Timeline for Is there a high-concept explanation for why "simplicial" leads to "homotopy-theoretic"?
Current License: CC BY-SA 2.5
21 events
| when toggle format | what | by | license | comment | |
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| S Jun 2, 2017 at 2:26 | history | suggested | Colectivo |
Added homotopy tag
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| Jun 2, 2017 at 2:05 | review | Suggested edits | |||
| S Jun 2, 2017 at 2:26 | |||||
| Jan 21, 2017 at 22:30 | answer | added | Ronnie Brown | timeline score: 5 | |
| Mar 16, 2011 at 3:23 | answer | added | Emerton | timeline score: 29 | |
| Mar 16, 2011 at 1:43 | answer | added | Peter Arndt | timeline score: 17 | |
| Mar 15, 2011 at 17:17 | comment | added | Akhil Mathew | @Harry: thanks for the explanation! | |
| Mar 15, 2011 at 17:12 | vote | accept | Akhil Mathew | ||
| Mar 15, 2011 at 13:15 | answer | added | Jacob Lurie | timeline score: 106 | |
| Mar 15, 2011 at 12:49 | comment | added | Harry Gindi | @John: We can say that for $K$ ranging over the class of all CW complexes or cubical complexes as well. | |
| Mar 15, 2011 at 11:42 | comment | added | John Klein | @Akhil: a short comment. The relevance and ubiquity of $\Delta$ is closely related to the fact that a weak (homotopy) equivalence $X \to Y$ of topological spaces is detected by taking homotopy classes of the form $[K,-]$ where $K$ is a simplicial complex. If we used homotopy equivalence instead, we would not be so lucky. | |
| Mar 15, 2011 at 10:22 | comment | added | Harry Gindi | Because it's not exactly precise and I didn't want to spend the time to make it precise =(. | |
| Mar 15, 2011 at 10:11 | comment | added | Ketil Tveiten | Harry, why didn't you just type that up as an answer? | |
| Mar 15, 2011 at 9:53 | comment | added | Harry Gindi | It's really that simplicial sets form a better category than topological spaces. It is completely possible (up to problems with bousfield-localization type stuff) to replace simplicial presheaves with presheaves of spaces, and simplicially enriched categries with categories enriched in spaces as well, but this will just make our lives harder. | |
| Mar 15, 2011 at 9:49 | comment | added | Harry Gindi | The reason why we use simplicially-enriched categories, simplicial presheaves, etc. is that simplicial objects form a topos and their model structure is cartesian and proper. This makes them much easier to enrich over than, say the category of topological spaces, which is, up to homotopy, pretty defective in comparison. We could do everything with sheaves of topological spaces and topologically-enriched categories, but topological spaces are not combinatorial, they do not form a topos, they're only cartesian closed if you $k$-ify everything all the time. | |
| Mar 15, 2011 at 9:42 | comment | added | Harry Gindi | As for why one would do all of this simplicial stuff, it's purely technical and an "accident" of history. It turns out that the homotopy theory of cubical sets is very hard to get right, and also, Dan Kan realized that the singular complex associated with a space has a lifting property with respect to particular class of maps resembling cubical versions of the cofibration-equivalences of Serre, but again, once we have some cubical intuition, we can actually figure out what is going on at the level of simplicial sets. | |
| Mar 15, 2011 at 9:38 | comment | added | Harry Gindi | However, cubical complexes are triangulable, and so we now have a simplicial model for the space. But because the homotopy theory of CW complexes was totally combinatorial in nature, we see how the combinatorial information of the associated cubical/simplicial complex determines the the homotopy type. This means that we can work with actual combinatorial objects instead of models of combinatorial objects as spaces. However, to get this information without the substrate it was grown on, we need maps to keep track of the degenerate simplices. | |
| Mar 15, 2011 at 9:31 | comment | added | Harry Gindi | Let $T$ be a space, and let $CT$ be a CW approximation of $T$. The idea is that the homotopy theory of CW complexes is totally combinatorial in nature. We can always replace homotopies with straight line homotopies, etc. Then realize that spheres and cubes are identical and notice that we can straight-line homotope attaching maps to actual cubical maps in a way that preserves the homotopy type of the CW complex. We notice, however, that all cubical complexes are still CW complexes, and that we can further approximate CW complexes by cubical complexes. | |
| Mar 15, 2011 at 8:59 | comment | added | Gjergji Zaimi | I feel some of the answers and comments from mathoverflow.net/questions/691/simplicial-objects are relevant here as well. Especially the test categories viewpoint, the comparison of "homotopical mathematics" vs. "ordinary mathematics", and the other reason for (co)simplicial objects to appear in "nature" mentioned in Reid's comment. | |
| Mar 15, 2011 at 8:47 | answer | added | Jonathan Chiche | timeline score: 44 | |
| Mar 15, 2011 at 2:34 | answer | added | Qiaochu Yuan | timeline score: 15 | |
| Mar 15, 2011 at 2:24 | history | asked | Akhil Mathew | CC BY-SA 2.5 |