Timeline for Localization of a symmetric monoidal category at a single morphism
Current License: CC BY-SA 3.0
18 events
| when toggle format | what | by | license | comment | |
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| Jan 16, 2013 at 13:15 | comment | added | Martin Brandenburg | Somehow all my questions are still open ... | |
| Apr 5, 2012 at 7:56 | vote | accept | Martin Brandenburg | ||
| Apr 3, 2012 at 20:52 | comment | added | Theo Johnson-Freyd | My understanding is that set-theoretic difficulties come in when inverting a class of morphisms. For instance, inverting all topological maps which are identity on homotopy groups. | |
| Apr 3, 2012 at 20:50 | comment | added | Theo Johnson-Freyd | @David: The point is that Martin is inverting precisely one morphism $f: x \to y$, for fixed $x,y$. Then if you also fix $z$, there is just a set of zig-zags of shape $y \overset{f^{-1}}\to x \to z$, namely the set $\hom(x,z)$. I agree that there is a class of possible $z$s, but the problem is not that $C_f$ has a class of objects, but whether for fixed objects $z,w$ is there a set of zig-zags connecting them. | |
| Apr 3, 2012 at 16:50 | comment | added | David White | @Theo. I hope I'm not misreading your question. The way I normally think of it is that you can't apply the relations you want if the collection is not a set. The collection of all zigzags $z \rightarrow y \rightarrow x \rightarrow \dots \rightarrow w$ is not a set in general. For instance, if $C=Set$ then you have an entire class worth of zigzags just of the form $y\rightarrow x \rightarrow z$, since there should be a class worth of $z$ you can choose. Is this what you were asking? | |
| Apr 3, 2012 at 14:31 | history | edited | Martin Brandenburg | CC BY-SA 3.0 |
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| Apr 3, 2012 at 13:16 | history | edited | Martin Brandenburg | CC BY-SA 3.0 |
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| Apr 3, 2012 at 12:40 | history | edited | Martin Brandenburg | CC BY-SA 3.0 |
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| Apr 3, 2012 at 8:49 | comment | added | Martin Brandenburg | @Ralph: Weibel only talks about left+right multiplicative systems, and $\{f\}$ doesn't have this property. See also my comment after Question 1. | |
| Apr 3, 2012 at 8:32 | comment | added | Ralph | Don't know if it helps, but there is a criterium in Weibel (Intr. Hom. Alg.) 10.3.6 which he calls "locally small multiplicative system". | |
| Apr 3, 2012 at 8:23 | comment | added | Martin Brandenburg | @Theo: Hm, right. But doesn't this contradict Mike's comment? | |
| Apr 3, 2012 at 8:16 | history | edited | Martin Brandenburg | CC BY-SA 3.0 |
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| Apr 3, 2012 at 4:34 | comment | added | Theo Johnson-Freyd | On question 1: Maybe I'm being dumb. How can localizing at a single morphism $f:x\to y$ force you out of the universe? If $z,w$ are objects in $C_f$, a morphism $z\to w$ in $C_f$ is a word whose letters are composable morphisms in $C$ or $f^{-1}$, but modulo relations. After contracting all composable morphisms, you are left with a string of the form $z \to y \to x \to y \to x \to \dots \to x \to w$, where all of the $y \to x$ maps are $f^{-1}$ and all other maps are in $C$. For fixed $z,w$, it seems that there is just a set of these (if $C$ is locally small). | |
| Apr 3, 2012 at 0:42 | answer | added | Peter Arndt | timeline score: 5 | |
| Apr 2, 2012 at 23:23 | answer | added | David White | timeline score: 4 | |
| Apr 2, 2012 at 23:01 | history | edited | David White | CC BY-SA 3.0 |
Fixed a typo, corrected name of a reference
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| Apr 2, 2012 at 21:44 | comment | added | Mike Shulman | I don't think there is a necessary and sufficient condition you can impose on C and f to ensure that the localization is locally small. Model categories are one technique for this; enriched notions of homotopy are another. | |
| Apr 2, 2012 at 21:36 | history | asked | Martin Brandenburg | CC BY-SA 3.0 |