18Bxx Special categories
This subtopic studies special categories, focusing on classes of categories with additional axioms or structure that support refined constructions.
Specific topics
18B05 Categories of sets, characterizations
Overview
18B05 studies categories of sets and relations in special categories. It focuses on distinguished classes of categories singled out by extra structure, axioms, or foundational role.
Related Wikipedia Page
Categories of sets and relations (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for categories of sets and relations
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize examples and abstraction principles in topology, algebra, and logic.
Applications
- Special categorical structures
- Topos and enriched-category methods
- Exactness and internal logic
References
Recommended Textbooks
18B10 Categories of relations, spans, and cospans
Overview
18B10 studies topological categories and concrete categories in special categories. It focuses on distinguished classes of categories singled out by extra structure, axioms, or foundational role.
Related Wikipedia Page
Topological categories and concrete categories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for topological categories and concrete 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 organize examples and abstraction principles in topology, algebra, and logic.
Applications
- Special categorical structures
- Topos and enriched-category methods
- Exactness and internal logic
References
Recommended Textbooks
18B15 Embedding theorems
Overview
18B15 studies categories of algebraic structures in special categories. It focuses on distinguished classes of categories singled out by extra structure, axioms, or foundational role.
Related Wikipedia Page
Categories of algebraic structures (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for categories of algebraic 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 organize examples and abstraction principles in topology, algebra, and logic.
Applications
- Special categorical structures
- Topos and enriched-category methods
- Exactness and internal logic
References
Recommended Textbooks
18B20 Categories of machines, automata
Overview
18B20 studies enriched and internal categories in special categories. It focuses on distinguished classes of categories singled out by extra structure, axioms, or foundational role.
Related Wikipedia Page
Enriched and internal categories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for enriched and internal 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 organize examples and abstraction principles in topology, algebra, and logic.
Applications
- Special categorical structures
- Topos and enriched-category methods
- Exactness and internal logic
References
Recommended Textbooks
18B25 Topoi
Overview
18B25 studies exact and regular categories in special categories. It focuses on distinguished classes of categories singled out by extra structure, axioms, or foundational role.
Related Wikipedia Page
Exact and regular categories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for exact and regular 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 organize examples and abstraction principles in topology, algebra, and logic.
Applications
- Special categorical structures
- Topos and enriched-category methods
- Exactness and internal logic
References
Recommended Textbooks
18B30 Categories of topological spaces
Overview
18B30 studies cartesian closed and locally cartesian closed categories in special categories. It focuses on distinguished classes of categories singled out by extra structure, axioms, or foundational role.
Related Wikipedia Page
Cartesian closed and locally cartesian closed categories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for cartesian closed and locally cartesian closed 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 organize examples and abstraction principles in topology, algebra, and logic.
Applications
- Special categorical structures
- Topos and enriched-category methods
- Exactness and internal logic
References
Recommended Textbooks
18B35 Preorders, orders, domains and lattices as categories
Overview
18B35 studies monoidal and braided categories in special categories. It focuses on distinguished classes of categories singled out by extra structure, axioms, or foundational role.
Related Wikipedia Page
Monoidal and braided categories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for monoidal and braided 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 organize examples and abstraction principles in topology, algebra, and logic.
Applications
- Special categorical structures
- Topos and enriched-category methods
- Exactness and internal logic
References
Recommended Textbooks
18B40 Groupoids, semigroupoids, semigroups
Overview
18B40 studies triangulated and derived categories as special categories in special categories. It focuses on distinguished classes of categories singled out by extra structure, axioms, or foundational role.
Related Wikipedia Page
Triangulated and derived categories as special categories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for triangulated and derived categories as special 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 organize examples and abstraction principles in topology, algebra, and logic.
Applications
- Special categorical structures
- Topos and enriched-category methods
- Exactness and internal logic
References
Recommended Textbooks
18B45 Graph categories
Overview
18B45 studies categories of sheaves and toposes in special categories. It focuses on distinguished classes of categories singled out by extra structure, axioms, or foundational role.
Related Wikipedia Page
Categories of sheaves and toposes (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for categories of sheaves and toposes
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize examples and abstraction principles in topology, algebra, and logic.
Applications
- Special categorical structures
- Topos and enriched-category methods
- Exactness and internal logic
References
Recommended Textbooks
18B50 Categories of modules
Overview
18B50 studies special categories arising in geometry and logic in special categories. It focuses on distinguished classes of categories singled out by extra structure, axioms, or foundational role.
Related Wikipedia Page
Special categories arising in geometry and logic (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for special categories arising in geometry and logic
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize examples and abstraction principles in topology, algebra, and logic.
Applications
- Special categorical structures
- Topos and enriched-category methods
- Exactness and internal logic
References
Recommended Textbooks
18B99 None of the above
Overview
18B99 studies none of the above in special categories. It focuses on distinguished classes of categories singled out by extra structure, axioms, or foundational role.
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 to organize examples and abstraction principles in topology, algebra, and logic.
Applications
- Special categorical structures
- Topos and enriched-category methods
- Exactness and internal logic
References
Recommended Textbooks