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When I reached the homomorphism concept in group theory, I thought that "natural" is only an ordinary literal word without mathematical meaning. But I read somewhere that this word has a precise mathematical meaning in the theory of categories.

I am not familiar with category theory; only I know it deals with morphisms.

Could you explain me the meaning of "natural" in the category theory?

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1 Answer

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"Natural" can be given a precise meaning using the concept of natural transformations. For example, loosely speaking, for any group $G$ there is a "natural" homomorphism $[G, G] \to G$, where $[G, G]$ is the commutator subgroup of $G$, and similarly there is a "natural" homomorphism $G \to G/[G, G]$. In fact there is a "natural" short exact sequence

$$1 \to [G, G] \to G \to G/[G, G] \to 1.$$

Natural transformations allow you to remove the quote marks in everything I said above. The three constructions $[G, G], G, G/[G, G]$ above can be upgraded to three functors $\text{Grp} \to \text{Grp}$ from the category of groups to itself, and the morphisms above are the components of natural transformations between these functors. Explicitly, this means that if $f : G \to H$ is a group homomorphism, then there are two induced homomorphisms $[G, G] \to [H, H]$ and $G/[G, G] \to H/[H, H]$ such that a certain diagram commutes.

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