How is the boundary map $\partial: H_n(X, A) \to H_{n-1}(A, \emptyset)$ a natural transformation?

Question

I see it referred to as such here: https://ocw.mit.edu/courses/18-905-algebraic-topology-i-fall-2016/64c56d8bcc2967c1d289a61c959f3095_MIT18_905F16_lec11.pdf

But I am only able to see it as a morphism in Ab. I know that a natural transformation from $F$ to $G$ induces morphisms $F(X) \to G(X)$, but if I were to apply that to the functors $H_n \to H_{n-1}$, it would induce only morphisms $H_n(X, A) \to H_{n-1}(X, A)$ and $H_n(A, \emptyset) \to H_{n-1}(A, \emptyset)$, not $H_n(X, A) \to H_{n-1}(A, \emptyset)$.

What is the "natural transformation" being referred to in this case?

Answer 1

The homology functors $H_n$ are functors from the category $\mathbf{Top^2}$ of topological pairs $(X,A)$ and maps of pairs $f : (X,A) \to (Y,B)$ (i.e. maps $f : X \to Y$ such that $f(A) \subset B$) to the category $\mathbf{Ab}$ of abelian groups and homomorphisms.

Saying that $\partial : H_n(X, A) \to H_{n-1}(A, \emptyset)$ is a natural transformation is a little bit careless. So which functors are involved here?

Define a functor $\rho : \mathbf{Top^2} \to \mathbf{Top^2}$ by $$\rho(X,A) = (A,\emptyset) \text{ for the objects}, \\ \rho(f) = f \mid_A :A \to B\text{ for the morphisms} f : (X,A )\to (Y,B). $$

Then $\partial$ is a natural transformation $$\partial : H_n \to H_{n-1} \circ \rho $$ between the functors $H_n : \mathbf{Top^2} \to \mathbf{Ab}$ and $H_{n-1} \circ \rho : \mathbf{Top^2} \to \mathbf{Ab}$.

Answer 2

If I understand the question correctly it boils down to "How can I understand $H_{n}\left(X,A\right)$ and $H_{n-1}\left(A,\emptyset\right)$ in the suitable way as functors".

The trick is in projecting on the correct coordinate. You have essentially two bifunctors, which means after fixing one input they become ordinary functors. So the two functors you need to consider are: $H_{n}\left(X,\_\right)$ and $H_{n-1}\left(\_,\emptyset\right)$, i.e. relative cohomology where you vary the relative space, and total cohomology of the relative space. Then all the lecture should make sense.

This also makes intuitive sense, as the connecting morphism $\delta$ should relate relative homology varying in the subspace to the total homology of the varying subspace.

Finally, I am very aware that this is not the most detailed answer, but the comment section did not suffice, so I am happy about anyone improving on this statement!

Answer 3

Alternatively, you can think of the pair $(X,A)$ as a single object where the functors act on it, here $F(X,A)=H_n(X, A)$ and $G(X,A)=H_{n-1}(A)$ (so G just ignores X). Then for continuous function $f:(X,A)\to(Y,B)$, you can form induced maps F(f) and G(f), and naturality is about the connecting morphism commuting with the induced maps.