🔢Category Theory Unit 2 – Categories and Morphisms

Key Concepts and Definitions

• Category theory studies mathematical structures and relationships between them using abstract concepts like objects and morphisms
• Objects represent mathematical entities (sets, groups, vector spaces) while morphisms describe structure-preserving mappings between objects
• Categories consist of a collection of objects and morphisms that satisfy certain axioms (identity morphisms, associativity of composition)
• Isomorphisms are morphisms with two-sided inverses, indicating two objects are essentially the same within the category
• Functors map between categories, preserving the structure of objects and morphisms
  • Covariant functors preserve the direction of morphisms $(F(f): F(A) \to F(B)$ for $f: A \to B)$
  • Contravariant functors reverse the direction of morphisms $(F(f): F(B) \to F(A)$ for $f: A \to B)$
• Natural transformations provide a way to compare functors, establishing relationships between them

Objects and Arrows: The Basics

• In category theory, objects are abstract entities representing mathematical structures without specifying their internal details
• Arrows, also called morphisms, are structure-preserving mappings between objects within a category
• Every object has an identity morphism that maps the object to itself $(id_A: A \to A)$
• Morphisms can be composed if the codomain of one matches the domain of another $(g \circ f: A \to C$ for $f: A \to B, g: B \to C)$
  • Composition is associative: $(h \circ g) \circ f = h \circ (g \circ f)$
• Commutative diagrams visually represent the equality of different compositions of morphisms
• Initial objects have exactly one morphism to every other object in the category
• Terminal objects have exactly one morphism from every other object in the category

Types of Morphisms

• Monomorphisms (monos) are left-cancellable morphisms: $f \circ g_1 = f \circ g_2 \implies g_1 = g_2$ (generalizes injectivity)
• Epimorphisms (epis) are right-cancellable morphisms: $g_1 \circ f = g_2 \circ f \implies g_1 = g_2$ (generalizes surjectivity)
• Isomorphisms are morphisms with two-sided inverses: $f: A \to B$ is an isomorphism if there exists $g: B \to A$ such that $g \circ f = id_A$ and $f \circ g = id_B$
  • Isomorphic objects are considered equivalent within the category
• Endomorphisms are morphisms from an object to itself $(f: A \to A)$
• Automorphisms are isomorphisms from an object to itself (invertible endomorphisms)
• Split monomorphisms have a left inverse $(g \circ f = id_A$ for $f: A \to B, g: B \to A)$
• Split epimorphisms have a right inverse $(f \circ g = id_B$ for $f: A \to B, g: B \to A)$

Properties of Categories

• Categories are defined by the objects and morphisms they contain, along with the axioms they satisfy
• Composition of morphisms must be associative: $(h \circ g) \circ f = h \circ (g \circ f)$
• Every object must have an identity morphism that serves as a neutral element for composition: $f \circ id_A = f = id_B \circ f$
• Some categories have additional properties or structures (products, coproducts, exponentials, limits, colimits)
• Duality principle states that for every categorical concept, there is a dual concept obtained by reversing the direction of morphisms
  • Example: initial and terminal objects, monomorphisms and epimorphisms, limits and colimits
• Universal properties characterize objects and morphisms by their relationships with other objects in the category
• Yoneda lemma establishes a correspondence between an object and the functor it represents, providing a powerful tool for studying categories

Functors and Natural Transformations

• Functors are structure-preserving mappings between categories, consisting of two components:
  • 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
• Functors preserve identity morphisms and composition: $F(id_A) = id_{F(A)}$ and $F(g \circ f) = F(g) \circ F(f)$
• Natural transformations provide a way to compare functors by establishing a family of morphisms between their corresponding objects
  • For functors $F, G: \mathcal{C} \to \mathcal{D}$, a natural transformation $\eta: F \Rightarrow G$ assigns to each object $A$ in $\mathcal{C}$ a morphism $\eta_A: F(A) \to G(A)$ in $\mathcal{D}$
  • These morphisms must satisfy the naturality condition: for any morphism $f: A \to B$ in $\mathcal{C}$, $\eta_B \circ F(f) = G(f) \circ \eta_A$
• Natural isomorphisms are natural transformations whose components are all isomorphisms, indicating that two functors are essentially the same
• Adjoint functors are pairs of functors $(F: \mathcal{C} \to \mathcal{D}, G: \mathcal{D} \to \mathcal{C})$ with a special relationship expressed via natural isomorphisms $\text{Hom}_\mathcal{D}(F(A), B) \cong \text{Hom}_\mathcal{C}(A, G(B))$

Examples and Applications

• Set theory: objects are sets, morphisms are functions, and composition is function composition
  • Monomorphisms are injective functions, epimorphisms are surjective functions, and isomorphisms are bijective functions
• Group theory: objects are groups, morphisms are group homomorphisms, and composition is function composition
  • Isomorphisms are group isomorphisms, and automorphisms are group automorphisms
• Linear algebra: objects are vector spaces, morphisms are linear transformations, and composition is function composition
  • Monomorphisms are injective linear transformations, epimorphisms are surjective linear transformations, and isomorphisms are invertible linear transformations
• Topology: objects are topological spaces, morphisms are continuous functions, and composition is function composition
  • Monomorphisms are injective continuous functions, epimorphisms are surjective continuous functions, and isomorphisms are homeomorphisms
• Programming: objects are types (or data structures), morphisms are functions between types, and composition is function composition
  • Functors represent type constructors or parameterized types, and natural transformations represent polymorphic functions

Common Challenges and Misconceptions

• Category theory is highly abstract and can be challenging to grasp initially due to its focus on general structures and relationships
• Morphisms are not always functions; they can represent various structure-preserving mappings depending on the category
• Isomorphisms in a category do not necessarily correspond to equality of objects; they indicate equivalence within the context of the category
• Epimorphisms and monomorphisms in a general category may not have the same properties as surjective and injective functions in Set
• Functors preserve the structure of categories but may not preserve all properties of objects and morphisms
• Natural transformations compare functors but do not necessarily indicate that the functors are equivalent
• Applying category theory to specific mathematical disciplines requires understanding how the abstract concepts relate to the concrete structures in that discipline

Further Exploration and Resources

• "Category Theory for Scientists" by David I. Spivak provides an accessible introduction to category theory with examples from various scientific fields
• "Basic Category Theory" by Tom Leinster offers a concise and clear introduction to the fundamental concepts of category theory
• "Categories for the Working Mathematician" by Saunders Mac Lane is a classic textbook that provides a comprehensive treatment of category theory
• "Conceptual Mathematics: A First Introduction to Categories" by F. William Lawvere and Stephen H. Schanuel presents category theory using a conceptual approach with examples and exercises
• "Category Theory in Context" by Emily Riehl is a modern textbook that covers a wide range of topics in category theory with a focus on applications
• Explore advanced topics such as limits and colimits, adjunctions, monads, and topoi to deepen your understanding of category theory
• Apply category theory to your specific area of interest (e.g., computer science, physics, biology) to gain new insights and perspectives
• Participate in online communities, forums, or workshops dedicated to category theory to learn from others and share your knowledge