Timeline for What is the "universal problem" that motivates the definition of homotopy limits/colimits (and more generally "derived" functors)?
Current License: CC BY-SA 2.5
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| Jun 4, 2015 at 23:45 | comment | added | Aaron Mazel-Gee | Err, I suppose I'm using "(homotopically) terminal" to mean "the spaces of maps to it are all contractible", which goes exactly against your definition. Sorry about that, I hope it's still clear enough what I meant. | |
| Jun 4, 2015 at 23:07 | comment | added | Aaron Mazel-Gee | The answer to your question is no. There is an essentially unique way to extract an "$\infty$-category" from a relative category; in fact, Barwick--Kan defined a model structure on RelCat which is Quillen-equivalent to all the other models of "$\infty$-categories". So, your hope boils down to the claim that if $C$ is an $\infty$-category and $c \in C$ is an object which becomes terminal in $ho(C)$, then it is (homotopically) terminal in $C$ itself. Any nontrivial connected $A_\infty$-space $X$ gives rise to a counterexample, namely $\mathrm{pt}/\!/X$. | |
| Sep 10, 2010 at 17:30 | comment | added | Harry Gindi | This is great (if it works)! | |
| Sep 10, 2010 at 16:02 | history | edited | Tom Goodwillie | CC BY-SA 2.5 |
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| Sep 10, 2010 at 15:22 | history | answered | Tom Goodwillie | CC BY-SA 2.5 |