Timeline for products in a category without reference to objects or sources and targets
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 29 at 14:52 | vote | accept | Ben Sprott | ||
| Jun 26, 2011 at 10:40 | answer | added | Wouter Stekelenburg | timeline score: 4 | |
| Jun 25, 2011 at 13:34 | comment | added | Ryan Reich | @Ben: (continued) You might argue that I forgot to mention the identity arrows. That's true; if I mentioned them, it would be exactly the same as just telling you what all the objects are. Then the "words" that you want, which express composability, would tell me what the source and targets are for each arrow (just check which identity arrows something is composable with). There is no escaping it. | |
| Jun 25, 2011 at 13:32 | comment | added | Ryan Reich | @Ben: you can't present a category as just words and equations. For example, here are two categories presented normally: C has two objects x and y, and Hom(x,y) has two arrows f and g; D has four objects a, b, x, y and Hom(a,b) and Hom(x,y) each have a single arrow f, g respectively. In your language, there are only two letters, namely f and g, no other words, and no equations in either category. So how do you tell C from D? | |
| Jun 25, 2011 at 13:28 | comment | added | Ryan Reich | There is also this question: mathoverflow.net/questions/33541/…. | |
| Jun 25, 2011 at 11:43 | comment | added | Ben Sprott | Qiaochu, By "the same thing" I think we would both mean that two different presentations are actually presenting theories of the same "thing" ie the theory of categories. Recall, I am presenting a category as a long list of words and equations. In this kind of presentation, you "know" that two morphisms, $a,b$ can be composed if anywhere in the list of you see a word with a subword $ab$. You know two words $ab$, $cd$ can be composed if anywhere in the list of equations there is a word with subword $bc$. | |
| Jun 25, 2011 at 10:22 | comment | added | Tom Goodwillie | Qiaochu: You mean "understand which pairs of morphisms can be composed", right? | |
| Jun 25, 2011 at 5:14 | answer | added | Martin Brandenburg | timeline score: 3 | |
| Jun 25, 2011 at 4:30 | comment | added | Qiaochu Yuan | I don't understand what you mean by "without reference to objects or sources and targets." If you understand how to compose morphisms, that's the same thing as understanding objects, sources, and targets. | |
| Jun 25, 2011 at 2:58 | history | asked | Ben Sprott | CC BY-SA 3.0 |