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The Eilenberg Steenrod axioms are functors on the homotopy category of pairs of "spaces" $(X, A)$. Typically they are introduced when $X$ and $A$ are some sort of topological spaces. My question is can "spaces" be taken to mean objects in other categories, like for example an arbitrary model category $C$?

First, we would need to define what a pair $(X, A)$ means, where $X, A \in Ob(C)$. My guess/attempt at a definition is $(X, A)$ is a pair if there is a map $A \to X$ that is a cofibration in $C$. What would the empty set (in $Top$) correspond to in $C$? And then what about an excissive triad $(X; A, B)$?

I presume this has all been figured out; I am not really familiar with algebraic topology. Any references?

ykm
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    So, this might be a little much, as the nlab tends to be, for a newcomer to algebraic topology, but a brief discussion of cohomology theories in general is discussed at http://ncatlab.org/nlab/show/generalized+(Eilenberg-Steenrod)+cohomology, though it's in the language of $\infty$-categories. – Jonathan Beardsley Aug 30 '13 at 21:53
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    @JonBeardsley yeah, that's exactly what I was looking for. thanks! – ykm Sep 02 '13 at 06:59
  • Awesome! Glad that's helpful. – Jonathan Beardsley Sep 02 '13 at 15:02

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