Category theory

File size
10.2KB
Lines of code
197

Category theory

The branch of mathematics dealing with abstract structures and the relationships between them.

Covers...

  • Categories, Functors, and Natural Transformations
  • Monoids, Product and Sum Types
  • Limits and Colimits
  • Functor Categories and Adjunctions
  • Monads
  • Initial and Terminal Objects
  • Equivalence of Categories
  • Yoneda Lemma
  • Representable Functors

Definitions

  1. Category

    • A category $\mathcal{C}$ consists of:
      • Objects: Denoted by symbols such as $A, B, C, \ldots$
      • Morphisms (or Arrows): Denoted by $f: A \to B$, representing a relationship from object $A$ to object $B$
      • Composition: For morphisms $f: A \to B$ and $g: B \to C$, the composition $g \circ f: A \to C$
      • Identity Morphisms: For each object $A$, there is an identity morphism $id_A: A \to A$ such that $id_A \circ f = f$ and $g \circ id_A = g$ for all $f$ and $g$ involving $A$
      • Associativity: Composition of morphisms is associative: $(h \circ g) \circ f = h \circ (g \circ f)$

  2. Functor

    • A functor $F: \mathcal{C} \to \mathcal{D}$ between categories $\mathcal{C}$ and $\mathcal{D}$ consists of:
      • A mapping of objects: For each object $A \in \mathcal{C}$, there is an object $F(A) \in \mathcal{D}$
      • A mapping of morphisms: For each morphism $f: A \to B$ in $\mathcal{C}$, there is a morphism $F(f): F(A) \to F(B)$ in $\mathcal{D}$
      • Preservation of composition and identity: $F(g \circ f) = F(g) \circ F(f)$ and $F(id_A) = id_{F(A)}$

  3. Natural Transformation

    • A natural transformation $\eta$ between functors $F, G: \mathcal{C} \to \mathcal{D}$ consists of:
      • A collection of morphisms $\eta_A: F(A) \to G(A)$ for each object $A \in \mathcal{C}$
      • Such that for any morphism $f: A \to B$ in $\mathcal{C}$, the diagram Diagram commutes

  4. Monoid

    • A monoid $(M, \cdot, e)$ is a set $M$ equipped with a binary operation $\cdot: M \times M \to M$ and an identity element $e \in M$ such that:
      • Associativity: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $a, b, c \in M$
      • Identity: $a \cdot e = a$ and $e \cdot a = a$ for all $a \in M$

  5. Product and Sum Types

    • Product Types: Denoted by $A \times B$, represent pairs of elements from sets $A$ and $B$.
    • Sum Types: Denoted by $A + B$, represent elements that belong to either set $A$ or set $B$.

  6. Limits and Colimits

    • Limit: The limit of a diagram (a functor from an index category to $\mathcal{C}$) is an object that captures the idea of the “best approximation” to the diagram.
    • Colimit: Dual to limits, capturing the “coarsest” structure that can be mapped to from the diagram.

  7. Functor Categories

    • The category Fun$(\mathcal{C}, \mathcal{D})$ consists of all functors from $\mathcal{C}$ to $\mathcal{D}$ and natural transformations between them.

  8. Adjunctions

    • An adjunction between categories $\mathcal{C}$ and $\mathcal{D}$ consists of a pair of functors $F: \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{C}$ such that:
      • For each pair of objects $A \in \mathcal{C}$ and $B \in \mathcal{D}$, there is a natural isomorphism between \text{Hom}_{\mathcal{D}}(F(A), B) and \text{Hom}_{\mathcal{C}}(A, G(B)).

  9. Monads

    • A monad on a category $\mathcal{C}$ consists of:
      • A functor $T: \mathcal{C} \to \mathcal{C}$
      • Two natural transformations: $\eta: 1_{\mathcal{C}} \to T$ (unit) and $\mu: T^2 \to T$ (multiplication) satisfying certain coherence conditions.

  10. Initial and Terminal Objects

    • Initial Object: An object $I$ in a category $\mathcal{C}$ such that for every object $A \in \mathcal{C}$, there exists a unique morphism from $I$ to $A$.
    • Terminal Object: An object $T$ in a category $\mathcal{C}$ such that for every object $A \in \mathcal{C}$, there exists a unique morphism from $A$ to $T$.

  11. Equivalence of Categories

    • Two categories $\mathcal{C}$ and $\mathcal{D}$ are said to be equivalent if there exist functors $F: \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{C}$ such that $G \circ F$ is naturally isomorphic to the identity functor on $\mathcal{C}$, and $F \circ G$ is naturally isomorphic to the identity functor on $\mathcal{D}$.

  12. Yoneda Lemma

    • The Yoneda Lemma states that for any category $\mathcal{C}$, object $A \in \mathcal{C}$, and functor $F: \mathcal{C} \to \textbf{Set}$, there is a natural isomorphism \text{Nat}(\text{Hom}_{\mathcal{C}}(-, A), F) \cong F(A) where:
      • \text{Hom}_{\mathcal{C}}(-, A) is the functor that maps an object X to the set of morphisms \text{Hom}_{\mathcal{C}}(X, A).
      • \text{Nat}(\text{Hom}_{\mathcal{C}}(-, A), F) denotes the set of natural transformations from \text{Hom}_{\mathcal{C}}(-, A) to F.

  13. Representable Functors

    • A functor $F: \mathcal{C} \to \textbf{Set}$ is called representable if it is naturally isomorphic to the hom-functor $\text{Hom}_{\mathcal{C}}(C, -)$ for some object $C \in \mathcal{C}$.

More on