Recently, I’ve been learning basic category theory from Tom Leinster’s book “Basic Category Theory”.
I thought, wouldn’t it be nice if we could always justify our feeling that a certain map is “natural” or “canonical” in terms of category theory?
The following is an attempt to explain why we feel that the natural surjection $G\rightarrow G/N$ is “natural”, where $G$ is a group and $N$ is a normal subgroup.
Let $\mathbf{NormSbGrp}$ be the category where:
- An object is a pair $(N,G)$, where $G$ is a group and $N\triangleleft G$ is a normal subgroup.
- A morphism between $(N,G)$ and $(K,H)$ will be denoted by $\dot{\varphi}$, where $G\xrightarrow{\varphi}H$ is a group homomorphism such that $\varphi(N)\subseteq K$.
There are two functors from $\mathbf{NormSbGrp}$ to $\mathbf{Grp}$ that we want to compare.
- The forgetful functor $F$, where $F(N,G)=G$ and $F(\dot{\varphi})=\varphi$.
- The quotienting functor $Q$, where $Q(N,G)=G/N$ and $Q(\dot{\varphi})=(gN\mapsto\varphi(g)K)$. For the latter map to be well-defined, we need $\varphi(N)\subseteq K$.
It is easily checked that the canonical surjection $\pi:G\rightarrow G/N$ gives a natural transformation between $F$ and $Q$.
I am tempted to think that behind every natural-seeming map in mathematics is a natural transformation, but I haven’t learned enough mathematics to be sure of that.