology
References
Recommended Textbooks
18N55 Globular categories
Overview
18N55 studies higher-topos theoretic methods in higher categories. It generalizes category theory to morphisms between morphisms and beyond, capturing coherence and higher-dimensional algebraic structure.
Related Wikipedia Page
Higher-topos theoretic methods (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for higher-topos theoretic methods
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in topology, algebraic geometry, and modern homotopy theory to encode higher-order equivalences.
Applications
- Higher-dimensional category theory
- Coherence and strictification
- Applications to homotopy and topology
References
Recommended Textbooks
18N60 $\infty$-categories and higher structures
Overview
18N60 studies applications of higher categories in higher categories. It generalizes category theory to morphisms between morphisms and beyond, capturing coherence and higher-dimensional algebraic structure.
Related Wikipedia Page
Applications of higher categories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for applications of higher categories
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in topology, algebraic geometry, and modern homotopy theory to encode higher-order equivalences.
Applications
- Higher-dimensional category theory
- Coherence and strictification
- Applications to homotopy and topology
References
Recommended Textbooks
18N99 None of the above
Overview
18N99 studies none of the above in higher categories. It generalizes category theory to morphisms between morphisms and beyond, capturing coherence and higher-dimensional algebraic structure.
Related Wikipedia Page
None of the above (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for none of the above
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in topology, algebraic geometry, and modern homotopy theory to encode higher-order equivalences.
Applications
- Higher-dimensional category theory
- Coherence and strictification
- Applications to homotopy and topology
References
Recommended Textbooks