Timeline for Is it reasonable to define `poset homotopy' as a `natural transformation of posets'?
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| Apr 14, 2011 at 20:04 | comment | added | Mike Shulman | @John: Natural transformations are not necessarily invertible, so the relation of "the existence of a natural transformation" is not symmetric. Also, it's true that if you invert the functors which induce weak homotopy equivalences of nerves, then you get a homotopy category equivalent to that of spaces, but I don't think that's the same thing as quotienting by the equivalence relation generated by natural transformations; not all objects in the Thomason model structure are fibrant and cofibrant. | |
| Apr 11, 2011 at 23:38 | comment | added | John Klein | Using posets is too restrictive. If one replaces posets by (small) categories, then "natural transformation" is an equivalence relation. Furthermore, with respect to this definition, the homotopy category of (small) categories is equivalent to the homotopy category of spaces. | |
| Apr 11, 2011 at 20:29 | comment | added | Mikola | Thanks for the response! I am still in the process of trying to teach myself algebraic topology and end up spending a lot of time blundering around trying to relate it back to things that I already know. Maybe another way to do this construction would be to more directly parallel the usual definition by replacing the cartesian product with [0,1] by a cartesian product with a span; {0, t, 1} where 0 < t and 1 < t. Of course at this point I am probably grasping at straws and so I should perhaps just stop now and try to think a bit more before I say anything else... | |
| Apr 11, 2011 at 20:21 | vote | accept | Mikola | ||
| Apr 11, 2011 at 20:03 | history | answered | Dylan Thurston | CC BY-SA 3.0 |