18Axx General category theory
This subtopic studies general category theory, covering foundational notions such as functors, natural transformations, limits, and adjunctions.
Specific topics
18A05 Definitions and generalizations in category theory
Overview
18A05 studies foundations of categories, functors, natural transformations in general category theory. It provides the basic language of objects, arrows, universal properties, and functorial constructions used throughout modern mathematics.
Related Wikipedia Page
Foundations of categories, functors, natural transformations (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for foundations of categories, functors, natural transformations
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to compare mathematical structures through morphisms, limits, adjunctions, and formalized universal behavior.
Applications
- Foundational category language
- Universal properties and adjunctions
- Functorial and monadic constructions
References
Recommended Textbooks
18A10 Graphs, diagram schemes, precategories
Overview
18A10 studies categories of sets and classes in general category theory. It provides the basic language of objects, arrows, universal properties, and functorial constructions used throughout modern mathematics.
Related Wikipedia Page
Categories of sets and classes (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for categories of sets and classes
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to compare mathematical structures through morphisms, limits, adjunctions, and formalized universal behavior.
Applications
- Foundational category language
- Universal properties and adjunctions
- Functorial and monadic constructions
References
Recommended Textbooks
18A15 Functor categories, comma categories
Overview
18A15 studies universal properties and limits in general category theory. It provides the basic language of objects, arrows, universal properties, and functorial constructions used throughout modern mathematics.
Related Wikipedia Page
Universal properties and limits (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for universal properties and limits
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to compare mathematical structures through morphisms, limits, adjunctions, and formalized universal behavior.
Applications
- Foundational category language
- Universal properties and adjunctions
- Functorial and monadic constructions
References
Recommended Textbooks
18A20 Epimorphisms, monomorphisms, special classes of morphisms
Overview
18A20 studies adjoints, equivalences, and monads in general category theory. It provides the basic language of objects, arrows, universal properties, and functorial constructions used throughout modern mathematics.
Related Wikipedia Page
Adjoints, equivalences, and monads (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for adjoints, equivalences, and monads
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to compare mathematical structures through morphisms, limits, adjunctions, and formalized universal behavior.
Applications
- Foundational category language
- Universal properties and adjunctions
- Functorial and monadic constructions
References
Recommended Textbooks
18A22 Special properties of functors
Overview
18A22 studies kan extensions and related constructions in general category theory. It provides the basic language of objects, arrows, universal properties, and functorial constructions used throughout modern mathematics.
Related Wikipedia Page
Kan extensions and related constructions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for kan extensions and related constructions
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to compare mathematical structures through morphisms, limits, adjunctions, and formalized universal behavior.
Applications
- Foundational category language
- Universal properties and adjunctions
- Functorial and monadic constructions
References
Recommended Textbooks
18A23 Natural morphisms, dinatural morphisms
Overview
18A23 studies presentations of categories in general category theory. It provides the basic language of objects, arrows, universal properties, and functorial constructions used throughout modern mathematics.
Related Wikipedia Page
Presentations of categories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for presentations of categories
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to compare mathematical structures through morphisms, limits, adjunctions, and formalized universal behavior.
Applications
- Foundational category language
- Universal properties and adjunctions
- Functorial and monadic constructions
References
Recommended Textbooks
18A25 Functor categories, comma categories (general)
Overview
18A25 studies size issues and foundations in general category theory. It provides the basic language of objects, arrows, universal properties, and functorial constructions used throughout modern mathematics.
Related Wikipedia Page
Size issues and foundations (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for size issues and foundations
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to compare mathematical structures through morphisms, limits, adjunctions, and formalized universal behavior.
Applications
- Foundational category language
- Universal properties and adjunctions
- Functorial and monadic constructions
References
Recommended Textbooks
18A30 Limits and colimits, universal constructions
Overview
18A30 studies enriched categories and modifications in general category theory. It provides the basic language of objects, arrows, universal properties, and functorial constructions used throughout modern mathematics.
Related Wikipedia Page
Enriched categories and modifications (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for enriched categories and modifications
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to compare mathematical structures through morphisms, limits, adjunctions, and formalized universal behavior.
Applications
- Foundational category language
- Universal properties and adjunctions
- Functorial and monadic constructions
References
Recommended Textbooks
18A35 Categories admitting limits, distributive laws
Overview
18A35 studies higher-categorical foundations in general category theory. It provides the basic language of objects, arrows, universal properties, and functorial constructions used throughout modern mathematics.
Related Wikipedia Page
Higher-categorical foundations (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for higher-categorical foundations
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to compare mathematical structures through morphisms, limits, adjunctions, and formalized universal behavior.
Applications
- Foundational category language
- Universal properties and adjunctions
- Functorial and monadic constructions
References
Recommended Textbooks
18A40 Adjoint functors, triples, monads
Overview
18A40 studies miscellaneous category-theoretic foundations in general category theory. It provides the basic language of objects, arrows, universal properties, and functorial constructions used throughout modern mathematics.
Related Wikipedia Page
Miscellaneous category-theoretic foundations (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for miscellaneous category-theoretic foundations
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to compare mathematical structures through morphisms, limits, adjunctions, and formalized universal behavior.
Applications
- Foundational category language
- Universal properties and adjunctions
- Functorial and monadic constructions
References
Recommended Textbooks