Timeline for How to think about model categories?
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 24, 2009 at 17:12 | comment | added | Ilya Nikokoshev | @Rezk: indeed! I stumbled on this myself. | |
| Oct 24, 2009 at 16:35 | comment | added | Charles Rezk | There are several answers and comments here that suggest that "CW-complexes" is a model category. This isn't quite right (model categories need to be closed under (at least finite) limits and colimits). The correct statement is: topological spaces have a model category where weak equivalences are weak equivalences, fibrations are Serre fibrations, and the class of cofibrant objects includes CW-complexes. | |
| Oct 23, 2009 at 21:53 | comment | added | Ilya Nikokoshev | I wish I could accept both answers, yours and Tolland's -- the answers nicely complement each other :) | |
| Oct 23, 2009 at 21:51 | comment | added | Ilya Nikokoshev | Dold-Kan: yes, and that's quite readable in Lurie, I recommend it. | |
| Oct 23, 2009 at 21:45 | comment | added | Eric Wofsey | Incidentally, the case of chain complexes and derived categories is actually a special case of "simplicial X". By the Dold-Kan theorem, simplicial objects in an abelian category are equivalent to (nonnegatively graded) chain complexes, and the standard model structure on simplicial stuff is then the standard projective model structure on chain complexes. In this way you can think of more general simplicial objects as versions of chain complexes and derived categories for nonabelian categories. | |
| Oct 23, 2009 at 21:12 | comment | added | Charles Siegel | Ahh. I've not been clear on that point. Thanks Eric. And Ilya, my understanding is that there are quite a few model categories, but those are the paradigm examples, and the ones that people care most about, but people also want to see which properties of homotopy follow completely algebraically, sort of like how they want to understand what properties of (co)homology follow algebraically, so we look at things like abelian categories, exact and derived functors, and the Eilenberg-Steenrod axioms. | |
| Oct 23, 2009 at 21:09 | comment | added | Eric Wofsey | Derived categories themselves are not model categories; they are the homotopy categories of a model structure on the category of chain complexes (the same way there is a homotopy category of spaces). | |
| Oct 23, 2009 at 21:09 | comment | added | Ilya Nikokoshev | Indeed, those are the typical examples, but I'm sure there would be no need for the notion if these were the only two. :) | |
| Oct 23, 2009 at 21:05 | history | answered | Charles Siegel | CC BY-SA 2.5 |