Timeline for The definition of a group object is wrong?
Current License: CC BY-SA 3.0
16 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| May 31, 2011 at 15:31 | answer | added | Qiaochu Yuan | timeline score: 1 | |
| May 31, 2011 at 14:57 | vote | accept | Qiaochu Yuan | ||
| May 30, 2011 at 21:30 | comment | added | Andrew Stacey | My first reaction to this is to say: "You silly twisted boy, you". | |
| May 30, 2011 at 19:12 | answer | added | Ryan Reich | timeline score: 4 | |
| May 30, 2011 at 18:50 | comment | added | Qiaochu Yuan | @Andrej: yes, I considered this as well, but it's not completely clear to me that the inverse of a generalized point ought to be a generalized point as opposed to a "generalized anti-point." In any case there appears to be something interesting going on here and perhaps someone knows the appropriate terminology for it. | |
| May 30, 2011 at 18:48 | comment | added | Ryan Reich | You don't need the functor of points to prove the uniqueness of the inverse, though, you just do the diagram-chasing version of $h = h1 = h(gk) = (hg) k = 1k = k$ to show that inverses $h$ and $k$ of the same $g$ are equal. However, I prefer the functor of points for this as well. | |
| May 30, 2011 at 18:44 | comment | added | Qiaochu Yuan | @Ryan: yes, the functor of points is precisely the reason I always thought the usual definition of a group object made sense. On the other hand if one thinks of inversion as a special case of adjunction in $2$-categories then $\text{Vect}$ (thought of as a $2$-category with one object) starts to seem like evidence against this. | |
| May 30, 2011 at 18:22 | comment | added | Ryan Reich | I also want to note that inversion is only seemingly a structure even when given as a morphism. If you interpret the structure of a monoid object as being a factorization of the functor of points through the category of monoids, then the property of being a group object is the property of that factorization taking values in the full subcategory of groups. This gives a natural inversion on the functor of points and, thus, on the object itself. On the category side, the inversion map (if it exists) is unique, so its existence is merely a property. | |
| May 30, 2011 at 18:21 | comment | added | Andrej Bauer | Global points are evil, as you note. So we should take generalized points in your "correct" definition, but then we get back the usual notion of group. The vector space example looks to me more like some sort of a $*$-autonomous category. | |
| May 30, 2011 at 18:08 | comment | added | Sridhar Ramesh | Note that the example with (not necessarily commutative) rings is just like your vector-space example, in that the antimorphisms are the contravariant functors: specifically, viewing a ring as a one-object, abelian-group enriched category, the desired anti-morphisms between rings are the contravariant enriched functors (as the notion of "opposite ring" is just a special case of the general notion of "opposite category"). | |
| May 30, 2011 at 17:48 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| May 30, 2011 at 17:41 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
added 533 characters in body; added 2 characters in body
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| May 30, 2011 at 17:39 | comment | added | Qiaochu Yuan | Oh, right. That makes sense. | |
| May 30, 2011 at 17:36 | comment | added | Ryan Reich | Presumably, F fixes the terminal object 1, so Hom(1, G) = Hom(F(1), F(G)) = Hom(1, F(g)), right? | |
| May 30, 2011 at 17:26 | history | asked | Qiaochu Yuan | CC BY-SA 3.0 |