Timeline for Definition of "simplicial complex"
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 17 at 19:32 | comment | added | Allen Hatcher | An updated link to the appendix of my book is pi.math.cornell.edu/~hatcher/AT/ATapp.pdf | |
| Mar 14 at 13:59 | comment | added | Calvin Khor | (The above link to Allen's appendix is broken; here is a copy on the Internet Archive) | |
| Apr 30, 2015 at 12:42 | comment | added | Ingo Blechschmidt | @user4676: It is true that the category of semisimplicial sets has all limits and all colimits. However, these do not behave as one would geometrically expect. This can already be seen with empty limits: The terminal object is not the one-point space, but an infinite-dimensional sphere. The deeper reason for this bad behaviour is the paucity of morphisms in the category of semisimplicial sets. Also note that the geometric realization functor from semisimplicial sets to topological spaces does not preserve (co-)limits. | |
| Apr 27, 2010 at 11:42 | comment | added | user4676 | The appendix is nicely written. It seems to me that the main reason for introducing simplicial sets is: $\Delta$-complexes (= contravariant functors from the category $\Delta'$ of order-preserving injections to the category of sets) "do not allow quotient constructions". How does this go well with the fact that the category of functors $\Delta'op\to Sets$ has colimits? | |
| Nov 21, 2009 at 3:22 | comment | added | S. Carnahan♦ | It's an illuminating exercise to work out by hand the product of two 1-simplices as a simplicial set. The nondegenerate 2-simplices seem to appear by magic. | |
| Nov 21, 2009 at 2:18 | comment | added | Alicia Garcia-Raboso | A nice account of the build-up from simplicial complexes to simplicial sets passing by Delta complexes is given in the excellent "An elementary illustrated introduction to simplicial sets", available at arxiv.org/abs/0809.4221 | |
| Nov 21, 2009 at 0:45 | comment | added | Greg Kuperberg | Also, since this is the accepted answer, here again is a Allen's appendix: math.cornell.edu/~hatcher/AT/ATsimplicial.pdf | |
| Nov 21, 2009 at 0:43 | comment | added | Greg Kuperberg | Degeneracy maps simply guarantee that the degenerate simplices exist. They are important because when you have a simplicial map, the simplices that collapse ought to be sent to something. The degenerate simplices also provide the automatic subdivision of the Cartesian product of two simplices. Finally in a simplicial group or similar, the non-degenerate simplices are poorly behaved: They are subsets that are not subgroups. | |
| Nov 20, 2009 at 23:05 | vote | accept | Kevin H. Lin | ||
| Nov 20, 2009 at 21:53 | comment | added | Peter Arndt | Good choice in your book (which is great anyway): When introducing simplicial sets, people rarely ever bother to motivate the degeneracy maps (and I don't find it easy to see what they are good for), so I find it more honest to go for Delta complexes until you need more structure... | |
| Nov 20, 2009 at 19:02 | history | answered | Allen Hatcher | CC BY-SA 2.5 |