🧮Topos Theory Unit 2 – Functors and Natural Transformations

Key Concepts and Definitions

• Functors map objects and morphisms between categories while preserving composition and identity
• Natural transformations provide a way to compare functors by mapping between them in a structure-preserving manner
• Functor categories have functors as objects and natural transformations as morphisms, forming a new category
• Isomorphisms in functor categories are natural isomorphisms, indicating naturally isomorphic functors
• Adjoint functors are pairs of functors (left and right adjoints) that form a special relationship characterized by natural isomorphisms
  • The unit and counit of an adjunction are natural transformations that satisfy certain coherence conditions
• Representable functors are functors naturally isomorphic to a hom-functor, providing a link between objects and morphisms
• The Yoneda lemma states that the set of natural transformations between a functor and a representable functor is isomorphic to the set of elements of the functor at the representing object

Historical Context and Development

• Functors were introduced by Eilenberg and MacLane in 1945 as part of their work on establishing category theory as a unified framework for mathematics
• Natural transformations were first defined by Eilenberg and MacLane in 1945 to compare functors and study their relationships
• Grothendieck's work on algebraic geometry in the 1950s and 1960s heavily relied on functors and natural transformations, showcasing their power in unifying various mathematical concepts
• Kan extensions, introduced by Kan in 1958, generalize the notion of extending a functor along a natural transformation
  • Kan extensions play a crucial role in the theory of adjoint functors and the construction of limits and colimits
• The concept of adjoint functors was introduced by Kan in 1958, providing a way to relate functors through natural isomorphisms
• The Yoneda lemma, proved by Yoneda in 1954, establishes a deep connection between functors and representable functors, becoming a fundamental tool in category theory

Categories and Functors Revisited

• Categories consist of objects and morphisms between them, with composition and identity morphisms satisfying associativity and unit laws
• Functors are structure-preserving maps between categories, mapping objects to objects and morphisms to morphisms
• Functors preserve composition, meaning $F(g \circ f) = F(g) \circ F(f)$ for any composable morphisms $f$ and $g$
• Functors preserve identity morphisms, so $F(id_A) = id_{F(A)}$ for any object $A$ in the domain category
• Functors can be composed, forming a new functor that maps from the domain of the first to the codomain of the second
  • The composition of functors is associative, and the identity functor acts as the unit of composition
• Functors can be full, faithful, or essentially surjective, describing their behavior on morphisms and objects
• Equivalence of categories is defined using functors that are full, faithful, and essentially surjective, establishing a strong notion of similarity between categories

Properties of Functors

• Functors preserve isomorphisms, meaning if $f: A \to B$ is an isomorphism, then $F(f): F(A) \to F(B)$ is also an isomorphism
• Full functors are surjective on morphisms, i.e., for any objects $A, B$ in the domain category and morphism $g: F(A) \to F(B)$ in the codomain category, there exists a morphism $f: A \to B$ such that $F(f) = g$
• Faithful functors are injective on morphisms, meaning if $F(f) = F(g)$ for morphisms $f, g: A \to B$, then $f = g$
• Essentially surjective functors are surjective on objects up to isomorphism, i.e., for any object $C$ in the codomain category, there exists an object $A$ in the domain category such that $F(A) \cong C$
• Functors preserve monomorphisms and epimorphisms, but not necessarily injective or surjective morphisms
  • A monomorphism is a morphism $f: A \to B$ such that for any object $C$ and morphisms $g, h: C \to A$, if $f \circ g = f \circ h$, then $g = h$
  • An epimorphism is a morphism $f: A \to B$ such that for any object $C$ and morphisms $g, h: B \to C$, if $g \circ f = h \circ f$, then $g = h$
• Functors preserve commutative diagrams, meaning if a diagram commutes in the domain category, its image under the functor commutes in the codomain category

