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Can you make the category whose objects are pairs of spaces $(X,A)$, and morphisms the obvious diagrams, into a model category? Of course I want this to be done in a meaningful way, that is, agreeing with the adjoint functors $X\mapsto (X,\emptyset)$ and $(X,A)\mapsto X$?

There might be some intuitive reason that it is wrong to expect this, but I don't see it yet.


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Are you restricting $A$ to be a subspace? If you don't, then you can use a model structure on the category of functors $\{\cdot \to \cdot\} \to X$, such as the projective or injective model structure. – Tyler Lawson Apr 28 '13 at 5:35
That sounds interesting! Is there a good reference for that? Do you know what goes wrong if I restrict to subspaces or $A\to X$ a cofibration? – Joseph Victor Apr 28 '13 at 6:16
up vote 7 down vote accepted

The answer has to be no. There's just no good homotopy theoretic way to talk about subspaces, because up to homotopy every map is an inclusion (via the mapping space construction). So even before you run into the completeness issue Karol raises you have a more fundamental issue of what the homotopy category would be. It seems to me that there's no good category theoretic/homotopy theoretic way to pick out a space and a subspace of it; the closest you can do is Tyler's comment and Karol's answer. Since neither of them mentioned this, I'll mention that it's also known as the Arrow Category $Arr(C)$ (in this case $Arr(Top)$). It's a well-studied object, but on the surface seems very different from what you were asking for. However, the lens of homotopy can't see the difference between an object in $Arr(Top)$ and an inclusion, so I guess that's the best you can do. The other obvious idea (taking the product model structure on $Top \times Top$ and then placing some restriction so the only pairs $(X,A)$ you get have $A\subset X$) fails for the same reason.

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I feared as much, ah well. – Joseph Victor Apr 29 '13 at 20:19

As Tyler Lawson points out you can use the category of all diagrams on $[1]$. Then the projective and injective model structures are both instances of Reedy model structures. This is discussed in Section 5.2 of Hovey's Model Categories and will work with both Quillen's and Strøm's model structures. (In fact it works with completely arbitrary model category.)

I'm not sure what happens if you want to restrict to the category of pairs of spaces. However, if you insist that your pairs are cofibered, then the first problem you run into is that an uncountable product of (Hurewicz or Serre) cofibrations is not necessarily a cofibration again. So I suspect that his category might not be complete.

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It's also worth pointing out that in the projective model structure on the category of arrows the cofibrant objects are exactly cofibrations between cofibrant objects of the original model category. – Karol Szumiło Apr 28 '13 at 8:47
I just picked u the Hovey book. Thanks for the reference! – Joseph Victor Apr 29 '13 at 20:19

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