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Category theory, a branch of mathematics that abstractly studies structures and systems, offers a unifying framework for understanding and formalizing relationships and transformations between mathematical structures. Here's a comprehensive list of various branches and topics within category theory:

1. Basic Concepts of Category Theory¶

Categories, Objects, and Morphisms: Fundamental elements of category theory where categories encapsulate mathematical structures, objects are the elements within these structures, and morphisms are the arrows representing functions between objects.
Functors and Natural Transformations: Functors map objects and morphisms from one category to another, while natural transformations are mappings between functors maintaining structural consistency.
Isomorphisms, Monomorphisms, and Epimorphisms: Isomorphisms are morphisms that can be reversed, monomorphisms are injective morphisms, and epimorphisms are surjective morphisms, each providing different ways of relating objects in categories.
Limits and Colimits: Concepts generalizing mathematical constructions like products and unions (limits) and sums and intersections (colimits) in category theory.
Universal Properties: Properties defining an object in terms of its relationships with other objects, often through unique morphisms.

2. Types of Categories¶

Small and Large Categories: Small categories have a set of morphisms, while large categories have a class, differentiating the scope of these categories.
Concrete Categories: Categories equipped with a faithful functor to the category of sets, grounding abstract category theory concepts in set theory.
Abelian Categories: Categories where morphisms form abelian groups, allowing the definition and study of concepts like exact sequences.
Topoi (Topos Theory): Categories resembling the category of sets, providing a framework for generalized set theory and logic.
Monoidal Categories: Categories equipped with a tensor product and a unit object, enabling the study of structures that can be 'multiplied.'
Symmetric and Braided Monoidal Categories: Extensions of monoidal categories, where objects and morphisms maintain symmetrical or braided relationships under tensor product.

3. Functorial Concepts¶

Covariant and Contravariant Functors: Covariant functors preserve the direction of morphisms between categories, whereas contravariant functors reverse them.
Representable Functors: Functors that are naturally isomorphic to a Hom functor, linking objects of a category to sets.
Equivalence of Categories: A pair of functors between two categories that are inverses of each other, up to natural isomorphism.
Adjoint Functors: Pairs of functors connecting two categories, where one functor is left adjoint and the other is right adjoint, formalizing a general notion of 'inverse operations.'
Yoneda Lemma: A fundamental result providing a way to represent objects in a category in terms of all morphisms into them.

4. Homological Algebra¶

Chain Complexes: Sequences of abelian groups connected by homomorphisms, fundamental in studying algebraic topology and homology.
Exact Sequences: Sequences of groups and homomorphisms where the image of one homomorphism equals the kernel of the next, critical in understanding algebraic structures.
Homology and Cohomology: Methods for assigning algebraic invariants to topological spaces, capturing their properties and relationships.
Derived Functors: Functors applied to objects in homological algebra, providing deeper insights into algebraic structures.
Ext and Tor Functors: Derived functors used to study extensions of modules and torsion in modules, respectively.

5. Higher Category Theory¶

n-Categories: Categories with morphisms between morphisms, extending up to 'n' levels, enriching the theory with higher-dimensional arrows.
Infinity-Categories: Categories extended to have morphisms between all levels, providing a framework for studying structures in infinite dimensions.
Higher-Dimensional Algebra: Algebraic structures within higher category theory, exploring relationships in dimensions beyond the classical scope.
Grothendieck's Pursuing Stacks: Grothendieck's work on applying higher category theory to homotopy and algebraic geometry.

6. Algebraic Topology¶

Functorial Treatment of Homotopy Theory: Applying category theory to study continuous transformations of topological spaces.
Spectral Sequences: Tools in algebraic topology for computing homology and cohomology groups by breaking down complex problems into simpler stages.
Model Categories: Categories equipped with structures enabling the study of homotopy theory abstractly.
Categorical Framework for Cohomology Theories: A category-theoretic approach to understanding and unifying various cohomology theories.

7. Algebraic Geometry¶

Schemes and Sheaves: Schemes extend the concept of algebraic varieties, and sheaves provide a way to study local properties globally.
Stacks: Generalizations of schemes, enabling the study of geometric objects with extra structure like group actions.
Derived Algebraic Geometry: An approach to algebraic geometry using derived categories, enabling the study of geometric objects and their deformations.

