What does it mean by "the diagram commutes by naturality"?

Question

I am confused with the term "The square commutes by naturality of the Mayer-Vietoris sequence". Can anyone explain this? Thanks!

Answer 1

For the particular case of the Mayer-Vietoris sequence, "naturality" formally means that if you have spaces $X_1$ and $X_2$ with open covers $(A_1,B_1)$ and $(A_2,B_2)$ along with a map $f:X_1\rightarrow X_2$ that has $f(A_1)\subseteq A_2$ and $f(B_1)\subseteq f(B_2)$ then we can "connect" the Mayer-Vietoris sequences for these two space as follows and everything commutes: $$\require{AMScd} \begin{CD} H_{n+1}(X_1) @>>> H_{n}(A_1\cap B_1) @>>> H_n(A_1)\oplus H_n(B_1) @>>> H_n(X_1)\\ @VVV @VVV @VVV @VVV\\ H_{n+1}(X_2) @>>> H_n(A_2\cap B_2) @>>> H_n(A_2)\oplus H_n(B_2) @>>> H_n(X_2) \end{CD}$$ where the horizontal maps are the maps in the Mayer-Vietoris sequence and the vertical maps are the various induced maps $f_*$ between the sets. This relation is not so hard to prove directly from definition.

The proof you cite is leaving out a significant amount of detail. To apply naturality to the example you give, you take $X_1=X_2=S^{n+1}$ and $A_1=A_2$ to be the northern hemisphere and $B_1=B_2$ to be the southern hemisphere (nothing that $Sf$ must preserve these sets) and then apply naturality to the map $Sf$. Then, note that $Sf$ is homotopic to $f$ on $A_1\cap B_1 \cong S^n$.

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In a bit more generality, it's worth knowing that "natural" is a more general category theoretic term referring to natural transformations, in case you should encounter the term elsewhere - there's far too much to be said about this field than can reasonably fit in an answer, but basically the notion of "natural transformation" is a useful way to describe things like the Mayer-Vietoris sequence which are defined purely on the space $X_1$ and the cover $(A_1,B_1)$ and do not make any arbitrary choices that might mess up nice logic - proving that the earlier diagram commutes is a good way to get a feel for what this property is like.