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Definition

Functor

Category Theory · category.tex
A functor $F$ is a map of categories that preserves commutative diagrams. In particular, given two categories, Category $\mathcal{C}$, $\mathcal{D}$, a functor $F : \mathcal{C} \to \mathcal{D}$ satisfies the following properties:
  1. Associates each object $X \in \operatorname{Ob}(\mathcal{C})$ to an object $F(X) \in \operatorname{Ob}(\mathcal{D})$
  2. Associates each morphism $f : X \to Y$ in $\mathcal{C}$ in $\mathcal{C}$ to a morphism $F(f) : F(X) \to F(Y)$ in $\mathcal{D}$ such that the following conditions hold: \begin{enumerate}
  3. $F(\operatorname{id}_X) = \operatorname{id}_{F(X)}$
  4. $F(g \circ f) = F(g) \circ F(f)$ for all morphisms $f : X \to Y$ and $g : Y \to Z$ in $\mathcal{C}$.
\end{enumerate} That is, functors must preserve identity morphisms and composition of morphisms.
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Dependency Graph
flowchart LR classDef current fill:#6366f1,color:#fff,stroke:#4f46e5 nf2c6bdf5["Category"] n39e872fb["Functor"]:::current n384e2b5d["Functors Preserve Isomorphisms"] n297b3591["Definition"] n76f014fe["Lemma"] nf2c6bdf5 --> n39e872fb n39e872fb --> n384e2b5d n39e872fb --> n297b3591 n39e872fb --> n76f014fe n39e872fb --> n76f014fe n39e872fb --> n297b3591 click nf2c6bdf5 "../objects/f2c6bdf5.html" "_self" click n384e2b5d "../objects/384e2b5d.html" "_self" click n297b3591 "../objects/297b3591.html" "_self" click n76f014fe "../objects/76f014fe.html" "_self"