Timeline for Why do we need model categories?
Current License: CC BY-SA 3.0
20 events
| when toggle format | what | by | license | comment | |
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| Jan 2, 2018 at 8:08 | comment | added | Aaron Mazel-Gee | I think there's some really great usage of model categories for explicit computations in the world of synthetic differential geometry -- I learned about this in Urs Schreiber's short note ncatlab.org/schreiber/files/dcct170811.pdf. It has also been necessary to use model categories to study algebras over "necessarily algebraic" operads (such as the Lie and Poisson operads), though as of fairly recently there is a theory of enriched $\infty$-operads (arxiv.org/abs/1707.08049). | |
| Dec 6, 2017 at 19:07 | answer | added | Edouard | timeline score: 7 | |
| Dec 5, 2017 at 3:03 | review | Close votes | |||
| Dec 5, 2017 at 21:59 | |||||
| Dec 3, 2017 at 11:54 | comment | added | the L | One reason we need model categories: to do calculations in the homotopy category of a model category. That is, given a category and a set of morphisms we wish to invert, if you know that these morphisms are the weak equivalnces of a model category, it becomes much easier to do calculations in the localized category. | |
| Nov 30, 2017 at 12:27 | answer | added | მამუკა ჯიბლაძე | timeline score: 7 | |
| Nov 29, 2017 at 2:19 | answer | added | Dmitri Pavlov | timeline score: 17 | |
| Nov 27, 2017 at 19:10 | answer | added | AAK | timeline score: 24 | |
| Nov 27, 2017 at 19:05 | comment | added | Asaf Karagila♦ | For the same reasons we need model citizens, I guess. :) | |
| Nov 27, 2017 at 17:00 | answer | added | user13113 | timeline score: 14 | |
| Nov 27, 2017 at 15:54 | review | Close votes | |||
| Nov 27, 2017 at 17:37 | |||||
| Nov 27, 2017 at 15:43 | comment | added | Simon Henry | In my opinion these are different questions: The other ones was explicitly about whether or not $\infty$-categories could replace model categories, which is a rather different issue than why do we need model categories in the first place, why this specific definition and whether they appears outside of algebraic topology. | |
| Nov 27, 2017 at 15:37 | comment | added | David White | Possible duplicate of Do we still need model categories? | |
| Nov 27, 2017 at 15:36 | answer | added | David White | timeline score: 10 | |
| Nov 27, 2017 at 15:35 | history | edited | Qfwfq | CC BY-SA 3.0 |
edited title
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| Nov 27, 2017 at 15:33 | comment | added | Dylan Wilson | (contd) It has become increasingly accepted that a suitable candidate for what "homotopy theories" are is the notion of an $\infty$-category. Please note I am making no comment or claims on the other parts of your question. | |
| Nov 27, 2017 at 15:33 | comment | added | Dylan Wilson | I would like to address the "philosophical" part of your question. People often forget about 0.3-0.4 in the introduction to Quillen's book inventing model categories: he himself distinguishes between a "model category" and the "homotopy theory" it models, hence the name! He then goes on to lament the absence of a satisfactory theory for what the actual "homotopy theory" should be. I think he viewed his theory just as a convenient formalism for carrying over arguments from homotopy theory to other contexts. (contd) | |
| Nov 27, 2017 at 15:28 | answer | added | Simon Henry | timeline score: 18 | |
| Nov 27, 2017 at 15:25 | answer | added | Lennart Meier | timeline score: 38 | |
| Nov 27, 2017 at 15:06 | comment | added | Ali Caglayan | Peter May gives an excellent answer to this question that might be worth reading. | |
| Nov 27, 2017 at 14:50 | history | asked | Megan | CC BY-SA 3.0 |