Timeline for A canonical and categorical construction for geometric realization
Current License: CC BY-SA 2.5
16 events
| when toggle format | what | by | license | comment | |
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| Aug 27, 2010 at 8:55 | answer | added | Todd Trimble | timeline score: 10 | |
| Aug 25, 2010 at 21:51 | comment | added | some guy on the street | I'm getting a sense, especially from remarks about recovering the unit interval from nonsense, that an unasked question is "why work in $Top$?" --- btw, the unit interval seems to be a terminal coalgebra for a reasonable functor, as observed by Peter Freyd. | |
| Aug 25, 2010 at 21:18 | answer | added | some guy on the street | timeline score: 0 | |
| Aug 25, 2010 at 20:06 | answer | added | Grigory M | timeline score: 0 | |
| Jun 20, 2010 at 13:56 | history | edited | David Carchedi | CC BY-SA 2.5 |
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| Jun 19, 2010 at 12:43 | comment | added | Steven Gubkin | Lets think about the "geometric realization of the nerve" functor from Cat to Top. Each of these are 2-categories, with the arrow and the interval playing an important role in each respectively (the 2-morphisms are determined by maps from $I \to B^A$ in both cases). I am not really conversant with higher category theory yet. But say we could come up with a nice reason why this (psuedo?)functor should be taking the arrow to the interval. Would that be good enough for you? It would reduce the question to "Why homotopy with the unit interval?". | |
| Jun 18, 2010 at 23:45 | answer | added | David Carchedi | timeline score: 1 | |
| Jun 18, 2010 at 1:25 | comment | added | David Carchedi | @Harry, the singular complex functor USES the definition of the standard n-simplices. But, having the definition of each standard n-simplex (and the maps between them) gives us the functor $\Delta \to Top$ which if we left-Kan extend we get geometric realization. So, this doesn't gain us anything. | |
| Jun 17, 2010 at 17:25 | comment | added | Harry Gindi | Well, you can construct it as an adjoint to the total singular complex functor (which you can define by hand). | |
| Jun 17, 2010 at 16:09 | answer | added | Tom Goodwillie | timeline score: 8 | |
| Jun 17, 2010 at 13:29 | history | edited | David Carchedi | CC BY-SA 2.5 |
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| Jun 17, 2010 at 9:48 | comment | added | David Carchedi | As a cute "observation", we can use the topology on the unit interval as a "seed" to produce all the standard n-simplices. The topologically-enriched free commutative monoid on the unit interval is the disjoint union all the standard n-simplices. If this somehow fit in, it would be nice. | |
| Jun 17, 2010 at 9:45 | comment | added | David Carchedi | I wouldn't be surprised, but, perhaps I have too much faith in category theory. As a side note, really you want the topology of the compact unit interval to be spat out, not the reals. | |
| Jun 16, 2010 at 17:13 | answer | added | Steven Gubkin | timeline score: 6 | |
| Jun 16, 2010 at 15:01 | comment | added | Steven Gubkin | It would be very surprising to me if a nice categorical construction could start off with finite linear orders, increasing functions, and somehow spit out the topology of the real numbers. | |
| Jun 16, 2010 at 12:45 | history | asked | David Carchedi | CC BY-SA 2.5 |