Timeline for Smashing with a cw-complex preserves weak equivalences between well-pointed spaces
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Oct 6, 2012 at 9:04 | comment | added | Christian Wimmer | 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. | |
| Oct 6, 2012 at 0:33 | comment | added | Peter May | 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. | |
| Oct 5, 2012 at 20:47 | comment | added | Christian Wimmer | 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. | |
| Oct 5, 2012 at 20:24 | comment | added | Peter May | Fernando is of course correct, but I preferred to answer in context. | |
| Oct 5, 2012 at 18:46 | history | answered | Peter May | CC BY-SA 3.0 |