I want to consider those $\eta_\bullet^t$ as natural transformations between functors. Those functors are the functors TOP->ABEL associated to $S_\bullet(X)$ and $S_\bullet(X\times I)$, right?
Hmm, I was thinking about the functors $S_q$ sending $X$ to $S_q(X)$ (the q-th group of chains) and $f:X\to Y$ to the induced map $S_q(X)\to S_q(Y)$
What you mean is instead the functor that sends $X$ to $S_\bullet(X)$ and $f:X\to Y$ to the associated chain map $S_\bullet(X)\to S_\bullet(Y)$, right?
Oh, lol, I didn't notice the square and was like "but doesn't it just follow from the cone construction? I guess those $\eta_\bullet^t$ must be useful for something!" feels dumb
Ok, that's very nice as far as 9.3.2 is concerned then! However I don't see why the $\eta_\bullet^t$ are natural transformations (probably because my experience with natural transformation is limited to our conversation from the other day)
So you have two functors from TOP to chain complexes of abelian groups, one sends $X$ to $S_\bullet(X)$ and the other one sends $X$ to $S_\bullet(X \times I)$
Right, and they send a continuous map $f:X\to Y$ maps between topological spaces to the induced chain maps $f_\bullet:S_\bullet(X)\to S_\bullet(Y)$ and $(f\times \mathrm{Id})_\bullet:S_\bullet(X\times I)\to S_\bullet(Y\times I)$
and tom Dieck claims that the $\eta_\bullet^t$ are natural transformations, the reason for naturality is basically just that the square $\require{AMScd} \begin{CD} X @>>{\eta^t_X}> X \times I\\ @V{f}VV @VV{f \times \mathrm{Id}}V\\ Y @>>{\eta^t_Y}> Y \times I \end{CD}$ commutes
I'm writing $\eta^t_X$ to keep track of which space where're considering the map $\eta$
So this diagram above says that $\eta^t$ (without going to the induced map on singular chains) is a natural transformation between the identity functor on TOP and the functor which sends $X \mapsto X \times I$ and $f:X \to Y$ to $f \times \mathrm{id}:X \times I \to Y \times I$
you have to check that it commutes, but that's basically obvious
and then you get that $\eta_\bullet^t$ is a natural transformation from the functor $X \mapsto S_\bullet(X)$ to the functor $X \mapsto S_\bullet(X \times I)$
that's a general principle: if $\eta: F \Rightarrow G$ is a natural transformation between parallel functors $F,G: \mathcal{C} \to \mathcal{D}$ and $H$ is any functor $H: \mathcal{D} \to \mathcal{E}$, then you can apply $H$ to each morphism in $\eta$ (since $\eta$ is a collection of morphisms satisfying some conditions) and you get that $H(\eta)$ is a natural transformation $H \circ F \Rightarrow H \circ G$
and the reason for this very abstract thing is just that functors preserve commutative diagrams, as you said
So in our example $\mathcal{C}=\mathcal{D}=\mathbf{TOP}$ and $\mathcal{E}=\mathbf{CH}(\Bbb Z)$, the category of chain complexes of abelian groups and $F=\mathrm{Id}_{\mathbf{Top}}$ and $G: X \mapsto X \times I, (f:X \to Y) \mapsto (f \times \mathrm{Id}: X \times I \to Y \times I)$ and $H: \mathbf{Top} \to \mathbf{CH}(\Bbb Z), X \mapsto S_\bullet(X)$
and $\eta=\eta^t$
that was quite an abstract explanation, but I think it's in line with tom Dieck's style :P
From skimming it looks like the point of 9.3.3 is that homotopic maps induce chain homotopic maps on the chain complexes and we'll use that to conclude that homotopic maps induce the same homomorphisms of homology groups, seems promising! I'll read the details now, thanks a lot for your help
@MatheinBoulomenos I'm afraid I don't know what those are but I'll reach them eventually
I'm kinda blackboxing the results he cites from Ch11 at the moment and postponing them for later
I looked up some of them from here, especially the snake lemma, because tom Dieck doesn't explain the connecting homomorphism when constructing the homology exact sequence in Ch9
Hi @Balarka, welcome back, this is the category theory chat room now
@AlessandroCodenotti yeah, showing that homotopic maps induce the same homomorphism has two parts: the algebraic part about chain complexes which is very easy. the more difficult part is that a continuous homotopy induces a chain homotopy but that's a consequence of 9.3.3
@AlessandroCodenotti tbf tom Dieck probably learned the snake lemma in the Kindergarden
random cohomology for quantum nerds
random cohomology for quantum nerds