By well-pointed i mean that the inclusion of the base point is a h-cofibration, weak equivalences are the usual weak homotopy equivalences between spaces. this is claimed as part of theorem 6.9 (i) in model categories of diagram spectra but as far as i can see without reference. can anyone point me to some place in the literature or indicate where this statement comes from ?
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$\begingroup$ There's an elementary proof by CW-induction, a nice exercise. $\endgroup$– Fernando MuroCommented Oct 5, 2012 at 19:03
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$\begingroup$ What would the induction step look like, smashing source and target with a cellular pushout and then use some sort of gluing lemma ? $\endgroup$– Christian WimmerCommented Oct 5, 2012 at 20:59
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$\begingroup$ Yes, induction on dimension using gluing. It doesn't use function spaces, just stays in the cofibrant side of homotopy theory. $\endgroup$– Fernando MuroCommented Oct 6, 2012 at 9:18
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2 Answers
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I would like to just comment but don't see how. Christian, here's an answer to your last question. For based spaces, you can just do what you want by hand, as Fernando suggested, taking care to use disjoint basepoints to make your attaching maps of $A$ based. You are right to complain that the interplay of $h$ and $q$-model structures is not obvious. Based spaces are of course spaces over and under a point. In ``Parametrized homotopy theory'', Sigurdsson and I generalize to parametrized spaces, which are spaces over and under a give space, and then the combination of $h$, $q$, and related model structures is surprisingly delicate. In that book, the answer to your original question is axiomatized in a general model categorical context in 5.4.1 (see (v)) and the axioms are verified for parametrized spaces in 5.4.9. But that is like hitting a thumb tack with a sledge hammer. Maybe it will help to add that in the direct argument you do need to know that a wedge of weak equivalences is a weak equivalence, and that uses the well-pointed hypothesis.
answered Oct 6, 2012 at 13:51
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$\begingroup$ Thanks again for answering (and to Fernando for suggesting the elementary argument). It's helpful to know where the well-pointed assumption is used. $\endgroup$ Commented Oct 6, 2012 at 17:21
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As usual, I apologize for excess concision. Let $A$ be a based CW complex, $X$ and $Y$ well-pointed spaces, $f\colon X\to Y$ a (weak) equivalence. The first claim in 6.9(i) is that $[X\wedge A,Z] \cong [X,F(A,Z)]$, and a proof is indicated. Since $f$ clearly induces a bijection on the right side, it must induce a bijection on the left side. By Yoneda (take $Z = X\wedge A$ to find an inverse to $f\wedge id$ in the homotopy category), that means that $f\wedge id\colon X\wedge A \to Y\wedge A$ is a weak equivalence.
answered Oct 5, 2012 at 18:46
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$\begingroup$ Fernando is of course correct, but I preferred to answer in context. $\endgroup$ Commented Oct 5, 2012 at 20:24
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$\begingroup$ Ah, thank you. I did not consider this bijection because i don't know how to proof it. One might try to show that if p:C->X is a cofibrant approximation (of diagram spaces or just spaces depending on the setting), then p/\A:C/\A->X/\A is again a cofibrant approximation when X is well-pointed, but this seems like circular reasoning. $\endgroup$ Commented Oct 5, 2012 at 20:47
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2$\begingroup$ Sorry for concision again. The adjunction is best checked model theoretically. We have the $q$-model structure on based spaces. Fixing $A$, we have the evident point-set level adjunction. By a quick use of the adjunction to check the lifting properties, we see that the functor $F(A,-) preserves (Serre) fibrations and acyclic fibrations. Therefore the adjunction is a Quillen adjunction and so descends to homotopy categories. In answer to your question above, yes, and the gluing lemma is itself best proven model categorically. See e.g. Section 17.2 of ``More Concise'' for details. $\endgroup$ Commented Oct 6, 2012 at 0:33
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$\begingroup$ I understand the argument using the Quillen adjunction when the spaces are q-cofibrant (in the pointed Quillen model structure), because then the left derived functor of the adjunction is given by smashing with A (on objects). What i dont understand is how this works under the assumption that X and Y are h-cofibrant, since then one has to q-cofibrantly replace before smashing. $\endgroup$ Commented Oct 6, 2012 at 9:04