Timeline for What do people mean by "subcategory"?
Current License: CC BY-SA 2.5
15 events
| when toggle format | what | by | license | comment | |
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| May 13, 2010 at 16:21 | vote | accept | Ben Wieland | ||
| May 12, 2010 at 23:35 | comment | added | Pete L. Clark | @Ben: I was confused. I don't want groups to be a subcategory of sets either. See my response below for clarification. | |
| May 12, 2010 at 23:33 | answer | added | Pete L. Clark | timeline score: 3 | |
| May 12, 2010 at 22:54 | answer | added | Theo Johnson-Freyd | timeline score: 8 | |
| May 12, 2010 at 22:41 | comment | added | Peter LeFanu Lumsdaine | @Xandi: That's surely no crazier than the fact that $\mathbb{Q}$ "is" a subset of $\mathbb{N}$ in the categorical sense; it's just another illustration of how "$A$ is a sub-foo of $B$" is almost always a statement not just about $A$ and $B$ but also about an implicit inclusion map. :-P | |
| May 12, 2010 at 22:38 | comment | added | Ben Wieland | Pete Clark, if you want groups to be a subcategory of sets, do you want (as in nLab or PLL's answer) each discrete category (ie, sets) to be a subcategory of the terminal category? | |
| May 12, 2010 at 22:20 | answer | added | Peter LeFanu Lumsdaine | timeline score: 2 | |
| May 12, 2010 at 21:59 | answer | added | some guy on the street | timeline score: 4 | |
| May 12, 2010 at 21:34 | answer | added | Tilman | timeline score: 6 | |
| May 12, 2010 at 21:34 | comment | added | Peter LeFanu Lumsdaine | The question "should Grp really be a subcategory of Set?" is a problem under either definition: with a bit of hacking, we can find a category equivalent to Grp as a Mac Lane subcat of Set. Define, say, an "ersatz group" to be a set X containing a unique element x (call it the "code" of X) such that x is a pair (X\{x},\mu), where \mu is a multiplication making X\{x} a group; and a map or these is a function preserving the code and giving a group homomorphism on the rest. I like your question, but I don't think the "Grp" example really motivates it as well as one might initially think. | |
| May 12, 2010 at 21:02 | answer | added | Andrew Stacey | timeline score: 5 | |
| May 12, 2010 at 20:08 | comment | added | Xandi Tuni | Haha, it's even crazier than that: <Groups> is a subcategory of <Sets>, but also, <Sets> is a subcategory of <Groups>, via the "free group generated by X"--functor. | |
| May 12, 2010 at 19:58 | comment | added | Pete L. Clark | "I don't want groups to be a subcategory of sets." Why not? (I do.) | |
| May 12, 2010 at 19:39 | history | asked | Ben Wieland | CC BY-SA 2.5 |