Timeline for Why are injective modules more complicated than projective modules?
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Jul 27, 2012 at 3:28 | comment | added | Qiaochu Yuan | @David: yes, I should've mentioned that. I also made a thinko about injective sets which has been corrected. | |
| Jul 27, 2012 at 3:27 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Jul 27, 2012 at 3:20 | comment | added | David Roberts♦ | Every object of $Set$ is projective iff you assume the axiom of choice. See also arxiv.org/abs/1111.5180, "Are There Enough Injective Sets?", where they authors say that every non-empty set is injective in the category of ZF-sets. | |
| Jul 27, 2012 at 3:16 | comment | added | David Roberts♦ | I guess the adjective 'compact' implicitly means topological, but if one forget there was a topology, then compact = finite. Just being silly :-) don't mind me. | |
| Jul 27, 2012 at 3:14 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Jul 27, 2012 at 3:03 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Jul 27, 2012 at 2:58 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Jul 27, 2012 at 2:49 | comment | added | Qiaochu Yuan | @Steven: ah, gotcha. I'll include an edit addressing this. | |
| Jul 27, 2012 at 2:30 | comment | added | Steven Landsburg | Qiaochu: Yes, I understand that. But I thought that another part of your answer was that in the categories we often choose to look at (module categories) projectives might look simpler, though in the opposite categories it's the injectives that look simpler. That leaves the question of why it's the projectives that are simpler in the categories we're naturally led to look at. | |
| Jul 27, 2012 at 2:09 | comment | added | Qiaochu Yuan | @Steven: part of my answer is that I disagree with the premise of the question (if interpreted as "why are injective objects more complicated than projective objects?") because injectivity and projectivity are dual. Injective objects in $\text{Ab}^{op}$ are exactly as complicated as projective objects in $\text{Ab}$. | |
| Jul 27, 2012 at 2:02 | comment | added | Steven Landsburg | This seems more like an answer to the question "Why are projectives and injectives not equally complicated?", as opposed to "Why are injectives more complicated than projectives?" | |
| Jul 27, 2012 at 1:56 | comment | added | paul garrett | A paraphrase: while reversing arrows is abstractly nothing-at-all, for concrete-ish categories (not necessarily refering, as "concrete" sometimes does, to categories whose objects meaningful underlying sets) the concrete-ish interpretation of the dual category is typically significantly different. The simplest case is that projective $\mathbb Z$-modules are easily given by $\mathbb Z$ and sums thereof, while injective $\mathbb Z$ modules are trickier both to exhibit, and trickier to prove are truly injective. I suppose in a different universe things might be more symmetrical? | |
| Jul 27, 2012 at 1:54 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Jul 27, 2012 at 1:45 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Jul 27, 2012 at 1:38 | comment | added | Qiaochu Yuan | @David: I don't follow. A finite group is also a compact abelian group with the discrete topology. (But thanks for reminding me I forgot to include Hausdorff.) | |
| Jul 27, 2012 at 1:37 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Jul 27, 2012 at 1:35 | comment | added | David Roberts♦ | compact abelian topological groups. But I guess finite groups are interesting too... | |
| Jul 27, 2012 at 1:32 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Jul 27, 2012 at 1:27 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Jul 27, 2012 at 1:05 | history | answered | Qiaochu Yuan | CC BY-SA 3.0 |