Categorical functors

Overview and basic definition

Functors are ways to pass between categories, or “domains of discourse” in mathematics. Functors preserve any property that can be stated purely categorically.

Definition: Functor

Let $\mathsf{C},\mathsf{D}$ be categories. A functor $F:\mathsf{C}\to \mathsf{D}$ consists of the following data:

1. For each $X\in \mathsf{Ob}(\mathsf{C})$, there exists an object $F(X)\in \mathsf{Ob}(\mathsf{D})$.
2. For each morphism $f\in \mathsf{C}(X,Y)$, there exists a morphism $F(f):F(X)\to F(Y)$ in $\mathsf{D}(F(X),F(Y))$.

This data is required to satisfy the following properties, which essentially require that functors preserve structures between categories:

• Compositions map to compositions: $F(g\circ f)=F(g)F(f)$.
• Identities map to identities: $F(1_X)=1_Y$.

Proposition: Functors preserve isomorphism

If $X,Y$ are isomorphic objects of a category $\mathsf{C}$ and $F:\mathsf{C}\to \mathsf{D}$ is a functor, then $F(X),F(Y)$ are isomorphic in $\mathsf{D}$ as well.

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Examples

Forgetful functors