Category Theory

Core idea

This section covers the foundations of category theory, providing a unified language for describing mathematical structures and the relationships between them.

Basic Structures

Categories and Morphisms

• Category
• Object
• Morphism
• Composition
• Identity morphism

Types of Morphisms

• Isomorphism
• Monomorphism
• Epimorphism
• Endomorphism
• Automorphism

Subcategories and Duality

• Subcategory
• Full subcategory
• Opposite category

Functors and Natural Transformations

• Functor
• Contravariant functor
• Natural transformation
• Natural isomorphism
• Equivalence of categories

Limits and Colimits

Universal Constructions

• Limit
• Colimit

Products and Coproducts

• Product
• Coproduct

Equalizers and Pullbacks

• Equalizer
• Coequalizer
• Pullback
• Pushout

Representability and Yoneda

• Representable functor
• Yoneda embedding

Adjunctions

• Adjoint functors
• Unit of an adjunction
• Counit of an adjunction

Additive and Abelian Categories

Additive Structure

• Additive category
• Kernel
• Cokernel

Abelian Categories

• Abelian category
• Exact sequence

Exactness of Functors

• Left exact functor
• Right exact functor
• Exact functor

Axioms

• Category axioms
• Abelian category axioms

Lemmas

• Yoneda lemma

---

Triangles and shifts

• Preadditive category
• Initial object
• Zero object
• Shift functor
• Triangle in a shifted category
• Distinguished triangle
• Pretriangulated category
• Octahedral axiom
• Triangulated category