🔶Intro to Abstract Math Unit 11 – Category Theory Fundamentals

What's Category Theory?

• Mathematical framework studies abstract structures and relationships between them
• Provides a unified language to describe mathematical concepts across various branches (algebra, topology, geometry)
• Focuses on objects, morphisms (arrows) between objects, and composition of morphisms
• Emphasizes the relationships and transformations between mathematical structures rather than their internal details
• Originated in algebraic topology in the 1940s by Samuel Eilenberg and Saunders Mac Lane
• Has found applications in computer science, physics, and other fields beyond pure mathematics
• Allows for the generalization and abstraction of mathematical concepts, revealing deep connections and analogies

Key Concepts and Definitions

• Category: consists of objects, morphisms between objects, and a composition operation on morphisms satisfying associativity and identity laws
• Object: abstract entities within a category with no specified internal structure
• Morphism (arrow): represents a structure-preserving map or transformation between objects in a category
• Composition: operation that combines two compatible morphisms to create a new morphism, satisfying associativity
  • Associativity: $(f \circ g) \circ h = f \circ (g \circ h)$ for morphisms $f: A \to B$, $g: B \to C$, and $h: C \to D$
• Identity morphism: special morphism $1_A: A \to A$ for each object $A$ in a category, satisfying $1_A \circ f = f$ and $f \circ 1_A = f$ for any morphism $f: A \to B$
• Isomorphism: morphism $f: A \to B$ with an inverse morphism $g: B \to A$ such that $f \circ g = 1_B$ and $g \circ f = 1_A$
• Commutative diagram: visual representation of objects and morphisms in a category, where paths between objects compose to give the same morphism

Basic Building Blocks

• Objects: serve as the nodes or vertices in a category, representing abstract mathematical structures
• Morphisms: serve as the edges or arrows between objects, representing structure-preserving maps or transformations
• Composition: fundamental operation in a category, allowing the combination of compatible morphisms
  • Compatibility: morphisms $f: A \to B$ and $g: B \to C$ can be composed to form $g \circ f: A \to C$
• Identity morphisms: special morphisms that act as the identity element for composition, leaving other morphisms unchanged when composed
• Commutative diagrams: visual tool for reasoning about the relationships between objects and morphisms in a category
  • Commutativity: paths between objects in a diagram compose to give the same morphism
• Examples of categories:
  • Set: objects are sets, morphisms are functions between sets
  • Grp: objects are groups, morphisms are group homomorphisms
  • Top: objects are topological spaces, morphisms are continuous functions

Functors and Natural Transformations

• Functor: structure-preserving map between categories, consisting of:
  • Object map: assigns to each object in the source category an object in the target category
  • Morphism map: assigns to each morphism in the source category a morphism in the target category
  • Preserves composition and identity morphisms
• Covariant functor: preserves the direction of morphisms ($F(f: A \to B) = F(f): F(A) \to F(B)$)
• Contravariant functor: reverses the direction of morphisms ($F(f: A \to B) = F(f): F(B) \to F(A)$)
• Natural transformation: morphism between functors, providing a way to compare and relate different functors
  • Component: morphism $\alpha_A: F(A) \to G(A)$ in the target category for each object $A$ in the source category
  • Naturality condition: $\alpha_B \circ F(f) = G(f) \circ \alpha_A$ for every morphism $f: A \to B$ in the source category
• Examples of functors:
  • Forgetful functor: maps algebraic structures to their underlying sets (Grp → Set)
  • Fundamental group functor: maps topological spaces to their fundamental groups (Top → Grp)

Universal Constructions

• Universal property: characterizes an object in a category by its relationships with other objects via morphisms
• Initial object: object $I$ with exactly one morphism $I \to A$ for every object $A$ in the category
• Terminal object: object $T$ with exactly one morphism $A \to T$ for every object $A$ in the category
• Product: object $A \times B$ with morphisms (projections) $\pi_1: A \times B \to A$ and $\pi_2: A \times B \to B$, satisfying a universal property
  • Universal property of product: for any object $C$ with morphisms $f: C \to A$ and $g: C \to B$, there exists a unique morphism $h: C \to A \times B$ such that $\pi_1 \circ h = f$ and $\pi_2 \circ h = g$
• Coproduct (sum): dual concept to product, object $A \sqcup B$ with morphisms (injections) $i_1: A \to A \sqcup B$ and $i_2: B \to A \sqcup B$, satisfying a universal property
• Pullback (fiber product): generalizes the concept of inverse image or preimage in various settings
• Pushout: dual concept to pullback, generalizing the concept of gluing or amalgamation

Limits and Colimits

• Limit: universal construction that generalizes the concept of products, pullbacks, and other "universal objects"
  • Cone: consists of an object (vertex) and a family of morphisms (projections) to the objects in a diagram, commuting with the morphisms in the diagram
  • Limit cone: terminal object in the category of cones over a given diagram
• Colimit: dual concept to limit, generalizing the concept of coproducts, pushouts, and other "co-universal objects"
  • Cocone: consists of an object (vertex) and a family of morphisms (injections) from the objects in a diagram, commuting with the morphisms in the diagram
  • Colimit cocone: initial object in the category of cocones over a given diagram
• Examples of limits and colimits:
  • Product is the limit of a discrete diagram (diagram with no morphisms between objects)
  • Coproduct is the colimit of a discrete diagram
  • Pullback is the limit of a diagram with two morphisms $f: A \to C$ and $g: B \to C$
  • Pushout is the colimit of a diagram with two morphisms $f: C \to A$ and $g: C \to B$
• Limits and colimits provide a unified framework for constructing and reasoning about universal objects in a category

Applications in Math and Beyond

• Algebraic topology: functors and natural transformations used to study topological spaces and their algebraic invariants
  • Homology and cohomology theories can be formulated as functors
  • Eilenberg-Steenrod axioms characterize homology theories using category-theoretic language
• Algebraic geometry: schemes and sheaves can be studied using category-theoretic tools
  • Grothendieck's approach to algebraic geometry heavily relies on categories and functors
• Representation theory: categories used to study the representations of algebraic structures (groups, algebras)
  • Representations can be seen as functors from the category of the algebraic structure to the category of vector spaces
• Logic and foundations: category theory provides an alternative foundation for mathematics
  • Topos theory: generalizes set theory and provides a framework for studying various mathematical concepts
• Computer science: category theory used in the study of programming language semantics, type theory, and database theory
  • Functors and monads used to model computational effects and data structures
  • Categorical semantics of programming languages

Common Pitfalls and Tips

• Focusing too much on the objects and neglecting the morphisms and their composition
  • Remember that the morphisms and their composition are the essential part of a category
• Confusing the direction of morphisms, especially when dealing with contravariant functors
  • Pay attention to the domain and codomain of morphisms and how they change under functors
• Forgetting to check the required properties (associativity, identity, functoriality, naturality)
  • Always verify that the defined structures satisfy the necessary axioms and conditions
• Overcomplicating diagrams and proofs
  • Aim for clarity and simplicity in diagrammatic reasoning and proofs
  • Break down complex diagrams into smaller, more manageable parts
• Not exploring the connections and analogies between different areas of mathematics
  • Category theory is a powerful tool for unifying and relating various mathematical concepts
  • Look for similarities and patterns across different fields and try to express them using categorical language
• Neglecting the importance of examples and counterexamples
  • Use concrete examples to gain intuition and understanding of abstract concepts
  • Construct counterexamples to test conjectures and identify the limits of certain statements