18Dxx Categorical structures
This subtopic studies categorical structures, including internal objects, fibrations, and enriched constructions that organize mathematics at a structural level.
Specific topics
18D05 Double categories, $2$-categories, bicategories
Overview
18D05 studies functor categories and diagram categories in categorical structures. It studies the internal structural features of categories themselves, including limits, adjunctions, enrichment, and higher-dimensional coherence.
Related Wikipedia Page
Functor categories and diagram categories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for functor categories and diagram 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 formalize and compare categorical constructions across algebra, topology, and logic.
Applications
- Limits and adjunctions
- Enrichment and monoidal structure
- Higher-dimensional category theory
References
Recommended Textbooks
18D10 Monoidal, symmetric monoidal and braided categories
Overview
18D10 studies natural transformations and coherence in categorical structures. It studies the internal structural features of categories themselves, including limits, adjunctions, enrichment, and higher-dimensional coherence.
Related Wikipedia Page
Natural transformations and coherence (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for natural transformations and coherence
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to formalize and compare categorical constructions across algebra, topology, and logic.
Applications
- Limits and adjunctions
- Enrichment and monoidal structure
- Higher-dimensional category theory
References
Recommended Textbooks
18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.)
Overview
18D15 studies adjunctions and monads in categorical structures. It studies the internal structural features of categories themselves, including limits, adjunctions, enrichment, and higher-dimensional coherence.
Related Wikipedia Page
Adjunctions and monads (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for adjunctions 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 formalize and compare categorical constructions across algebra, topology, and logic.
Applications
- Limits and adjunctions
- Enrichment and monoidal structure
- Higher-dimensional category theory
References
Recommended Textbooks
18D20 Enriched categories
Overview
18D20 studies limits, colimits, and completeness in categorical structures. It studies the internal structural features of categories themselves, including limits, adjunctions, enrichment, and higher-dimensional coherence.
Related Wikipedia Page
Limits, colimits, and completeness (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for limits, colimits, and completeness
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to formalize and compare categorical constructions across algebra, topology, and logic.
Applications
- Limits and adjunctions
- Enrichment and monoidal structure
- Higher-dimensional category theory
References
Recommended Textbooks
18D25 Strong functors, strong adjunctions
Overview
18D25 studies factorization systems and exactness in categorical structures. It studies the internal structural features of categories themselves, including limits, adjunctions, enrichment, and higher-dimensional coherence.
Related Wikipedia Page
Factorization systems and exactness (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for factorization systems and exactness
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to formalize and compare categorical constructions across algebra, topology, and logic.
Applications
- Limits and adjunctions
- Enrichment and monoidal structure
- Higher-dimensional category theory
References
Recommended Textbooks
18D30 Fibered categories
Overview
18D30 studies enriched and internal categorical structures in categorical structures. It studies the internal structural features of categories themselves, including limits, adjunctions, enrichment, and higher-dimensional coherence.
Related Wikipedia Page
Enriched and internal categorical structures (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for enriched and internal categorical structures
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to formalize and compare categorical constructions across algebra, topology, and logic.
Applications
- Limits and adjunctions
- Enrichment and monoidal structure
- Higher-dimensional category theory
References
Recommended Textbooks
18D35 Structured objects in a $2$-category
Overview
18D35 studies closed categories and tensor structures in categorical structures. It studies the internal structural features of categories themselves, including limits, adjunctions, enrichment, and higher-dimensional coherence.
Related Wikipedia Page
Closed categories and tensor structures (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for closed categories and tensor structures
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to formalize and compare categorical constructions across algebra, topology, and logic.
Applications
- Limits and adjunctions
- Enrichment and monoidal structure
- Higher-dimensional category theory
References
Recommended Textbooks
18D40 Internal categories, diagrams
Overview
18D40 studies monoidal and symmetric monoidal categories in categorical structures. It studies the internal structural features of categories themselves, including limits, adjunctions, enrichment, and higher-dimensional coherence.
Related Wikipedia Page
Monoidal and symmetric monoidal categories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for monoidal and symmetric monoidal 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 formalize and compare categorical constructions across algebra, topology, and logic.
Applications
- Limits and adjunctions
- Enrichment and monoidal structure
- Higher-dimensional category theory
References
Recommended Textbooks
18D50 Operads, multicategories, modular operads
Overview
18D50 studies higher-dimensional categorical structures in categorical structures. It studies the internal structural features of categories themselves, including limits, adjunctions, enrichment, and higher-dimensional coherence.
Related Wikipedia Page
Higher-dimensional categorical structures (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for higher-dimensional categorical structures
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to formalize and compare categorical constructions across algebra, topology, and logic.
Applications
- Limits and adjunctions
- Enrichment and monoidal structure
- Higher-dimensional category theory
References
Recommended Textbooks
18D60 $n$-categories for $n \geq 3$
Overview
18D60 studies 2-categories and bicategories in categorical structures. It studies the internal structural features of categories themselves, including limits, adjunctions, enrichment, and higher-dimensional coherence.
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 to formalize and compare categorical constructions across algebra, topology, and logic.
Applications
- Limits and adjunctions
- Enrichment and monoidal structure
- Higher-dimensional category theory
References
Recommended Textbooks
18D70 Cartesian closed categories, topos-theoretic methods
Overview
18D70 studies miscellaneous categorical structures in categorical structures. It studies the internal structural features of categories themselves, including limits, adjunctions, enrichment, and higher-dimensional coherence.
Related Wikipedia Page
Miscellaneous categorical structures (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for miscellaneous categorical structures
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to formalize and compare categorical constructions across algebra, topology, and logic.
Applications
- Limits and adjunctions
- Enrichment and monoidal structure
- Higher-dimensional category theory
References
Recommended Textbooks