11.1 Categories and Morphisms

Categories and morphisms form the backbone of category theory, introducing a powerful framework for studying mathematical structures. This abstract approach unifies various mathematical concepts, allowing us to analyze relationships between objects across different fields.

By focusing on objects and the arrows (morphisms) connecting them, we can explore common patterns in mathematics. This perspective helps us understand similarities between seemingly unrelated areas, providing insights into the underlying structures of mathematical systems.

Categories and Their Components

Defining Categories

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• A category consists of a collection of objects and a collection of morphisms (arrows) between those objects
  • The objects and morphisms must satisfy certain properties to form a valid category (closure under composition, associativity, identity morphisms)
• For each pair of objects A and B in a category, there is a set denoted Hom(A,B) consisting of the morphisms from A to B
  • If f is a morphism in Hom(A,B), it is denoted as f: A → B, where A is the domain and B is the codomain of f
  • Example: In the category of sets (Set), objects are sets and morphisms are functions between sets

Identity Morphisms and Composition

• For each object A in a category, there exists an identity morphism, denoted as idA or 1A, which is a morphism from A to itself
  • The identity morphism satisfies the property that for any morphism f: A → B, idB ∘ f = f and f ∘ idA = f
  • Example: In the category of groups (Grp), the identity morphism for each group is the identity element of the group
• Morphisms in a category are closed under composition
  • If f: A → B and g: B → C are morphisms, then there exists a morphism g ∘ f: A → C, called the composition of f and g
  • Composition is associative: (h ∘ g) ∘ f = h ∘ (g ∘ f) for any morphisms f: A → B, g: B → C, and h: C → D
  • Example: In the category of vector spaces over a field K (Vect_K), composition of linear transformations is matrix multiplication

Properties and Types of Morphisms

Monomorphisms and Epimorphisms

• A morphism f: A → B is called a monomorphism (or monic) if for any object C and morphisms g1, g2: C → A, f ∘ g1 = f ∘ g2 implies g1 = g2
  • Monomorphisms are the categorical generalization of injective functions
  • Example: In Set, monomorphisms are injective functions
• A morphism f: A → B is called an epimorphism (or epic) if for any object C and morphisms g1, g2: B → C, g1 ∘ f = g2 ∘ f implies g1 = g2
  • Epimorphisms are the categorical generalization of surjective functions
  • Example: In Set, epimorphisms are surjective functions

Isomorphisms, Endomorphisms, and Automorphisms

• A morphism f: A → B is called an isomorphism if there exists a morphism g: B → A such that g ∘ f = idA and f ∘ g = idB
  • The morphism g is called the inverse of f and is denoted as f^(-1)
  • Isomorphisms are the categorical generalization of bijective functions
• A morphism f: A → B is called an endomorphism if A = B, i.e., the domain and codomain of f are the same object
  • The set of all endomorphisms of an object A is denoted as End(A)
  • Example: In the category of rings (Ring), endomorphisms are ring homomorphisms from a ring to itself
• A morphism f: A → B is called an automorphism if it is both an endomorphism and an isomorphism
  • The set of all automorphisms of an object A forms a group under composition, denoted as Aut(A)
  • Example: In Grp, automorphisms are group isomorphisms from a group to itself

Constructing and Analyzing Categories

Examples of Categories

• The category Set has sets as objects and functions between sets as morphisms
  • The composition of morphisms in Set is the usual composition of functions, and the identity morphism for each set is the identity function
• The category Grp has groups as objects and group homomorphisms as morphisms
  • The composition of morphisms in Grp is the usual composition of group homomorphisms, and the identity morphism for each group is the identity homomorphism
• The category Top has topological spaces as objects and continuous functions as morphisms
  • The composition of morphisms in Top is the usual composition of continuous functions, and the identity morphism for each topological space is the identity function

More Examples of Categories

• The category Vect_K has vector spaces over a field K as objects and linear transformations as morphisms
  • The composition of morphisms in Vect_K is the usual composition of linear transformations, and the identity morphism for each vector space is the identity linear transformation
• The category Pos has partially ordered sets as objects and order-preserving functions (monotone functions) as morphisms
  • The composition of morphisms in Pos is the usual composition of order-preserving functions, and the identity morphism for each partially ordered set is the identity function
  • Example: The set of real numbers with the usual order (ℝ, ≤) is an object in Pos, and the absolute value function |·|: ℝ → ℝ is a morphism in Pos

Isomorphism in Categories

Isomorphic Objects

• Two objects A and B in a category are said to be isomorphic if there exists an isomorphism f: A → B between them
  • Isomorphic objects have the same structure and properties within the context of the category
  • Example: In Vect_K, two vector spaces are isomorphic if and only if they have the same dimension
• If A and B are isomorphic, then there exists an inverse morphism f^(-1): B → A such that f^(-1) ∘ f = idA and f ∘ f^(-1) = idB
  • The inverse morphism "undoes" the action of the isomorphism
  • Example: In Set, if f: A → B is a bijective function, then its inverse f^(-1): B → A is also a bijective function

Properties of Isomorphisms

• Isomorphisms satisfy the reflexive, symmetric, and transitive properties:
  • Reflexive: For any object A, idA: A → A is an isomorphism
  • Symmetric: If f: A → B is an isomorphism, then f^(-1): B → A is also an isomorphism
  • Transitive: If f: A → B and g: B → C are isomorphisms, then g ∘ f: A → C is also an isomorphism
• In many categories, isomorphic objects are considered to be "essentially the same" within the context of the category
  • Isomorphisms preserve the structure and properties of objects
  • Example: In Grp, two groups are isomorphic if and only if they have the same group structure (same multiplication table up to relabeling of elements)