Natural Transformations Explained

• A natural transformation $\alpha: F \Rightarrow G$ is a family of morphisms between two functors $F, G: \mathcal{C} \to \mathcal{D}$ that commutes with the action of the functors on morphisms
• For each object $A$ in $\mathcal{C}$, a natural transformation assigns a morphism $\alpha_A: F(A) \to G(A)$ in $\mathcal{D}$
• The naturality condition requires that for any morphism $f: A \to B$ in $\mathcal{C}$, the following diagram commutes: $$\begin{CD} F(A) @>\alpha_A>> G(A)\\ @VF(f)VV @VVG(f)V\\ F(B) @>>\alpha_B> G(B) \end{CD}$$
• Natural transformations can be composed vertically (when the codomain of one matches the domain of the other) and horizontally (when the functors are composable)
  • Vertical composition of natural transformations is associative and has an identity natural transformation as the unit
  • Horizontal composition of natural transformations is associative and compatible with vertical composition
• A natural isomorphism is a natural transformation where each component $\alpha_A$ is an isomorphism, indicating that the functors $F$ and $G$ are naturally isomorphic
• The Yoneda lemma establishes a natural isomorphism between the set of natural transformations from a representable functor to any functor $F$ and the set $F(A)$, where $A$ is the representing object

Functor Categories and Their Significance

• A functor category $[\mathcal{C}, \mathcal{D}]$ has functors $F: \mathcal{C} \to \mathcal{D}$ as objects and natural transformations between them as morphisms
• Composition of morphisms in a functor category is given by the vertical composition of natural transformations
• The identity morphism on a functor $F$ in a functor category is the identity natural transformation $id_F$, where each component $(id_F)_A$ is the identity morphism on $F(A)$
• Functor categories allow for the study of functors and natural transformations using the tools and techniques of category theory
  • Limits, colimits, and adjunctions in functor categories provide insights into the behavior of functors and natural transformations
• The Yoneda embedding is a fully faithful functor from a category $\mathcal{C}$ to the functor category $[\mathcal{C}^{op}, \mathbf{Set}]$, revealing the deep connection between objects and their representable functors
• Presheaves on a category $\mathcal{C}$ are functors $F: \mathcal{C}^{op} \to \mathbf{Set}$, forming the functor category $[\mathcal{C}^{op}, \mathbf{Set}]$, which plays a crucial role in sheaf theory and topos theory
• The category of small categories, $\mathbf{Cat}$, is a functor category where objects are small categories and morphisms are functors between them

Applications in Mathematics and Beyond

• Algebraic topology uses functors to study topological spaces by associating algebraic objects (groups, rings, modules) to them, enabling the use of algebraic techniques to solve topological problems
  • Homology and cohomology theories are functors from the category of topological spaces to the category of abelian groups or modules
• Representation theory employs functors to study the representations of algebraic structures (groups, algebras, Lie algebras) in terms of linear transformations on vector spaces
• Sheaf theory and topos theory heavily rely on functors and natural transformations to study the local-to-global properties of mathematical objects
  • Sheaves are functors from the category of open sets of a topological space (with inclusions as morphisms) to the category of sets or abelian groups
  • A topos is a category that behaves like the category of sets, characterized by the existence of certain limits, colimits, and exponential objects
• Functorial constructions in algebraic geometry, such as the spectrum of a ring or the scheme associated to a ring, allow for the study of geometric objects using algebraic tools
• In mathematical physics, functors are used to describe the relationships between different physical theories and to study the behavior of physical systems under symmetry transformations
• Category theory and its concepts, including functors and natural transformations, have found applications in computer science, particularly in the areas of type theory, programming language semantics, and database theory

Common Challenges and Misconceptions

• Understanding the difference between a functor and a function can be challenging, as functors map between categories while functions map between sets
• Remembering the direction of composition for functors and natural transformations can be confusing, especially when dealing with contravariant functors or functor categories
• Distinguishing between different types of morphisms (monomorphisms, epimorphisms, isomorphisms) and their preservation under functors requires careful attention to definitions
• Grasping the concept of naturality and the commutative diagrams involved in natural transformations may take time and practice
  • It's essential to understand that naturality is a property of the entire family of morphisms, not just individual components
• Applying the Yoneda lemma and understanding its implications can be challenging, as it involves the interplay between functors, representable functors, and natural transformations
• Working with functor categories and their morphisms (natural transformations) requires a solid understanding of both functors and natural transformations, as well as the ability to visualize and manipulate commutative diagrams
• Recognizing the connections between functors, natural transformations, and other categorical concepts such as limits, colimits, and adjunctions is crucial for a deep understanding of category theory and its applications
• Overcoming the initial abstraction and embracing the unifying power of category theory may require a shift in mathematical thinking and a willingness to explore new concepts and perspectives