I've only recently picked up literature about category theory, but the definition of a functor on Wikipedia seems redundant to me. I've paraphrased it here:
Given categories $C$ and $D$, then a mapping $F : C \rightarrow D$ is a functor iff:
- $F$ assigns to every object $X \in C$ an object $F(X) \in D$, (left-total for objects)
- $F$ assigns to every arrow $f : X \rightarrow Y \in C$ an arrow $F(f) : F(X) \rightarrow F(Y) \in D$, (left-total for arrows)
- For all arrows $f : X \rightarrow Y$ and $g : Y \rightarrow Z$ in $C$, $F(g \circ f) = F(g) \circ F(f)$, (homomorphism) and
- For every identity arrow $1_X : X \rightarrow X$ in $C$, $F(1_X) = 1_{F(X)}$. (preservation of identity)
From what I understand, it can be proven that every identity arrow in a category is unique. Combined with this theorem, I think the following proof is correct:
- Reflexivity of equality: $\forall f\in C: f = f$
- Left-totality for arrows: $\forall f\in C: F(f) = F(f)$
- Introduce left-identity: $\forall f\in C: F(1_Y \circ f) = F(f)$ (given $f:X \rightarrow Y$)
- Homomorphism: $\forall f\in C: F(1_Y) \circ F(f) = F(f)$ (given $f:X \rightarrow Y$)
- Because of uniqueness of identity, $F(1_Y)$ must be the identity arrow for $F(Y)$, so it follows that $F(1_Y) = 1_{F(Y)}$.
Am I missing something here? Does preservation of identity follow from other laws?
Best Answer
You've shown that $F(1_Y) \circ F(f) = F(f)$ for each $\mathcal{C}$-arrow $f : X \to Y$. However, in order to show that $F(1_Y)=1_{FY}$ you need to show that $F(1_Y) \circ g = g$ for each $\mathcal{D}$-arrow $g:Z \to FY$. These two notions won't necessarily be the case unless every $\mathcal{D}$-arrow into $FY$ is of the form $F(f)$ for some $\mathcal{C}$-arrow $f$.
For instance, consider the categories $\mathcal{C}$ and $\mathcal{D}$, which both have just one object, the set $X=\{ 0,1 \}$, and
Let $F : \mathcal{C} \to \mathcal{D}$ take $1_X$ to $f$. You'll see that all the axioms are satisfied except the requirement that $F(1_X)=1_{FX}=1_X$.