Timeline for What are the fibrant objects in the injective model structure?
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 22, 2009 at 22:41 | comment | added | Reid Barton | I got the term "cellular model" from Cisinski's papers. I haven't seen it elsewhere. | |
| Dec 22, 2009 at 21:03 | comment | added | Mark Hovey | @philip314: I was replying to Reid's use of the term "cellular". It is true that this is new usage to me, but maybe Jacob Lurie introduced it. @reid: Take any Bousfield localization; the generating cofibrations are the same, but the generating trivial cofibrations get completely out of control in general. You need a cardinality argument to get them. | |
| Dec 22, 2009 at 1:25 | comment | added | user2146 | I am a bit surprised about the use of "cellular in your post. How does it correspond with the (more usual?) meaning of cellular (cf. ncatlab.org/nlab/show/cellular+model+category). What you write reminds me to a characterisation of cofibrations as retracts of transfinite compositions of push outs of generating cofibrations that is valid in a cofibrantly generated model category with small objects. This characterisation is a (collection of) lemma(-ta) numbered with 2.1.x in Hovey's book. | |
| Dec 22, 2009 at 1:20 | comment | added | Reid Barton | OK, I was worried that might be the case. I was hoping I could get such a cellular model by forming the pushout product of a cellular model for C with the set of horn inclusions. That seems to work at least some of the time. Do you have an example where it fails? | |
| Dec 22, 2009 at 0:58 | comment | added | Mark Hovey | What you need is a cellular model of the levelwise trivial cofibrations, not a cellular model of the monomorphisms. This is in general very difficult, and depends heavily on your category C. | |
| Dec 21, 2009 at 22:18 | comment | added | Reid Barton | Well, you can look in Cisinski's thesis Les préfaisceaux comme modèle des types d'homotopie available at his web site www-math.univ-paris13.fr/~cisinski/publications.html for the terminology (Definition 1.2.26). By Lemma 2.3.2 what I wrote in the first paragraph is true if you replace fibration by acyclic fibration everywhere. But I don't really know whether it's true as written in general. | |
| Dec 21, 2009 at 21:16 | comment | added | Chris Schommer-Pries | Can you elaborate on this? I think I can find the subset S which "generates" the monomorphism in the cases I'm thinking about (which you can probably guess, Reid). Do you know any references for this besides HTT? | |
| Dec 21, 2009 at 18:15 | history | answered | Reid Barton | CC BY-SA 2.5 |