Timeline for Fields aren't group objects in Ab, so what are they?
Current License: CC BY-SA 3.0
14 events
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| Nov 17, 2013 at 8:06 | comment | added | Martin Brandenburg | "if we want to generalize the definition of a field to other categories similar to CRing" ... we could look for example at Yves Diers book Categories of commutative algebras, where field objects in arbitrary so-called Zariski categories are defined. Actually just by the condition that they are simple. But the nlab article on fields also explains how to internalize the notion of a field using pullbacks. | |
| Nov 17, 2013 at 8:03 | comment | added | Martin Brandenburg | As for the last paragraph: Actually there is a well-known notion of "graded field" (defined for example by the condition that every graded module is graded-free), and every such is isomorphic to $K[x,x^{-1}]$ for some field $K$. | |
| Nov 16, 2013 at 1:46 | comment | added | Qiaochu Yuan | @Kris: sure. I don't think having a zero object is that big a deal anyway. (There's no particular reason to be scared of non-unital rings. In fact the category of non-unital rings is equivalent to the category of augmented rings, which is in some sense a very natural category to look at, e.g. it is the algebraic analogue of looking at pointed spaces.) | |
| Nov 16, 2013 at 0:25 | comment | added | user11863 | @Qiaochu: and relax the zero object to terminal? (I'd just like to avoid non-unital rings at all costs) the "zeroness" only seems to matter in the semisimple case | |
| Nov 15, 2013 at 23:39 | comment | added | Qiaochu Yuan | @Kris: the zero ring is not a zero object in the category of unital rings. It is only the terminal object, while the initial object is $\mathbb{Z}$. | |
| Nov 15, 2013 at 22:48 | comment | added | user11863 | Wouldn't it be a more obvious fix to allow 0 = 1? | |
| Nov 15, 2013 at 22:44 | vote | accept | CommunityBot | ||
| Nov 15, 2013 at 22:02 | comment | added | Fernando Muro | Yes, indeed.... | |
| Nov 15, 2013 at 21:52 | comment | added | Qiaochu Yuan | Hmm. Well, we get a zero object if we move to non-unital rings so this doesn't seem like a big deal. | |
| Nov 15, 2013 at 21:44 | comment | added | Fernando Muro | nLab defines simple objects for categories with zero objects, and the considers the category of unital rings with $0\neq 1$, which has no zero object. | |
| Nov 15, 2013 at 21:40 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Nov 15, 2013 at 21:37 | comment | added | Qiaochu Yuan | By the way, I don't understand the nLab's claim that simple rings aren't the simple objects in $\text{Ring}$. I must be missing something obvious. Can anyone explain this? | |
| Nov 15, 2013 at 21:20 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Nov 15, 2013 at 21:03 | history | answered | Qiaochu Yuan | CC BY-SA 3.0 |