Timeline for What is a cogroup and what are coactions?
Current License: CC BY-SA 4.0
22 events
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| Apr 5 at 18:42 | comment | added | Jonathan Beardsley | It's maybe worth mentioning that in the ∞-categorical context (and probably elsewhere, I just don't know of it) there is a notion of "universal coendomorphism object," and so a map of coalgebras Coend(X)→C is indeed a coaction. See, for instance, Aras Ergus' thesis here: aergus.net/academic/documents | |
| Apr 3 at 22:48 | comment | added | David Roberts♦ | @DavidWhite why not just write 0 instead of {0}? The latter is not the initial object in Set, but the terminal object. | |
| Apr 3 at 21:50 | comment | added | David White | @DavidRoberts I think, for a beginner, it's best to work in pointed sets, so we disallow the empty group. So, the initial and terminal objects agree. But, I agree with you that, once you move to a general cocartesian category, this distinction matters, so I edited accordingly. Thanks! | |
| Apr 3 at 21:49 | history | edited | David White | CC BY-SA 4.0 |
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| Apr 3 at 21:24 | comment | added | David Roberts♦ | The coproduct of no objects is not {0} but the initial object... | |
| Apr 3 at 14:01 | history | edited | David White | CC BY-SA 4.0 |
Fixed typo
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| Apr 3 at 13:19 | vote | accept | CommunityBot | ||
| Apr 3 at 11:56 | comment | added | Paul Taylor | Yes, writing MO answers at 3am isn't a good idea. You have changed the wrong place. | |
| Apr 3 at 11:34 | comment | added | Nikola Tomić | There is still a typo for the unit you wrote $\varnothing$ instead of $*$ I think. | |
| Apr 3 at 11:27 | history | edited | David White | CC BY-SA 4.0 |
added 215 characters in body
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| Apr 3 at 11:26 | comment | added | David White | @PaulTaylor Yes, you're right. I fixed it, and also clarified the situation in categories other than Set. I think this also answers the first point David Roberts raised, and I fixed the typo of writing "identity" instead of "inverse." Chalk it up to being startled awake and then thinking it was a good idea to dash off a MO answer at 3am. | |
| Apr 3 at 10:11 | comment | added | David Roberts♦ | Why do you insist to use the coproduct for a cogroup? If {0} is the codomain of the comultiplication, then you should use the product, not coproduct. And your identity and coidentity should be inverse and coinverse. [edit: corrected autocorrect-induced typo] | |
| Apr 3 at 8:44 | comment | added | HenrikRüping | Cant you also dualize the notion of a category to a cocategory, where you have a cocomposition of morphisms and then the (co?)automorphisms of each (co?)object form a cogroup. Then maybe there is an analogous statement. | |
| Apr 3 at 8:01 | comment | added | Paul Taylor | I think you mean the initial object ($\emptyset$ in $\mathbf{Set}$), not $\{0\}$, unless you intend to be specific to Abelian categories. | |
| Apr 3 at 7:51 | vote | accept | CommunityBot | ||
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| Apr 3 at 7:51 | vote | accept | CommunityBot | ||
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| Apr 3 at 7:48 | comment | added | YCor |
It's not a personal taste, just standard typing. $Hom$ ($Hom$) is formatted as a string of individual letters, like in $e=mc^2$, so is ill-formated, with random spacing between letters, its only reason to be frequent is indeed that it's the quickest to be typed.
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| Apr 3 at 7:40 | comment | added | David White | @YCor Ok, I resized coproduct. I personally don't see a big issue with using $Hom$, etc., but if you hate it feel free to do the declare math thing. It's 3:40am here and I'm going to bed! | |
| Apr 3 at 7:39 | history | edited | David White | CC BY-SA 4.0 |
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| Apr 3 at 7:30 | comment | added | YCor |
$\coprod$ ($\coprod$) is sized as a operator, not as a binary operator, where $\sqcup$($\sqcup$) might be more suitable. (I assume it means coproduct in the category of sets [=disjoint union], not of pointed sets [=disjoint union gluing basepoints] or groups [free product]). By the way you can start the post with $\DeclareMathOperator\Hom{Hom}$ and then use \Hom, and similarly \Aut.
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| Apr 3 at 7:25 | history | edited | David White | CC BY-SA 4.0 |
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| Apr 3 at 7:16 | history | answered | David White | CC BY-SA 4.0 |