18Mxx Monoidal and enriched categories
This subtopic studies monoidal and enriched categories, emphasizing tensor-like structures and enriched hom-objects for advanced categorical modeling.
Specific topics
18M05 Monoidal categories, symmetric monoidal categories
Overview
18M05 studies monoidal categories and tensor products in monoidal and enriched categories. It studies categories equipped with tensor products or enriched hom-objects, providing the algebraic language for categorical symmetry and multilinear constructions.
Related Wikipedia Page
Monoidal categories and tensor products (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for monoidal categories and tensor products
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in tensor categories, operads, quantum algebra, and enriched semantics.
Applications
- Tensor and enriched category theory
- Braided and compact closed structures
- Applications to quantum algebra and topology
References
Recommended Textbooks
18M10 Closed categories
Overview
18M10 studies closed and compact closed categories in monoidal and enriched categories. It studies categories equipped with tensor products or enriched hom-objects, providing the algebraic language for categorical symmetry and multilinear constructions.
Related Wikipedia Page
Closed and compact closed categories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for closed and compact 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 in tensor categories, operads, quantum algebra, and enriched semantics.
Applications
- Tensor and enriched category theory
- Braided and compact closed structures
- Applications to quantum algebra and topology
References
Recommended Textbooks
18M15 Enriched categories
Overview
18M15 studies braided and symmetric monoidal categories in monoidal and enriched categories. It studies categories equipped with tensor products or enriched hom-objects, providing the algebraic language for categorical symmetry and multilinear constructions.
Related Wikipedia Page
Braided and symmetric monoidal categories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for braided 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 in tensor categories, operads, quantum algebra, and enriched semantics.
Applications
- Tensor and enriched category theory
- Braided and compact closed structures
- Applications to quantum algebra and topology
References
Recommended Textbooks
18M20 Traced monoidal categories
Overview
18M20 studies enriched categories in monoidal and enriched categories. It studies categories equipped with tensor products or enriched hom-objects, providing the algebraic language for categorical symmetry and multilinear constructions.
Related Wikipedia Page
Enriched categories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for enriched 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 tensor categories, operads, quantum algebra, and enriched semantics.
Applications
- Tensor and enriched category theory
- Braided and compact closed structures
- Applications to quantum algebra and topology
References
Recommended Textbooks
18M25 Compact and autonomous categories, duals
Overview
18M25 studies module categories and actegories in monoidal and enriched categories. It studies categories equipped with tensor products or enriched hom-objects, providing the algebraic language for categorical symmetry and multilinear constructions.
Related Wikipedia Page
Module categories and actegories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for module categories and actegories
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in tensor categories, operads, quantum algebra, and enriched semantics.
Applications
- Tensor and enriched category theory
- Braided and compact closed structures
- Applications to quantum algebra and topology
References
Recommended Textbooks
18M30 String diagram methods
Overview
18M30 studies higher monoidal structures in monoidal and enriched categories. It studies categories equipped with tensor products or enriched hom-objects, providing the algebraic language for categorical symmetry and multilinear constructions.
Related Wikipedia Page
Higher monoidal structures (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for higher monoidal structures
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in tensor categories, operads, quantum algebra, and enriched semantics.
Applications
- Tensor and enriched category theory
- Braided and compact closed structures
- Applications to quantum algebra and topology
References
Recommended Textbooks
18M35 Applied categorical structures
Overview
18M35 studies tensor categories in monoidal and enriched categories. It studies categories equipped with tensor products or enriched hom-objects, providing the algebraic language for categorical symmetry and multilinear constructions.
Related Wikipedia Page
Tensor categories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for tensor 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 tensor categories, operads, quantum algebra, and enriched semantics.
Applications
- Tensor and enriched category theory
- Braided and compact closed structures
- Applications to quantum algebra and topology
References
Recommended Textbooks
18M40 Categorical probability and stochastic processes
Overview
18M40 studies operads and multicategories in monoidal and enriched categories. It studies categories equipped with tensor products or enriched hom-objects, providing the algebraic language for categorical symmetry and multilinear constructions.
Related Wikipedia Page
Operads and multicategories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for operads and multicategories
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in tensor categories, operads, quantum algebra, and enriched semantics.
Applications
- Tensor and enriched category theory
- Braided and compact closed structures
- Applications to quantum algebra and topology
References
Recommended Textbooks
18M60 Categories in quantum mechanics
Overview
18M60 studies applications of monoidal categories in monoidal and enriched categories. It studies categories equipped with tensor products or enriched hom-objects, providing the algebraic language for categorical symmetry and multilinear constructions.
Related Wikipedia Page
Applications of monoidal categories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for applications of 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 in tensor categories, operads, quantum algebra, and enriched semantics.
Applications
- Tensor and enriched category theory
- Braided and compact closed structures
- Applications to quantum algebra and topology
References
Recommended Textbooks
18M65 Categories in computer science
Overview
18M65 studies applications of enriched categories in monoidal and enriched categories. It studies categories equipped with tensor products or enriched hom-objects, providing the algebraic language for categorical symmetry and multilinear constructions.
Related Wikipedia Page
Applications of enriched categories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for applications of enriched 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 tensor categories, operads, quantum algebra, and enriched semantics.
Applications
- Tensor and enriched category theory
- Braided and compact closed structures
- Applications to quantum algebra and topology
References
Recommended Textbooks
18M70 Universal algebra using categories
Overview
18M70 studies monoidal methods in topology and algebra in monoidal and enriched categories. It studies categories equipped with tensor products or enriched hom-objects, providing the algebraic language for categorical symmetry and multilinear constructions.
Related Wikipedia Page
Monoidal methods in topology and algebra (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for monoidal methods in topology and 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 tensor categories, operads, quantum algebra, and enriched semantics.
Applications
- Tensor and enriched category theory
- Braided and compact closed structures
- Applications to quantum algebra and topology
References
Recommended Textbooks
18M75 Topological field theories
Overview
18M75 studies categorical quantum structures in monoidal and enriched categories. It studies categories equipped with tensor products or enriched hom-objects, providing the algebraic language for categorical symmetry and multilinear constructions.
Related Wikipedia Page
Categorical quantum structures (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for categorical quantum structures
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in tensor categories, operads, quantum algebra, and enriched semantics.
Applications
- Tensor and enriched category theory
- Braided and compact closed structures
- Applications to quantum algebra and topology
References
Recommended Textbooks
18M80 Categorical aspects of topological quantum computing
Overview
18M80 studies computational aspects of monoidal categories in monoidal and enriched categories. It studies categories equipped with tensor products or enriched hom-objects, providing the algebraic language for categorical symmetry and multilinear constructions.
Related Wikipedia Page
Computational aspects of monoidal categories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for computational aspects of 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 in tensor categories, operads, quantum algebra, and enriched semantics.
Applications
- Tensor and enriched category theory
- Braided and compact closed structures
- Applications to quantum algebra and topology
References
Recommended Textbooks
18M85 Categorical aspects of quantum groups
Overview
18M85 studies miscellaneous monoidal and enriched categories in monoidal and enriched categories. It studies categories equipped with tensor products or enriched hom-objects, providing the algebraic language for categorical symmetry and multilinear constructions.
Related Wikipedia Page
Miscellaneous monoidal and enriched categories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for miscellaneous monoidal and enriched 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 tensor categories, operads, quantum algebra, and enriched semantics.
Applications
- Tensor and enriched category theory
- Braided and compact closed structures
- Applications to quantum algebra and topology
References
Recommended Textbooks