Timeline for Arrows, furnished by Yoneda
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Sep 29, 2022 at 17:34 | comment | added | Alec Rhea | @fosco Interesting! I flagged the question for conversion to CW since there is no one correct intended answer; if you’d like to elaborate a bit on your example in an answer I’d be interested to read more! | |
| Sep 29, 2022 at 12:03 | comment | added | fosco | I've been busy elsewhere :) now I see what you aim to gain! I'm surprised no one mentioned cohomology operations, en.wikipedia.org/wiki/Cohomology_operation that are usually defined as natural transformations between cohomology functors, and through Yoneda lemma equated to certain cohomology classes of classifying space: it's possible to say something in the latter picture, but good luck working with it for too long! | |
| Sep 29, 2022 at 9:02 | history | made wiki | Post Made Community Wiki by Asaf Karagila♦ | ||
| Sep 29, 2022 at 7:32 | history | edited | Alec Rhea | CC BY-SA 4.0 |
corrected typo
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| Sep 29, 2022 at 1:10 | history | edited | Alec Rhea | CC BY-SA 4.0 |
added 1 character in body
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| Sep 28, 2022 at 22:11 | answer | added | Qiaochu Yuan | timeline score: 8 | |
| Sep 28, 2022 at 18:36 | history | edited | Alec Rhea | CC BY-SA 4.0 |
added details in response to fosco's comment
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| Sep 26, 2022 at 23:57 | comment | added | Alec Rhea | (that identity needs to hold for all $i$ and $j$, obviously, but that shouldn't make it any more obtuse unless I'm missing something) | |
| Sep 26, 2022 at 23:50 | comment | added | Alec Rhea | @fosco It seems like showing naturality basically boils down to showing that for any arrow $g:A\to B$ in $\mathcal{C}$ and any arrow $([h_{ij}],[h'_{ij}]):B\to R^{n\times k}\times R^{k\times m}$ we have $$(\sum_{\ell<k}h_{i\ell}h'_{\ell j})\circ g=\sum_{\ell<k}(h_{i\ell}\circ g)(h'_{\ell j}\circ g)$$ which should be pretty straightforward, no? Without the Yoneda route we need to copy the matrices a bunch of times, hunt around for the appropriate row/column shuffling isomorphisms, collapse everything back down with pointwise operations, etc. unless I'm missing an easier route? | |
| Sep 26, 2022 at 21:16 | comment | added | fosco | Isn't checking that your $f$ is natural equally tedious than the "impractically long" definition? (And strictly speaking you also need to check something else: that the definition of $f_U$ is associative and unital in the appropriate sense). I'm not saying this is a bad idea, I'm just asking, what exactly do you gain? | |
| Sep 26, 2022 at 19:55 | history | asked | Alec Rhea | CC BY-SA 4.0 |