Timeline for $(\infty,1)$-categories and model categories
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 12, 2015 at 14:51 | answer | added | Ilan Barnea | timeline score: 4 | |
| Jun 23, 2010 at 19:35 | history | edited | Charles Rezk | CC BY-SA 2.5 |
backquotes on tex
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| May 23, 2010 at 11:31 | comment | added | David Carchedi | It's worth pointing out that, if you are given a simplicial model category, by taking the full simplicial category on the cofibrant and fibrant objects, you get a simplicial category which is equivalent to the Dwyner-Kan localization and moreover, each enriched Hom(X,Y) is Kan, hence it is a fibrant object in Bergner's model structure on simplicial categories. Therefore, the homotopy coherent nerve caries it to an actual $(\infty,1)$-category. | |
| Feb 26, 2010 at 9:33 | vote | accept | veit79 | ||
| Feb 24, 2010 at 18:17 | comment | added | Charles Rezk | Tyler's answer answers the question in your edit: take a model category C, with weak equivalences W, and form the Dwyer-Kan simplicial localization L(C,W). What you get is a simplicially enriched category, i.e., an infinty-1 category. | |
| Feb 24, 2010 at 12:45 | comment | added | Chris Schommer-Pries | Don't forget the cardinal rule in category theory: look at the morphisms. The classes of fibrations and cofibrations of a Quillen model category are not preserved under Quillen equivalence of model categories and so it is unfair to try to remember that exact structure. In contrast they will have the same induced $(\infty, 1)$-categories. I think, but am not sure, that if you look at the $(\infty,1)$-category of model categories (what you get by (derived) localizing at Quillen equivalences) then the functor F you describe is an embedding. | |
| Feb 24, 2010 at 11:36 | comment | added | veit79 | @Lasergun: Yes. | |
| Feb 24, 2010 at 11:36 | history | edited | veit79 | CC BY-SA 2.5 |
added 841 characters in body
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| Feb 23, 2010 at 23:19 | comment | added | Harry Gindi | Lurie's book is called higher topos theory. I assume that is what you're referencing? | |
| Feb 23, 2010 at 23:08 | answer | added | Reid Barton | timeline score: 9 | |
| Feb 23, 2010 at 22:53 | answer | added | Tyler Lawson | timeline score: 7 | |
| Feb 23, 2010 at 20:58 | comment | added | Andrew Stacey | Obligatory nLab link: ncatlab.org/nlab/show/higher+category+theory is probably a good place to start. | |
| Feb 23, 2010 at 20:14 | history | asked | veit79 | CC BY-SA 2.5 |