Timeline for Resolutions by Adapted Class of Objects and Model Categories
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 23, 2011 at 17:30 | answer | added | Emily Riehl | timeline score: 6 | |
| Jun 28, 2011 at 2:15 | comment | added | Mikhail Gudim | Maybe one should ask this question instead: Let $F:A \to B$ be an additive left-exact functor of abelian categories (do not assume that they have enough injectives / projectives. Suppose we are given a class of objects $R$ adapted to $F$. Is there a closed model category structure on $Com^+(A)$ such that weak equivalences are quasi-isomorphisms and $R$ is (or contains) the class of cofibrant objects. If the answer to this question is "yes", then I guess this gives an answer to the original question. But maybe the original question has other solutions...? | |
| Jun 28, 2011 at 2:05 | comment | added | Mikhail Gudim | Derived functors are defined by universal properties without any choices. Their existence is proven using cofibrant replacements which exist by axioms of closed model category (sorry, above by a "model category" I actually meant closed model category) and certain properties of the functor $F$ itself. | |
| Jun 28, 2011 at 1:55 | comment | added | Mikhail Gudim | Here is the definition of class of objects $R$ adapted to a left exact functor $F:A \to B$ (from Gelfand and Manin): $R$ is closed under finite direct sums, $F$ maps acyclic complex from $Com^+(R)$ into an acyclic complex, any object from $A$ is a subobject of an object from $R$. | |
| Jun 27, 2011 at 23:44 | comment | added | David White | This doesn't answer your question, but it's a comment that often it's better to define derived functors so they only depend on $\mathcal{C}$ not on a choice of cofibrant replacement. I suppose your way would allow easier computation but might lose a lot by making that choice. A good reference is this page of Hovey's book: books.google.com/… | |
| Jun 27, 2011 at 23:35 | comment | added | David White | Just to confirm, you're looking at $Ch(\mathcal{A})$ where $\mathcal{A}$ is an abelian category with enough projectives and injectives? Based on the article I linked above, those seem to be prerequisites for your fact about adapted objects being used to compute derived functors. I don't think a model category $\mathcal{C}$ needs to have enough projectives. In general cofibrant replacement is not the same as projective resolution. See e.g. mathoverflow.net/questions/10246/… | |
| Jun 27, 2011 at 23:14 | comment | added | David White | I had to look this up, so I figured I'd comment to help others. Adapted Object: ncatlab.org/nlab/show/class+of+adapted+objects | |
| Jun 27, 2011 at 22:39 | history | asked | Mikhail Gudim | CC BY-SA 3.0 |