Timeline for Definition of internal field objects
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Mar 13, 2015 at 23:07 | comment | added | Martin Brandenburg | $T \to X$ is a morphism in $\mathcal{C}$, obviously. $R$ is a ring object, not $X$. It is important that $T$ is an arbitrary object, not just $1$. | |
| Mar 13, 2015 at 13:07 | comment | added | sure | I'm not sure that your pullback condition is enough to have everyone but $0$ non invertible. Do you obviously require that $T \rightarrow X$ to be a morphism of ring object? Or maybe it is enough to have your pullback square for all $1 \rightarrow X$ arrows different than the $0: 1 \rightarrow X$ element? | |
| Mar 3, 2015 at 21:40 | history | edited | Martin Brandenburg | CC BY-SA 3.0 |
added 390 characters in body
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| Mar 3, 2015 at 19:22 | answer | added | Qiaochu Yuan | timeline score: 9 | |
| Mar 3, 2015 at 19:20 | answer | added | Paul Taylor | timeline score: 4 | |
| Mar 3, 2015 at 16:28 | comment | added | Laurent Moret-Bailly | Never thought about this before, but it does sound right. So, for instance, for any scheme $S$, $\mathbb{A}^1_S$ is a field object in the category of $S$-schemes. | |
| Mar 3, 2015 at 11:35 | comment | added | Zhen Lin | @მამუკაჯიბლაძე No, I don't think Martin's definition is especially similar to the notion of geometric field (which nLab calls "discrete field"). It's much closer to the definition via axiom F2. | |
| Mar 3, 2015 at 11:03 | comment | added | Zhen Lin | It appears to me your definition is a translation of "$x$ is invertible in a field if and only if $x \ne 0$", where you have interpreted $x \ne 0$ using the notion of a complementary subobject. This is a natural definition, I suppose. | |
| Mar 3, 2015 at 10:22 | comment | added | David Roberts♦ | Actually, I wonder if the condition "The pullback of $T\to X \xleftarrow{0} 1$ is initial" defines a tight apartness relation [1] on $X$? [1] ncatlab.org/nlab/show/apartness+relation | |
| Mar 3, 2015 at 10:02 | comment | added | მამუკა ჯიბლაძე | Several possible versions (most of them needing substantially more than finite limits to formulate) are studied in Johnstone's "Rings, Fields and Spectra" (J. Algebra 49, 1977, 238-260). The version you are interested in must be more or less what Johnstone calls geometric field. | |
| Mar 3, 2015 at 10:01 | comment | added | David Roberts♦ | Seems like the definition of a residue field object ncatlab.org/nlab/show/field | |
| Mar 3, 2015 at 9:41 | comment | added | S. Carnahan♦ | This might be the discussion you remember: mathoverflow.net/questions/3003/… | |
| Mar 3, 2015 at 9:17 | history | asked | Martin Brandenburg | CC BY-SA 3.0 |