Timeline for When does a "representable functor" into a category other than Set preserve limits?
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15 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| May 10, 2010 at 20:13 | comment | added | user2734 | @Qiaochu Yuan: Please note that I have corrected the part concerning your dynamical systems example. | |
| May 5, 2010 at 15:14 | vote | accept | Qiaochu Yuan | ||
| May 5, 2010 at 12:51 | answer | added | user2734 | timeline score: 7 | |
| May 1, 2010 at 18:59 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
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| May 1, 2010 at 18:19 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
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| May 1, 2010 at 18:17 | comment | added | Qiaochu Yuan | @Tilman: I didn't know that - thanks! I'll give a different example. | |
| May 1, 2010 at 13:03 | comment | added | Tilman | My comment was referring to OP's motivating example. | |
| May 1, 2010 at 10:48 | comment | added | Martin Brandenburg | @unknown (google): no you're right. | |
| May 1, 2010 at 10:10 | comment | added | user2734 | @Tilman: My comments referred only to the first part of the question. Is there something wrong with what I said? | |
| May 1, 2010 at 8:17 | comment | added | Tilman | The problem with your example is that the homotopy category doesn't even have limits. And even if you try to lift this into a model-category setting and use the homotopy limit, $\pi_1$ doesn't send those to limits (take a pullback of a fibration as an example, where you get a Mayer-Vietoris-like sequence involving all $\pi_i$). | |
| Apr 30, 2010 at 23:07 | comment | added | user2734 | (...cont'd) But $F\tau$ is such a lift, and hence we're done. Now, if I am not mistaken, for any $\tau$-algebra the forgetful functor to $\mathbf{Set}$ creates limits. (I once proved it for solving some exercise, but I'm not sure if my proof is correct, since it has been quite a while since I did it). | |
| Apr 30, 2010 at 23:07 | comment | added | user2734 | If the forgetful functor creates limits, then I think you get what you want. In detail: Let $J$ be an index category, and let $J\stackrel{T}{\to}C\stackrel{F}{\to}D\stackrel{U}{\to}\mathbf{Set}$ be functors. Suppose that $UF$ preserves limits and $U$ creates limits. Suppose that $\tau\colon \ell\to T$ is a limiting cone in $C$. Since $UF$ preserves limits, $UF\tau\colon UF\ell\to UFT$ is a limiting cone. As $U$ creates limits, there is a unique lifting of $UF\tau$ to a cone in $D$, and this cone is a limiting cone. (to be continued...) | |
| Apr 30, 2010 at 22:41 | history | edited | Harry Gindi | CC BY-SA 2.5 |
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| Apr 30, 2010 at 22:32 | history | asked | Qiaochu Yuan | CC BY-SA 2.5 |