🔢Category Theory Unit 15 – Applications to Algebra and Topology

Key Concepts and Definitions

• Categories consist of objects and morphisms between those objects satisfying composition and identity laws
• Objects in a category can represent various mathematical structures (sets, groups, topological spaces)
• Morphisms are structure-preserving maps between objects (functions, group homomorphisms, continuous maps)
• Composition of morphisms $f: A \rightarrow B$ and $g: B \rightarrow C$ is a morphism $g \circ f: A \rightarrow C$ that satisfies associativity
• Identity morphisms $id_A: A \rightarrow A$ exist for each object $A$ and satisfy left and right identity laws with composition
• Isomorphisms are morphisms $f: A \rightarrow B$ with an inverse $g: B \rightarrow A$ such that $f \circ g = id_B$ and $g \circ f = id_A$
• Commutative diagrams express equality of compositions of morphisms along different paths (e.g., $f \circ g = h \circ k$)
• Initial and terminal objects are unique up to isomorphism and have universal properties (e.g., empty set, singleton set)

Algebraic Structures in Category Theory

• Groups can be viewed as categories with a single object and all morphisms being isomorphisms (group elements)
• Composition of morphisms in a group category corresponds to the group operation
• Rings can be viewed as categories with a single object and morphisms representing ring elements
• Composition of morphisms in a ring category corresponds to ring multiplication
• Modules over a ring $R$ form a category with module homomorphisms as morphisms
• Abelian categories generalize categories of modules and have additional structure (kernels, cokernels, exact sequences)
• Monoids can be viewed as categories with a single object and a composition operation on morphisms
• Monoid homomorphisms are functors between monoid categories preserving the composition operation

Topological Spaces as Categories

• Topological spaces can be viewed as categories with open sets as objects and inclusions as morphisms
• Continuous maps between topological spaces are functors between the corresponding categories
• Homotopy equivalences are isomorphisms in the category of topological spaces and homotopy classes of maps
• The fundamental groupoid of a topological space has points as objects and homotopy classes of paths as morphisms
• Sheaves on a topological space $X$ form a category with sheaf morphisms as arrows
  • Sheaves assign data to open sets of $X$ and satisfy gluing conditions
  • Sheaf cohomology is an important invariant in algebraic geometry and complex analysis

Functors and Natural Transformations

• Functors are structure-preserving maps between categories that map objects to objects and morphisms to morphisms
• Functors preserve composition and identity morphisms (e.g., forgetful functors, free functors)
• Natural transformations are morphisms between functors that commute with the functors' action on morphisms
• Components of a natural transformation are morphisms in the codomain category indexed by objects of the domain category
• Natural isomorphisms are natural transformations with invertible components (e.g., double dual of a vector space)
• Adjoint functors are pairs of functors $(F, G)$ with a natural isomorphism between hom-sets $Hom(F(A), B) \cong Hom(A, G(B))$
• Adjunctions capture universal properties and are ubiquitous in mathematics (e.g., free-forgetful adjunctions)

Universal Properties and Constructions

• Universal properties characterize objects and morphisms by their relationships with other objects and morphisms
• Initial objects have a unique morphism to every other object in the category (e.g., empty set, trivial group)
• Terminal objects have a unique morphism from every other object in the category (e.g., singleton set, trivial group)
• Products are objects with projections to factors satisfying a universal property (generalize Cartesian products)
• Coproducts (sums) are dual to products and have injections from factors satisfying a universal property (generalize disjoint unions)
• Equalizers are limits of parallel morphisms and satisfy a universal property (generalize kernels)
• Coequalizers are colimits of parallel morphisms and satisfy a universal property (generalize quotients)
• Pullbacks (fiber products) are limits of diagrams and satisfy a universal property (generalize inverse images)
• Pushouts are colimits of diagrams and satisfy a universal property (generalize amalgamated sums)

Applications in Algebra

• Diagram chasing is a technique for proving algebraic statements using commutative diagrams and universal properties
• Snake lemma relates kernels and cokernels in a commutative diagram of exact sequences
• Five lemma proves isomorphisms between objects in a commutative diagram of exact sequences
• Short five lemma is a special case of the five lemma for short exact sequences
• Homological algebra studies chain complexes and their homology using categorical methods
  • Chain complexes are sequences of objects and morphisms satisfying boundary conditions
  • Homology measures the failure of exactness in a chain complex
• Derived functors (e.g., Tor, Ext) are obtained by applying a functor to a resolution and taking homology
• Spectral sequences are algebraic tools for computing homology and cohomology using filtrations and gradings

Applications in Topology

• Fundamental groupoid of a topological space captures homotopy classes of paths between points
• Van Kampen theorem computes the fundamental group of a space using a pushout diagram of fundamental groupoids
• Covering spaces correspond to functors from the fundamental groupoid to the category of sets
• Galois theory of covering spaces relates subgroups of the fundamental group to covering spaces
• Simplicial sets are functors from the simplex category to the category of sets and generalize simplicial complexes
• Singular homology and cohomology are defined using chain complexes of singular simplices (continuous maps from simplices)
• Čech cohomology is defined using limits of cochains over open covers and agrees with singular cohomology for nice spaces
• Eilenberg-Steenrod axioms characterize homology and cohomology theories for topological spaces using categorical properties

Advanced Topics and Current Research

• Higher category theory studies categories with morphisms between morphisms (2-categories) and beyond (n-categories)
• Homotopy type theory is a foundation for mathematics that combines type theory with homotopy-theoretic ideas
• Infinity categories (quasicategories) are simplicial sets satisfying weak composition laws and model higher categories
• Topos theory generalizes sheaves and provides a categorical framework for logic and geometry
• Monoidal categories have a tensor product operation on objects and morphisms compatible with composition
• Braided monoidal categories have a braiding isomorphism that swaps the order of the tensor product
• Symmetric monoidal categories have a symmetry isomorphism that makes the braiding involutive
• Enriched categories replace hom-sets with objects of a monoidal category (e.g., categories enriched over abelian groups)