Timeline for Constructions unique up to non-unique isomorphism
Current License: CC BY-SA 2.5
11 events
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| Jan 30, 2011 at 21:40 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
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| Jan 30, 2011 at 20:46 | comment | added | Harry Altman | Come to think of it, algebraic closure exhibits this dual characterization as well. It's the maximal algebraic extension or the minimal algebraically closed extension. It may seem a bit silly but it is valid. And we often define algebraic closure to be the extension that's both algebraic and algebraically closed just as injective hull is the extension that's both essential and injective. Chris Heunen points out below that bases of vector spaces meet the condition and well, we all know that one. Don't see how universal covers could be fit in though. Minimal simply connected makes sense how? | |
| Jan 30, 2011 at 16:09 | comment | added | Qiaochu Yuan | @Theo: the definition I'm working off of is just an object such that every object has at least one morphism out of (resp. at least one morphism into) it. I should amend my last comment: without some kind of restriction on the morphisms one only gets a category of weak objects (although one can always consider its core). In both of the above examples, the corresponding category has the property that the only endomorphisms of an object are automorphisms, and that's enough to get a groupoid of weak terminal/initial objects, I think. | |
| Jan 30, 2011 at 15:58 | comment | added | Theo Buehler | I don't understand your last comment. It seems to me that this is due to the fact that after all I'm not so sure what the good definition of "weak initial/terminal" object is, anyway. | |
| Jan 30, 2011 at 15:17 | comment | added | Qiaochu Yuan | @Steven: I guess that would give a groupoid which is not connected, but it's still possible to restrict attention to each of its connected components. | |
| Jan 30, 2011 at 15:00 | comment | added | Steven Landsburg | "You will get this kind of behavior in any situation where you have a weak universal object instead of a universal one" --- this is so only when the weak universal objects are all isomorphic, no? | |
| Jan 30, 2011 at 14:27 | comment | added | Harry Altman | Not to mention it's also the weak terminal object in the category of essential extensions of M (with embeddings for the maps). | |
| Jan 30, 2011 at 14:24 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
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| Jan 30, 2011 at 14:18 | comment | added | Theo Buehler | Nice observation! In fact, the Galois group of modules alluded to in example 3) is also of this type: it's the automorphism group of the weak initial object given by the injective hull inside the category of embeddings (monomorphisms) $M \to I$ into an injective, so it's analogous to example 1) and dual to 2). | |
| Jan 30, 2011 at 14:16 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
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| Jan 30, 2011 at 14:07 | history | answered | Qiaochu Yuan | CC BY-SA 2.5 |