18Exx Categories in geometry and topology
This subtopic studies categories in geometry and topology, where categorical methods encode spaces, maps, and invariants in a flexible abstract framework.
Specific topics
18E05 Preabelian categories
Overview
18E05 studies toposes and categorical logic in categories in geometry and topology. It uses categorical language to organize sheaves, toposes, descent, and geometric constructions in topology and geometry.
Related Wikipedia Page
Toposes and categorical logic (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for toposes and categorical 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 encode local-to-global principles and to study geometric objects via categorical methods.
Applications
- Sheaf and topos methods
- Descent and stacks
- Category theory in geometry and topology
References
Recommended Textbooks
18E10 Abelian categories, Grothendieck categories
Overview
18E10 studies sheaves and sheaf cohomology in categories in geometry and topology. It uses categorical language to organize sheaves, toposes, descent, and geometric constructions in topology and geometry.
Related Wikipedia Page
Sheaves and sheaf cohomology (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for sheaves and sheaf cohomology
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to encode local-to-global principles and to study geometric objects via categorical methods.
Applications
- Sheaf and topos methods
- Descent and stacks
- Category theory in geometry and topology
References
Recommended Textbooks
18E13 Protomodular categories, semi-abelian categories, Mal'tsev categories
Overview
18E13 studies descent theory in categories in geometry and topology. It uses categorical language to organize sheaves, toposes, descent, and geometric constructions in topology and geometry.
Related Wikipedia Page
Descent theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for descent theory
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to encode local-to-global principles and to study geometric objects via categorical methods.
Applications
- Sheaf and topos methods
- Descent and stacks
- Category theory in geometry and topology
References
Recommended Textbooks
18E15 Grothendieck categories
Overview
18E15 studies sites and grothendieck topologies in categories in geometry and topology. It uses categorical language to organize sheaves, toposes, descent, and geometric constructions in topology and geometry.
Related Wikipedia Page
Sites and Grothendieck topologies (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for sites and grothendieck topologies
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to encode local-to-global principles and to study geometric objects via categorical methods.
Applications
- Sheaf and topos methods
- Descent and stacks
- Category theory in geometry and topology
References
Recommended Textbooks
18E20 Categorical representation of boolean algebras and lattices
Overview
18E20 studies classifying toposes in categories in geometry and topology. It uses categorical language to organize sheaves, toposes, descent, and geometric constructions in topology and geometry.
Related Wikipedia Page
Classifying toposes (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for classifying 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 encode local-to-global principles and to study geometric objects via categorical methods.
Applications
- Sheaf and topos methods
- Descent and stacks
- Category theory in geometry and topology
References
Recommended Textbooks
18E25 Tensor products, torsion
Overview
18E25 studies categories in algebraic geometry in categories in geometry and topology. It uses categorical language to organize sheaves, toposes, descent, and geometric constructions in topology and geometry.
Related Wikipedia Page
Categories in algebraic geometry (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for categories in algebraic geometry
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to encode local-to-global principles and to study geometric objects via categorical methods.
Applications
- Sheaf and topos methods
- Descent and stacks
- Category theory in geometry and topology
References
Recommended Textbooks
18E30 Derived categories, triangulated categories
Overview
18E30 studies categories in topology and homotopy theory in categories in geometry and topology. It uses categorical language to organize sheaves, toposes, descent, and geometric constructions in topology and geometry.
Related Wikipedia Page
Categories in topology and homotopy theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for categories in topology and homotopy theory
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to encode local-to-global principles and to study geometric objects via categorical methods.
Applications
- Sheaf and topos methods
- Descent and stacks
- Category theory in geometry and topology
References
Recommended Textbooks
18E35 Localization of categories
Overview
18E35 studies derived categories in geometry and topology in categories in geometry and topology. It uses categorical language to organize sheaves, toposes, descent, and geometric constructions in topology and geometry.
Related Wikipedia Page
Derived categories in geometry and topology (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for derived categories in geometry and topology
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to encode local-to-global principles and to study geometric objects via categorical methods.
Applications
- Sheaf and topos methods
- Descent and stacks
- Category theory in geometry and topology
References
Recommended Textbooks
18E40 Torsion theories, radicals
Overview
18E40 studies stacks and higher sheaf theory in categories in geometry and topology. It uses categorical language to organize sheaves, toposes, descent, and geometric constructions in topology and geometry.
Related Wikipedia Page
Stacks and higher sheaf theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for stacks and higher sheaf theory
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to encode local-to-global principles and to study geometric objects via categorical methods.
Applications
- Sheaf and topos methods
- Descent and stacks
- Category theory in geometry and topology
References
Recommended Textbooks
18E45 Monomorphism classes, injective envelopes
Overview
18E45 studies geometric morphisms and continuous functors in categories in geometry and topology. It uses categorical language to organize sheaves, toposes, descent, and geometric constructions in topology and geometry.
Related Wikipedia Page
Geometric morphisms and continuous functors (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for geometric morphisms and continuous functors
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to encode local-to-global principles and to study geometric objects via categorical methods.
Applications
- Sheaf and topos methods
- Descent and stacks
- Category theory in geometry and topology
References
Recommended Textbooks
18E50 Module categories with extra structure
Overview
18E50 studies applications of categories in geometry and topology in categories in geometry and topology. It uses categorical language to organize sheaves, toposes, descent, and geometric constructions in topology and geometry.
Related Wikipedia Page
Applications of categories in geometry and topology (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for applications of categories in geometry and topology
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to encode local-to-global principles and to study geometric objects via categorical methods.
Applications
- Sheaf and topos methods
- Descent and stacks
- Category theory in geometry and topology
References
Recommended Textbooks
18E99 None of the above
Overview
18E99 studies none of the above in categories in geometry and topology. It uses categorical language to organize sheaves, toposes, descent, and geometric constructions in topology and geometry.
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 encode local-to-global principles and to study geometric objects via categorical methods.
Applications
- Sheaf and topos methods
- Descent and stacks
- Category theory in geometry and topology
References
Recommended Textbooks