🔢Elementary Algebraic Topology Unit 6 – Categories and Functors

Key Concepts and Definitions

• Categories consist of objects and morphisms between those objects, forming a directed graph-like structure
  • Objects can be thought of as nodes in the graph, while morphisms are the directed edges connecting them
• Morphisms are structure-preserving maps between objects, capturing the relationships and transformations within a category
• Composition of morphisms is a fundamental operation, allowing for the chaining of morphisms to create new ones
  • Composition is associative, meaning $(f \circ g) \circ h = f \circ (g \circ h)$
• Identity morphisms exist for each object, acting as the neutral element under composition
• Functors are structure-preserving maps between categories, mapping objects to objects and morphisms to morphisms
  • Functors preserve the composition and identity properties of the source category in the target category
• Natural transformations provide a way to compare and relate functors, forming a "morphism of functors"
• Isomorphisms are special morphisms that have an inverse, indicating a strong equivalence between objects

Historical Context and Motivation

• Category theory emerged in the 1940s, primarily through the work of Samuel Eilenberg and Saunders Mac Lane
• Initially developed as a language to unify and generalize various concepts in mathematics, particularly in algebraic topology
• Eilenberg and Mac Lane introduced categories, functors, and natural transformations in their seminal paper "General Theory of Natural Equivalences" (1945)
• The development of category theory was motivated by the need for a common framework to express and study mathematical structures and their relationships
• Category theory provided a way to abstract and generalize concepts, revealing deep connections between seemingly disparate areas of mathematics
• The language and tools of category theory have since found applications in various fields, including computer science, physics, and logic

Category Theory Basics

• A category $\mathcal{C}$ consists of a collection of objects $\text{Ob}(\mathcal{C})$ and a collection of morphisms $\text{Hom}(\mathcal{C})$
  • For each morphism $f$, there is a source object $\text{dom}(f)$ (domain) and a target object $\text{cod}(f)$ (codomain)
• Morphisms are closed under composition, meaning that for any two morphisms $f: A \to B$ and $g: B \to C$, there exists a composite morphism $g \circ f: A \to C$
• Composition of morphisms is associative: $(h \circ g) \circ f = h \circ (g \circ f)$
• For each object $A$, there exists an identity morphism $\text{id}_A: A \to A$ satisfying $f \circ \text{id}_A = f$ and $\text{id}_A \circ g = g$ for any morphisms $f: A \to B$ and $g: C \to A$
• Commutative diagrams are a visual tool for expressing equalities between compositions of morphisms
• Universal properties characterize objects and morphisms in terms of their relationships with other objects and morphisms in the category

Types of Categories

• Discrete categories have objects but no morphisms between distinct objects (only identity morphisms)
• Indiscrete categories have exactly one morphism between any pair of objects
• Monoids can be viewed as categories with a single object, where the morphisms correspond to the elements of the monoid
• Groupoids are categories where every morphism is an isomorphism
  • Groups are special cases of groupoids with a single object
• Preorders and partial orders form categories where objects are elements and morphisms represent the order relation
• Functor categories have functors as objects and natural transformations as morphisms
• Product categories are formed by the Cartesian product of two or more categories, with componentwise composition of morphisms

Functors: Bridging Categories

• Functors are structure-preserving maps between categories, capturing relationships and analogies between different mathematical contexts
• A functor $F: \mathcal{C} \to \mathcal{D}$ consists of:
  • An object function mapping objects of $\mathcal{C}$ to objects of $\mathcal{D}$
  • A morphism function mapping morphisms of $\mathcal{C}$ to morphisms of $\mathcal{D}$, preserving domains and codomains
• Functors preserve identity morphisms: $F(\text{id}_A) = \text{id}_{F(A)}$ for every object $A$ in $\mathcal{C}$
• Functors preserve composition of morphisms: $F(g \circ f) = F(g) \circ F(f)$ for any composable morphisms $f$ and $g$ in $\mathcal{C}$
• Examples of functors include forgetful functors (stripping away structure), free functors (generating free objects), and hom-functors (mapping objects to their hom-sets)
• Functors can be composed, leading to the notion of a functor category where functors are the objects and natural transformations are the morphisms

Important Examples in Topology

• The fundamental group functor $\pi_1$ assigns to each topological space $X$ its fundamental group $\pi_1(X)$
  • Continuous maps between spaces are sent to group homomorphisms between the corresponding fundamental groups
• Homology and cohomology functors ($H_n$ and $H^n$) associate algebraic objects (abelian groups or modules) to topological spaces
  • Induced maps on homology and cohomology capture important topological information and invariants
• The category of topological spaces $\mathbf{Top}$ has topological spaces as objects and continuous maps as morphisms
• The category of simplicial sets $\mathbf{sSet}$ is a combinatorial model for topological spaces, with simplicial sets as objects and simplicial maps as morphisms
• The singular simplicial complex functor $S: \mathbf{Top} \to \mathbf{sSet}$ relates topological spaces to their combinatorial counterparts
• The geometric realization functor $|\cdot|: \mathbf{sSet} \to \mathbf{Top}$ constructs topological spaces from simplicial sets

Applications to Algebraic Topology

• Category theory provides a unified language for expressing and studying various constructions in algebraic topology
• Functors and natural transformations capture the relationships between different algebraic invariants, such as homology and cohomology theories
• The Eilenberg-Steenrod axioms characterize homology theories as functors satisfying certain properties, allowing for a systematic study of homology
• Spectral sequences, which are important computational tools in algebraic topology, can be formulated using the language of categories and functors
• The notion of homotopy can be generalized to functors, leading to the concept of homotopy coherent diagrams and homotopy limits and colimits
• Grothendieck's theory of schemes and motives heavily relies on categorical concepts, providing a foundation for modern algebraic geometry

Common Challenges and Misconceptions

• Category theory is often perceived as abstract and difficult to grasp, requiring a shift in perspective from object-centric thinking to understanding the relationships between objects
• The terminology used in category theory can be intimidating and unfamiliar, leading to a steep learning curve for beginners
• It is important to remember that category theory is a tool for understanding and organizing mathematical concepts, not a replacement for other areas of mathematics
• While category theory provides a general framework, it is crucial to understand the specific context and properties of the categories being studied
• Overemphasis on categorical formalism can sometimes obscure the underlying mathematical ideas and intuition
• Category theory is not a "one-size-fits-all" solution, and not every mathematical problem or structure is best understood through a categorical lens
• Balancing the abstract concepts with concrete examples and applications is essential for developing a deep understanding of category theory and its role in algebraic topology