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The Eilenberg-Steenrod axioms state that a homology theory is a sequence of functors $H_n : \mathsf{Top} \times \mathsf{Top} \to \mathsf{Ab}$ satisfying some additional properties.

What I don't understand is how a homology functor can be defined on $\mathsf{Top} \times \mathsf{Top}$? In the case of singular homology, we can define $H_n(X, A)$ in the case where $\iota: A \hookrightarrow X$ is a subspace by taking the homology of the cokernel of $S_n(\iota): S_n(A) \to S_n(X)$. How can we extend this to a functor on all of $\mathsf{Top} \times \mathsf{Top}$?

Edit: Looks like it was actually just a mistake on the nLab page.

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    $\begingroup$ The category of pairs is not $\text{Top} \times \text{Top}$. $\endgroup$ – Qiaochu Yuan May 25 '15 at 4:45
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    $\begingroup$ @QiaochuYuan I think that is the reason for this question. I am also a little confused, because also almighty nLab seems to use the same notation, but somehow assuming that there necessarily exists an inclusion $U \hookrightarrow X$ for $(X,U)$, see ncatlab.org/nlab/show/… $\endgroup$ – Daniel Valenzuela May 25 '15 at 4:54
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    $\begingroup$ And also it should be "The category of topological pairs is not Top $\times$ Top $\endgroup$ – Daniel Valenzuela May 25 '15 at 4:56
  • $\begingroup$ @ Qiaochu Yuan that is what it says on the nlab article, which is why I was wondering. $\endgroup$ – ಠ_ಠ May 25 '15 at 5:04
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    $\begingroup$ The nLab is simply wrong, apparently... Since it's a wiki, anyone can fix it, though. By the way, you should look up subobject, it's not an "evil" notion. $\endgroup$ – Najib Idrissi May 25 '15 at 9:26
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As pointed out in the comments, there is an error in the nlab article so in fact the source category for the homology functors should not be $\mathsf{Top} \times \mathsf{Top}$. Instead, the source should be the subcategory whose objects are embeddings $\iota_A: A \hookrightarrow X$, and whose morphisms from $\iota_A: A \hookrightarrow X$ to $\iota_B: B \hookrightarrow Y$ should be continuous maps $f: X \to Y$ such that $f \circ \iota_A$ factors through $\iota_B: B \hookrightarrow Y$.

Equivalently, this can be describe as the category of pairs of spaces $(X, A)$ where $A$ is a subspace of $X$ and a morphism from $(X, A)$ to $(Y, B)$ is a continuous map $f: X \to Y$ such that $f(A) \subseteq B$.

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