Timeline for The definition of a group object is wrong?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Aug 24, 2012 at 2:43 | comment | added | Toby Bartels | Warning: The term ‘heteromorphism’ is also used in another sense: given an adjunction $F: C \rightleftarrows D :G$, an object $x$ of $C$, and an object $y$ of $D$, an $(F,G)$-heteromorphism from $x$ to $y$ is $C$-morphism from $x$ to $G(y)$, or equivalently a $D$-morphism from $F(x)$ to $y$. For example, using free groups and underlying sets, a heteromorphism from a set $x$ to a group $y$ is what ordinary mathematicians would call a ‘function’ from $x$ to $y$. Heteromorphisms cannot be composed with each other, but they are acted on by (homo)morphisms on either side. | |
| May 31, 2011 at 15:03 | comment | added | Qiaochu Yuan | The conceptual difficulty that motivated the first question has passed, so I don't need an example anymore. The situation is just that sometimes group objects in the usual sense form too small a collection of examples to include the ones of interest, so one weakens the definition as above. For example apparently all group objects in the usual sense in the category of Poisson manifolds have trivial Poisson bracket, and this is a shame, so one forgets the bracket. | |
| May 31, 2011 at 14:57 | vote | accept | Qiaochu Yuan | ||
| May 30, 2011 at 20:33 | comment | added | Ryan Reich | There's also an obvious concept of a "weak $(\cat{C}, S)$-group object", for lack of a better term, in which we give a monoid in $\cat{C}$ which is a group in $\cat{D}$ but with its inverse not necessarily lifting at all. I have no idea what that entails. | |
| May 30, 2011 at 20:30 | history | edited | Ryan Reich | CC BY-SA 3.0 |
fix
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| May 30, 2011 at 20:22 | comment | added | Ryan Reich | @Qiaochu: Yes. Basically, I have in mind that $\cat{C}$ is a "concrete in $\cat{D}$ category", like Poisson manifolds in sets or some such. @Ben: Of course! It's perfect. I shall edit immediately. | |
| May 30, 2011 at 19:56 | comment | added | Ben Webster♦ | By the way, when we discussed this (actually at tea) years ago, I think we decided an "anti-morphism" should obviously be called a "heteromorphism." | |
| May 30, 2011 at 19:30 | comment | added | Qiaochu Yuan | Oh, I see. This is quite nice. We need $S$ to preserve products, though, right? | |
| May 30, 2011 at 19:30 | history | edited | Ryan Reich | CC BY-SA 3.0 |
added 185 characters in body
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| May 30, 2011 at 19:12 | history | answered | Ryan Reich | CC BY-SA 3.0 |