Timeline for Homology theory constructed in a homotopy-invariant way
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Jul 24, 2012 at 21:40 | comment | added | Yosemite Sam | @DC Cisinski: could please expand a little bit on your comment? | |
| Jul 24, 2012 at 20:55 | history | edited | David White | CC BY-SA 3.0 |
Fixed typos, since it was on the front-page anyway
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| Jan 28, 2012 at 1:31 | comment | added | Spice the Bird | If you live in a based category, you might consider \textit{based} simplicies. What I mean by a based n-simplex is the standard n-simplex, with the vertices identified. Even though the n-simplices are not homotopically interesting, the based simplices are. They are all $K(\pi,1)''s$ where the group is a free group on the number of letters given by the dimension. | |
| Jan 27, 2012 at 21:54 | comment | added | D.-C. Cisinski | If you are happy to speak the language of $(\infty,1)$-categories, singular homology is the unique cocontinuous functor which sends the point to $\mathbf{Z}$. This characterization is also a construction of singular homology (which might be seen as a tiny baby version of the theory of Kan extensions in $(\infty,1)$-categories). | |
| Jan 27, 2012 at 17:43 | answer | added | Ronnie Brown | timeline score: 1 | |
| Oct 22, 2011 at 12:48 | comment | added | Tyler Lawson | @Guillaume, if you're looking for general ways to define "homology" that share some of the same properties you might look into either Andre-Quillen homology or Goodwillie calculus. | |
| Oct 22, 2011 at 9:36 | comment | added | Guillaume Brunerie | Thank for your answers, I think my question was rather to define homology in a way that would work in any $(\infty,1)$-topos (seen as a generalization of the homotopy category), rather than try to characterize the homotopy category. And I think Eilenberg-MacLane spaces and smash products are always definable there, so Tyler's answer and Dan's comment (for cohomology) are what I was searching for. | |
| Oct 22, 2011 at 9:26 | vote | accept | Guillaume Brunerie | ||
| Oct 18, 2011 at 16:05 | answer | added | Tyler Lawson | timeline score: 21 | |
| Oct 18, 2011 at 14:04 | answer | added | Fabian Lenhardt | timeline score: 9 | |
| Oct 18, 2011 at 13:12 | comment | added | Dan Petersen | You could construct a $K(\mathbf{Z},n)$ directly and define $H^n(X)$ as homotopy classes of maps into it. But I guess this is not what you are looking for. | |
| Oct 18, 2011 at 12:33 | history | asked | Guillaume Brunerie | CC BY-SA 3.0 |