2.1 Basic concepts of category theory

Category theory provides a unifying framework for mathematics, abstracting common structures across different fields. It introduces objects and morphisms, which generalize mathematical entities and their relationships, allowing for powerful analysis and comparison of diverse mathematical concepts.

The fundamental building blocks of categories—objects, morphisms, composition, and identity—form the basis for more complex structures. These elements enable us to study mathematical relationships in a abstract, yet rigorous way, revealing deep connections between seemingly unrelated areas of mathematics.

Category Fundamentals

Objects and Morphisms

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• Category comprises collection of objects and morphisms (arrows) between objects, satisfying specific axioms
• Objects represent abstract mathematical structures (sets, groups, topological spaces)
• Morphisms act as structure-preserving maps between objects, generalizing functions between sets
• Identity morphism id_A: A → A exists for every object A, serving as identity element under composition
• Composition of morphisms combines two compatible morphisms to create new morphism, satisfying associativity
• Denote composition of morphisms f: A → B and g: B → C as g ∘ f: A → C, applying from right to left
  • Example: In category of sets, composition (g ∘ f)(x) = g(f(x)) for x in domain of f

Axioms and Properties

• Categories must satisfy identity and associativity axioms
• Identity axiom ensures f ∘ id_A = f = id_B ∘ f for any morphism f: A → B
  • Example: In category of groups, identity morphism acts as identity function on group elements
• Associativity axiom requires (h ∘ g) ∘ f = h ∘ (g ∘ f) for composable morphisms f, g, and h
  • Example: In category of topological spaces, associativity holds for composition of continuous functions
• Morphisms generalize various mathematical concepts
  • Functions between sets
  • Group homomorphisms
  • Continuous maps between topological spaces
• Categories provide unified framework for studying diverse mathematical structures

Isomorphism in Categories

Definition and Properties

• Isomorphism in category defined as morphism f: A → B with two-sided inverse g: B → A
• Two-sided inverse satisfies g ∘ f = id_A and f ∘ g = id_B
• Isomorphisms generalize bijective functions in set theory to arbitrary categories
• Objects A and B considered isomorphic if isomorphism exists between them, denoted A ≅ B
• Isomorphisms preserve all categorical properties and structures
  • Example: In category of groups, isomorphic groups have same algebraic structure
• Existence of isomorphism establishes equivalence relation on objects in category
  • Reflexive: A ≅ A (identity morphism)
  • Symmetric: If A ≅ B, then B ≅ A (inverse morphism)
  • Transitive: If A ≅ B and B ≅ C, then A ≅ C (composition of isomorphisms)

Significance and Applications

• Isomorphisms allow treatment of isomorphic objects as essentially same within category
• Play crucial role in defining universal properties
  • Example: Universal property of product uniquely determined up to unique isomorphism
• Represent objects up to unique isomorphism
  • Example: In category of vector spaces, all n-dimensional vector spaces over field K isomorphic to K^n
• In concrete categories, isomorphisms often correspond to structure-preserving bijections
  • Example: In category of sets, isomorphisms are precisely bijective functions
• Isomorphisms used to establish categorical equivalences and adjunctions
  • Example: Galois connection between subgroups and subfields in field theory

Commutative Diagrams for Reasoning

Fundamentals of Commutative Diagrams

• Commutative diagrams graphically represent objects and morphisms in category
• Objects depicted as points, morphisms as arrows between points
• Diagram commutes if all paths between any two objects yield same composite morphism
• Provide powerful visual tool for expressing and proving complex categorical relationships
• Commutativity checked by comparing compositions of morphisms along different paths
  • Example: Square diagram commutes if f ∘ g = h ∘ k for morphisms f, g, h, k
• Essential in defining and working with universal properties
  • Products, coproducts, limits, colimits

Applications and Techniques

• Diagram chasing technique involves manipulating commutative diagrams to establish results
  • Example: Five Lemma in homological algebra proved using diagram chasing
• Express and verify naturality conditions in definition of natural transformations
  • Example: Naturality square for natural transformation between functors
• Used to define and study adjoint functors
  • Example: Unit and counit of adjunction represented as commutative diagrams
• Facilitate proofs in various areas of mathematics
  • Homological algebra
  • Category theory
  • Algebraic topology
• Simplify complex algebraic manipulations by reducing them to diagram chasing
  • Example: Proving associativity of tensor products using commutative hexagon diagram

Categories in Mathematics

Fundamental Categories

• Set: Objects are sets, morphisms are functions between sets
  • Example: Morphism from ℕ to ℝ could be function mapping natural numbers to their square roots
• Grp: Objects are groups, morphisms are group homomorphisms
  • Example: Homomorphism from (ℤ, +) to (ℝ*, ×) mapping n to e^n
• Top: Objects are topological spaces, morphisms are continuous functions
  • Example: Inclusion map from rationals ℚ to reals ℝ with standard topology

Advanced Categories

• Vect_K: Objects are vector spaces over field K, morphisms are linear transformations
  • Example: Linear transformation from ℝ² to ℝ³ represented by 3×2 matrix
• Ring: Objects are rings, morphisms are ring homomorphisms
  • Example: Canonical homomorphism from integers ℤ to integers modulo n, ℤ/nℤ
• Pos: Objects are partially ordered sets, morphisms are order-preserving functions
  • Example: Inclusion map between power sets P(A) and P(B) for A ⊆ B
• Categories constructed from other mathematical structures
  • Category of modules over ring
  • Category of schemes in algebraic geometry
  • Example: Category of R-modules for commutative ring R, generalizing vector spaces