Category Theory

A category \( \mathcal{C} \) consists of three mathematical entities:

• A class \( \text{ob}(\mathcal{C}) \) of objects

• A class \( \text{hom}(A , B) \) of morphisms \( A, B \in \text{ob}(\mathcal{C}) \)

  The \( \hom \) set satisfies the following properties:

  • Composition

  \[ \circ \colon \hom(A, B) \times \hom(B, C) \to \hom(A, C) \]

  • Existence of Identity

\[ \exists 1 \in \hom(A,A) . \forall f \in \hom(A,A) . 1 \circ f = f = f \circ 1 \]

• Associativity

\[ \forall f \in \hom(A,B) . \forall g \in \hom(B,C) . \forall h \in \hom(C,D) . (f \circ g) \circ h = f \circ (g \circ h) \]

A functor \( \mathcal{F} \) from a category \( \mathcal{C} \) to \( \mathcal{D} \) denoted \( \mathcal{F} \colon \mathcal{C} \to \mathcal{D} \)

is a map \( \text{ob}(\mathcal{C}) \to \text{ob}(\mathcal{D}) \) and \( \hom(A, B) \to \hom(\mathcal{F}(A), \mathcal{F}(D)) \) such that

• \( \mathcal{F}(1_A) = 1_{\mathcal{F}(A)} \)
• \( \mathcal{F}(f \circ g) = \mathcal{F}(f) \circ \mathcal{F}(g) \)

A natural transformation between two functors \( \mathcal{F} \) and \( \mathcal{G} \) is a system of morphisms

\( \eta(X) \colon \mathcal{F}(X) \to \mathcal{G}(X) \) such that the following diagram commutes

\[ \begin{xy} \xymatrix{ \mathcal{F}(X) \ar[d]^{\mathcal{F}(f)} \ar[r]^{\eta(X)} & \mathcal{G}(X) \ar[d]^{\mathcal{G}(f)} \\ \mathcal{F}(Y) \ar[r]^{\eta(Y)} & \mathcal{G}(Y) } \end{xy} \]