1.3 Examples of categories from various mathematical fields

Mathematics is full of fundamental categories that help us understand complex structures. Sets and functions, groups and homomorphisms, and topological spaces and continuous functions form the backbone of many mathematical concepts.

These categories provide a framework for studying relationships between objects and morphisms. By exploring vector spaces, linear transformations, and connections through category theory, we can unify diverse mathematical ideas and reveal hidden patterns across different fields.

Fundamental Categories in Mathematics

Sets and functions category

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• Category of sets, denoted as Set consists of:
  • Objects: Sets (collections of elements)
  • Morphisms: Functions between sets map elements from one set to another
    • Identity morphism: Identity function maps each element to itself in the same set
    • Composition: Function composition applies one function after another, satisfying associativity
• Set serves as a foundational example for understanding categories
  • Many mathematical structures can be viewed as special cases of sets with additional properties (groups, rings, fields)

Groups and homomorphisms category

• Category of groups, denoted as Grp consists of:
  • Objects: Groups (sets equipped with a binary operation satisfying group axioms)
    • Associativity: $(a * b) * c = a * (b * c)$ for all elements $a$, $b$, $c$ in the group
    • Identity: Unique identity element $e$ exists such that $a * e = e * a = a$ for all elements $a$ in the group
    • Inverses: Each element $a$ has a unique inverse $a^{-1}$ such that $a * a^{-1} = a^{-1} * a = e$
  • Morphisms: Group homomorphisms (functions between groups preserving the group structure)
    • Homomorphism property: $f(a * b) = f(a) * f(b)$ for all elements $a$, $b$ in the domain group
    • Identity morphism: Identity function on the underlying set maps each element to itself
    • Composition: Function composition of group homomorphisms preserves the homomorphism property
• Grp encapsulates the essential features of group theory within a categorical framework
  • Studying groups and their homomorphisms reveals symmetries and structural properties (symmetry groups, permutation groups)

Topological spaces and functions category

• Category of topological spaces, denoted as Top consists of:
  • Objects: Topological spaces (sets equipped with a topology)
    • Topology: Collection of subsets, called open sets, satisfying certain axioms
      • Union of any collection of open sets is open
      • Finite intersection of open sets is open
      • Empty set and the entire space are open
  • Morphisms: Continuous functions between topological spaces
    • Continuity: Preimages of open sets under the function are open in the domain space
    • Identity morphism: Identity function on the underlying set is continuous
    • Composition: Function composition of continuous functions is continuous
• Top captures the core concepts of topology, such as continuity and homeomorphisms
  • Homeomorphisms: Continuous functions with continuous inverses, establishing topological equivalence (circle and ellipse, Möbius strip and cylinder)

Vector Spaces and Connections

Vector spaces and transformations category

• Category of vector spaces over a field $K$, denoted as $[Vect_K](https://www.fiveableKeyTerm:vect_k)$ consists of:
  • Objects: Vector spaces over $K$ (sets equipped with vector addition and scalar multiplication)
    • Vector addition: Associative, commutative, and has an identity element (zero vector)
    • Scalar multiplication: Associative, distributive over vector addition, and has an identity element (scalar 1)
  • Morphisms: Linear transformations between vector spaces
    • Linearity: $f(a\vec{u} + b\vec{v}) = af(\vec{u}) + bf(\vec{v})$ for all vectors $\vec{u}$, $\vec{v}$ and scalars $a$, $b$
    • Identity morphism: Identity function on the underlying set is a linear transformation
    • Composition: Function composition of linear transformations is a linear transformation
• $Vect_K$ encodes the essential properties of linear algebra within a categorical setting
  • Studying vector spaces and linear transformations is fundamental in mathematics and physics (Euclidean spaces, function spaces)

Connections through category theory

• Functors: Structure-preserving maps between categories
  1. Map objects from one category to objects in another category
  2. Map morphisms from one category to morphisms in another category
  3. Preserve identity morphisms and composition
  • Functors relate different mathematical structures by mapping objects and morphisms consistently (forgetful functor from Grp to Set, fundamental group functor from Top to Grp)
• Natural transformations: Structure-preserving maps between functors
  • Componentwise morphisms between functors that commute with the functors' action on morphisms
  • Compare and relate functors, establishing connections between different constructions (natural isomorphism, equivalence of categories)
• Universal properties: Characterize objects and morphisms by their relationships with other objects
  • Products: Objects with morphisms to other objects, satisfying a universal property (Cartesian product of sets, direct product of groups)
  • Coproducts: Objects with morphisms from other objects, satisfying a universal property (disjoint union of sets, free product of groups)
  • Limits and colimits: Generalize products and coproducts to diagrams of objects and morphisms (pullbacks, pushouts)
  • Adjoint functors: Pairs of functors with a natural bijection between their sets of morphisms (free-forgetful adjunction, Galois connections)
• Duality: Reverse the direction of morphisms in a category
  • Opposite category: Obtained by reversing the direction of all morphisms
  • Dual concepts and theorems arise, revealing deep symmetries in mathematics (opposite group, dual vector space)
• Category theory unifies and abstracts common patterns across different branches of mathematics
  • Allows for the transfer of ideas and techniques between seemingly disparate fields (algebraic topology, algebraic geometry, representation theory)