. Categorical Logic - Cartesian Closed Categories: Categories in which any two objects have a product and exponential, important for modeling logical operations. - Internal Logic of Categories: A study of logic within the framework of a category, where propositions and proofs are objects and morphisms. - Topos as a Model of Set Theory: Topoi (or topoi) provide a generalized context for set theory, accommodating various logical and set-theoretical concepts. - Categorical Semantics of Type Theory: Using category theory to provide semantics for different types of mathematical logic and computer science type theories.

9. Enriched Category Theory¶

Categories Enriched over a Monoidal Category: Categories whose hom-sets carry additional structure, typically from a monoidal category.
Enriched Hom-sets: Hom-sets in an enriched category having structure beyond mere sets, such as topological spaces or groups.
Enriched Functors and Natural Transformations: Generalizations of functors and natural transformations in the context of enriched category theory.

10. Quantum Algebra and Category Theory¶

Monoidal Categories in Quantum Field Theory: Applying monoidal category theory to study the algebraic and categorical structures in quantum field theory.
Categorical Quantum Mechanics: A category-theoretic framework for understanding and formulating quantum mechanics.
Fusion Categories: Categories used in quantum algebra to study the properties of quantum symmetries and topological quantum field theories.

11. Computer Science Applications¶

Categories in Programming Language Semantics: Applying category theory to understand and formalize programming language constructs and their semantics.
Functorial Semantics of Computational Effects: Using functors to model and reason about computational effects like state, exceptions, and input/output.
Categorical Databases: Applying category theory principles to the design and implementation of databases.

12. Categorical Algebra¶

Groupoids and Groupoid Objects: Categories where all morphisms are isomorphisms, generalizing the concept of groups.
Categorical Groups and Group Actions: Groups and their actions viewed in the context of category theory, providing a more abstract perspective.
Tannaka Duality: A duality theory relating algebraic groups to certain categories, revealing deep connections between algebra and category theory.

13. Sheaf Theory and Grothendieck Topologies¶

Sheaves on a Topological Space: Tools for systematically tracking locally defined data across a space.
Sites and Grothendieck Topologies: Generalizations of topological spaces in category theory, enabling the definition of sheaves on more abstract structures.
Sheaf Cohomology: A method in algebraic topology for computing invariants of topological spaces, particularly in algebraic geometry.

14. Diagrams and Sketches¶

Diagram Categories: Categories formed from diagrams, which are collections of objects and morphisms, structuring complex mathematical relationships.
Limits and Colimits in Diagram Categories: Studying limits and colimits in the context of diagram categories to understand universal constructions.
Sketches and Theories: Frameworks in category theory for formalizing mathematical theories and structures.

15. Operads and Monads¶

Operad Theory: Study of operads, which are algebraic structures that model operations with multiple inputs.
Monads in Category Theory: Structures representing a type of endofunctor, central to modeling state, effects, and transformations in categories.
Algebras over a Monad: Algebraic structures defined over a monad, providing a unifying framework for various mathematical constructs.

16. Categorical Ring Theory¶

Additive Categories: Categories where morphisms form an abelian group, allowing the extension of linear algebra concepts to more abstract settings.
Module Categories: Categories whose objects behave like modules over a ring, providing a categorical approach to module theory.
Triangulated Categories: Categories equipped with an additional structure that allows for the study of algebraic structures derived from complex geometrical shapes.

17. Categorical Methods in Homotopy Theory¶

Simplicial Objects in Categories: Use of simplicial methods in category theory to study shapes and spaces through simplices and their relations.
Quillen Model Categories: A framework for abstract homotopy theory, allowing the study of homotopy relations in various mathematical contexts.
Homotopical Algebra: The application of homotopy theoretical methods to algebra, particularly in studying complex algebraic structures.

Category theory is known for its high level of abstraction, but it provides powerful tools and perspectives for understanding and connecting various areas of mathematics. It is particularly influential in fields like algebraic topology, algebraic geometry, and homological algebra, and its concepts are increasingly applied in computer science, physics, and logic.

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