Timeline for Understanding the definition of left homotopy as given in Quillen’s Homotopical algebra book
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Oct 19, 2022 at 23:46 | vote | accept | Praphulla Koushik | ||
| Oct 3, 2022 at 19:55 | answer | added | David White | timeline score: 3 | |
| Oct 3, 2022 at 19:47 | history | edited | David White | CC BY-SA 4.0 |
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| Oct 3, 2022 at 18:48 | comment | added | Praphulla Koushik | @AndyPutman yes, that is helpful. So, in case of topological spaces, the 3 out of 4 maps I mentioned above are obvious. So, we mention only one map; the homotopy map. Different in the sense all of them first discuss the notion of cylinder object and then talk about left homotopy… nlab page is ncatlab.org/nlab/show/homotopy+in+a+model+category | |
| Oct 3, 2022 at 18:42 | comment | added | Andy Putman | Here is one remark that might help you. The purpose of $\widetilde{A}$ is to be a replacement for $A \times I$. So for topological spaces, the maps he gives are the obvious ones, e.g. $\sigma\colon A \times I \rightarrow A$ is the projection onto the first factor, and the two maps $\partial_i\colon A \rightarrow A \times I$ are the inclusions onto $A \times 0$ and $A \times 1$. He's trying to sort out exactly what properties you need from these obvious maps when you generalize to other settings. | |
| Oct 3, 2022 at 18:37 | comment | added | Andy Putman | Do they define things differently? I avoid the n-lab since I've never found their explanations useful (it's like they apply a one-way filter to comprehensible mathematics, turning it into something that makes no sense). But all the sources I've read seem more-or-less the same. It might help if you spell out exactly what you think the differences are. | |
| Oct 3, 2022 at 18:16 | comment | added | Praphulla Koushik | @AndyPutman I think you misunderstood what I mean.. “ It's not that "more recent" books (by which I assume you mean more recent books on topology) define things differently. ”.. I don’t mean topology books. I mean books on model categories.. and also in the n-lab.. | |
| Oct 3, 2022 at 18:14 | history | edited | Praphulla Koushik | CC BY-SA 4.0 |
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| Oct 3, 2022 at 17:39 | comment | added | Praphulla Koushik | @AndyPutman I am in initial stages of understanding model categories… I think misunderstanding comes in first half of understanding something… :) I will read that notes. | |
| Oct 3, 2022 at 17:23 | comment | added | Andy Putman | So I think that the fact that you are asking #3 means that you have misunderstood the point of model categories. It's not that "more recent" books (by which I assume you mean more recent books on topology) define things differently. Rather, what Quillen is doing is setting up a machine that works in a vast number of cases beyond simply topological spaces. It wouldn't make sense for a book like Hatcher to write things model-categorically since he is only interested in spaces themselves, not more general things. Again: read Dwyer-Spalinski. | |
| Oct 3, 2022 at 17:19 | comment | added | Praphulla Koushik | @AndyPutman Thansk for the suggestion of that book. I will see that.. I thought the answer for part 2 would not be so simple.. do you have any thing to say for 1 and 3? | |
| Oct 3, 2022 at 17:17 | comment | added | Andy Putman | I think you would probably benefit from reading a more didactic exposition of model categories to help you appreciate why Quillen makes these definitions and how they are related to the more classical ones for topological spaces, which is probably too much for a brief answer. The easiest source I know of is Dwyer-Spalinski's "Homotopy theory and model categories", available here: math.jhu.edu/~eriehl/616-s16/DwyerSpalinski.pdf | |
| Oct 3, 2022 at 17:07 | history | asked | Praphulla Koushik | CC BY-SA 4.0 |