🧬Homological Algebra Unit 1 – Homological Algebra: Categories Intro

Key Concepts and Definitions

• Categories consist of objects and morphisms between those objects which satisfy certain axioms
• Objects in a category can be mathematical structures (sets, groups, rings) or more abstract entities
• Morphisms are structure-preserving maps between objects that can be composed associatively
• Identity morphisms exist for each object and act as the identity under composition
• Isomorphisms are morphisms with two-sided inverses, indicating two objects are essentially the same
  • Automorphisms are isomorphisms from an object to itself
• Initial and terminal objects are unique up to isomorphism and have universal properties
  • Initial objects have exactly one morphism to every other object
  • Terminal objects have exactly one morphism from every other object

Historical Context and Development

• Category theory emerged in the 1940s from the work of Samuel Eilenberg and Saunders Mac Lane
• Originally developed as a language to study algebraic topology and homology theories
• Gained prominence in the 1950s and 1960s as a unifying framework for various branches of mathematics
• Alexander Grothendieck's work in algebraic geometry heavily relied on and advanced category theory
• William Lawvere's categorical foundations for set theory and logic expanded the scope of the field
• Category theory has found applications in computer science, physics, and other disciplines
• The development of higher category theory and ∞-categories has pushed the boundaries of the subject

Category Theory Fundamentals

• A category consists of a collection of objects and morphisms satisfying identity and associativity axioms
• Composition of morphisms is an operation that takes two compatible morphisms and produces a third
  • Compatibility requires the codomain of the first morphism to match the domain of the second
• Associativity ensures that composing morphisms in different orders yields the same result when parenthesized correctly
• Identity morphisms compose with other morphisms to give back the original morphism
• Commutative diagrams express the equality of different compositions of morphisms
• Opposite or dual categories are formed by reversing the direction of all morphisms
• Subcategories are formed by selecting a subset of objects and morphisms from a larger category

Objects and Morphisms

• Objects in a category are not required to be sets and can have additional structure
• Morphisms capture the relationships and transformations between objects in a category
• Monomorphisms are left-cancellative morphisms generalizing one-to-one functions
  • Two morphisms $f, g : A \to B$ are equal if $h \circ f = h \circ g$ for all $h : B \to C$
• Epimorphisms are right-cancellative morphisms generalizing onto functions
  • Two morphisms $f, g : B \to C$ are equal if $f \circ h = g \circ h$ for all $h : A \to B$
• Isomorphisms are morphisms with two-sided inverses, capturing the notion of equivalence
• Zero objects are both initial and terminal, with the zero morphism between any two objects factoring through them
• Subobjects generalize subsets and subsystems, with monomorphisms representing inclusion

Functors and Natural Transformations

• Functors are structure-preserving maps between categories that send objects to objects and morphisms to morphisms
• Covariant functors preserve the direction of morphisms, while contravariant functors reverse them
• Functors preserve identity morphisms and composition, enabling the study of relationships between categories
• Natural transformations are morphisms between functors, capturing the notion of a "natural" family of morphisms
  • Components of a natural transformation are morphisms between the images of objects under the functors
  • Naturality squares express the compatibility between components and the functors' action on morphisms
• Natural isomorphisms are natural transformations with invertible components, indicating two functors are naturally equivalent
• Adjoint functors are pairs of functors $F : C \to D$ and $G : D \to C$ with a natural isomorphism between $Hom_D(F(-), -)$ and $Hom_C(-, G(-))$
  • Adjunctions capture universal properties and are ubiquitous in mathematics

Universal Properties and Constructions

• Universal properties characterize objects and morphisms by their relationships with other objects and morphisms
• Initial and terminal objects are the simplest examples of universal properties
• Products and coproducts generalize Cartesian products and disjoint unions, characterized by universal properties
  • Products have projection morphisms and a unique morphism from any other object satisfying compatibility conditions
  • Coproducts have injection morphisms and a unique morphism to any other object satisfying compatibility conditions
• Equalizers and coequalizers are limits and colimits that generalize subobjects and quotient objects
• Pullbacks and pushouts are universal constructions that generalize inverse images and quotients by equivalence relations
• Limits and colimits are universal objects that generalize products, coproducts, equalizers, and coequalizers
  • Limits are terminal objects in the category of cones over a diagram
  • Colimits are initial objects in the category of cocones under a diagram

Applications in Homological Algebra

• Homological algebra studies sequences of objects and morphisms with certain properties, such as exactness
• Chain complexes are sequences of objects and morphisms with the composition of consecutive morphisms being zero
  • Homology measures the failure of a sequence to be exact by taking quotients of kernels by images
• Exact functors preserve exact sequences and are essential for studying the behavior of homology under functors
• Derived functors are a way to "correct" non-exact functors and measure their deviation from exactness
  • Examples include Tor and Ext functors, which arise from the non-exactness of the tensor product and Hom functors
• Spectral sequences are powerful tools for computing homology groups by organizing information from various sources
• Derived categories are formed by localizing the category of chain complexes at quasi-isomorphisms
  • Quasi-isomorphisms are chain maps inducing isomorphisms on homology, capturing the notion of "homological equivalence"
• Triangulated categories axiomatize key properties of derived categories and provide a general framework for homological algebra

Common Challenges and Problem-Solving Strategies

• Identifying the correct category to work in and understanding its properties is crucial for problem-solving
• Constructing suitable morphisms and verifying their properties can be challenging
  • Universal properties and adjunctions can often guide the construction of morphisms
• Proving the existence and uniqueness of objects satisfying certain properties may require clever use of universal properties
• Diagram chasing is a technique for proving statements about morphisms by following elements through commutative diagrams
• Spectral sequence computations can be intricate and require careful bookkeeping and understanding of the underlying structure
• Derived functors and derived categories require a good grasp of homological algebra and localization techniques
• Generalizing results from familiar categories to more abstract settings can lead to new insights and connections
  • Analogy and abstraction are powerful tools for problem-solving in category theory
• Collaborating with others and seeking guidance from experts can help overcome challenges and provide new perspectives