18Cxx Categories and algebraic theories
This subtopic studies categories and algebraic theories, linking categorical language with universal algebra, syntax, and abstract model construction.
Specific topics
18C05 Equational categories
Overview
18C05 studies universal algebra and categories in categories and algebraic theories. It connects universal algebra with categorical language, showing how operations and equations can be encoded by monads, theories, and sketches.
Related Wikipedia Page
Universal algebra and categories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for universal algebra and 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 algebraic structures via categorical semantics and presentation theory.
Applications
- Lawvere theories and monads
- Categorical universal algebra
- Presentations of algebraic structure
References
Recommended Textbooks
18C10 Theories (e.g. algebraic theories), structure, and semantics
Overview
18C10 studies algebraic theories and monads in categories and algebraic theories. It connects universal algebra with categorical language, showing how operations and equations can be encoded by monads, theories, and sketches.
Related Wikipedia Page
Algebraic theories and monads (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for algebraic theories 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 organize algebraic structures via categorical semantics and presentation theory.
Applications
- Lawvere theories and monads
- Categorical universal algebra
- Presentations of algebraic structure
References
Recommended Textbooks
18C15 Monads and comonads
Overview
18C15 studies lawvere theories in categories and algebraic theories. It connects universal algebra with categorical language, showing how operations and equations can be encoded by monads, theories, and sketches.
Related Wikipedia Page
Lawvere theories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for lawvere theories
- 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 algebraic structures via categorical semantics and presentation theory.
Applications
- Lawvere theories and monads
- Categorical universal algebra
- Presentations of algebraic structure
References
Recommended Textbooks
18C20 Eilenberg-Moore and Kleisli constructions for monads
Overview
18C20 studies varieties and finitary monads in categories and algebraic theories. It connects universal algebra with categorical language, showing how operations and equations can be encoded by monads, theories, and sketches.
Related Wikipedia Page
Varieties and finitary monads (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for varieties and finitary 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 organize algebraic structures via categorical semantics and presentation theory.
Applications
- Lawvere theories and monads
- Categorical universal algebra
- Presentations of algebraic structure
References
Recommended Textbooks
18C30 Sketches and generalizations
Overview
18C30 studies categories of models and algebras in categories and algebraic theories. It connects universal algebra with categorical language, showing how operations and equations can be encoded by monads, theories, and sketches.
Related Wikipedia Page
Categories of models and algebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for categories of models and algebras
- 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 algebraic structures via categorical semantics and presentation theory.
Applications
- Lawvere theories and monads
- Categorical universal algebra
- Presentations of algebraic structure
References
Recommended Textbooks
18C35 Accessible and locally presentable categories
Overview
18C35 studies accessibility and algebraic theories in categories and algebraic theories. It connects universal algebra with categorical language, showing how operations and equations can be encoded by monads, theories, and sketches.
Related Wikipedia Page
Accessibility and algebraic theories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for accessibility and algebraic theories
- 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 algebraic structures via categorical semantics and presentation theory.
Applications
- Lawvere theories and monads
- Categorical universal algebra
- Presentations of algebraic structure
References
Recommended Textbooks
18C40 Structured objects in a category
Overview
18C40 studies sketches and algebraic presentations in categories and algebraic theories. It connects universal algebra with categorical language, showing how operations and equations can be encoded by monads, theories, and sketches.
Related Wikipedia Page
Sketches and algebraic presentations (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for sketches and algebraic presentations
- 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 algebraic structures via categorical semantics and presentation theory.
Applications
- Lawvere theories and monads
- Categorical universal algebra
- Presentations of algebraic structure
References
Recommended Textbooks
18C50 Categorical semantics of formal languages
Overview
18C50 studies categorical algebra and universal algebra in categories and algebraic theories. It connects universal algebra with categorical language, showing how operations and equations can be encoded by monads, theories, and sketches.
Related Wikipedia Page
Categorical algebra and universal algebra (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for categorical algebra and universal algebra
- 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 algebraic structures via categorical semantics and presentation theory.
Applications
- Lawvere theories and monads
- Categorical universal algebra
- Presentations of algebraic structure
References
Recommended Textbooks