1.1 Fundamentals of category theory

Category theory provides a powerful framework for understanding mathematical structures. It introduces objects and morphisms as building blocks, with functors and natural transformations connecting different categories. These concepts form the foundation for studying complex algebraic relationships.

Isomorphisms and equivalences play a crucial role in category theory. They allow us to compare objects within a category and entire categories themselves. This abstract approach enables mathematicians to uncover deep connections between seemingly unrelated areas of mathematics.

Categories and Morphisms

Definition and Components of a Category

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• Category consists of a collection of objects and morphisms between those objects
• Objects can be thought of as the "points" or "vertices" in the category (groups, vector spaces, topological spaces)
• Morphisms are the "arrows" or "edges" connecting objects in the category (group homomorphisms, linear maps, continuous functions)
• Composition operation assigns to each pair of morphisms $f: A \to B$ and $g: B \to C$ a morphism $g \circ f: A \to C$, called their composite
• Identity morphism for each object $A$, denoted as $id_A: A \to A$, satisfies $f \circ id_A = f$ and $id_B \circ f = f$ for any morphism $f: A \to B$

Properties and Diagrams in Category Theory

• Composition of morphisms is associative: $(h \circ g) \circ f = h \circ (g \circ f)$ for any morphisms $f: A \to B$, $g: B \to C$, and $h: C \to D$
• Commutative diagram is a graphical way to represent the equality of two compositions of morphisms
• In a commutative diagram, if two paths of morphisms starting from the same object and ending at the same object are equal (path independence)
• Commutative diagrams help visualize complex relationships between objects and morphisms in a category (diagram chasing)

Functors and Natural Transformations

Functors as Structure-Preserving Maps

• Functor is a map between categories that preserves the structure of the categories
• Functor $F: \mathcal{C} \to \mathcal{D}$ assigns to each object $A$ in $\mathcal{C}$ an object $F(A)$ in $\mathcal{D}$, and to each morphism $f: A \to B$ in $\mathcal{C}$ a morphism $F(f): F(A) \to F(B)$ in $\mathcal{D}$
• Functors preserve identity morphisms: $F(id_A) = id_{F(A)}$ for every object $A$ in $\mathcal{C}$
• Functors preserve composition of morphisms: $F(g \circ f) = F(g) \circ F(f)$ for any morphisms $f: A \to B$ and $g: B \to C$ in $\mathcal{C}$ (forgetful functor, free functor)

Natural Transformations as Morphisms of Functors

• Natural transformation is a "morphism" between functors, providing a way to compare and relate different functors
• Given two functors $F, G: \mathcal{C} \to \mathcal{D}$, a natural transformation $\alpha: F \Rightarrow G$ assigns to each object $A$ in $\mathcal{C}$ a morphism $\alpha_A: F(A) \to G(A)$ in $\mathcal{D}$
• Naturality condition: for any morphism $f: A \to B$ in $\mathcal{C}$, the following diagram commutes: $\alpha_B \circ F(f) = G(f) \circ \alpha_A$
• Natural transformations capture the idea of "natural" or "canonical" relationships between functors (natural isomorphism, Yoneda lemma)

Isomorphisms and Equivalences

Isomorphisms in Category Theory

• Isomorphism is a morphism $f: A \to B$ in a category that has an inverse morphism $g: B \to A$ such that $g \circ f = id_A$ and $f \circ g = id_B$
• Objects $A$ and $B$ are isomorphic if there exists an isomorphism between them, denoted as $A \cong B$
• Isomorphic objects have the same structure and properties within the context of the category (isomorphic groups, isomorphic vector spaces)
• Isomorphisms capture the notion of "sameness" or "equivalence" between objects in a category

Equivalence of Categories

• Equivalence of categories is a stronger notion than isomorphism between individual objects
• Two categories $\mathcal{C}$ and $\mathcal{D}$ are equivalent if there exist functors $F: \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{C}$ such that $G \circ F \cong id_{\mathcal{C}}$ and $F \circ G \cong id_{\mathcal{D}}$ via natural isomorphisms
• Equivalent categories have the same structure and properties, although their objects and morphisms may differ (equivalence between the category of finite-dimensional vector spaces and the category of matrices)
• Equivalence of categories allows for the transfer of results and insights between seemingly different mathematical domains