Timeline for What is the earliest definition given by a universal mapping property?
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 5, 2010 at 2:21 | comment | added | some guy on the street | eep! should have said "much later than the definition of supremum"... can you tell I'm a bit distracted today? | |
| May 5, 2010 at 1:52 | comment | added | Steven Gubkin | T ${ }$ | |
| May 5, 2010 at 1:52 | comment | added | Steven Gubkin | @KConrad: Note the following hack: | |
| May 5, 2010 at 0:55 | comment | added | KConrad | I see. [adding more to make this comment postable] | |
| May 4, 2010 at 23:39 | comment | added | some guy on the street | I'm really not clear on what you're getting at. True, the categorical way of thinking about posets (or even the abstract notion of a poset) come much later than the definition of a category or a universal object; all I mean by what I've said, is that the Supremum as originally defined is literally an object with a specific universal mapping property, despite the fact Cauchy/Dirichlet/Dedekind/Riemann weren't thinking in those terms when whoever first wrote the definition. BTW, if the Questioner accepts my answer, I'll delete it; this question shouldn't have a "final answer". | |
| May 4, 2010 at 21:14 | comment | added | KConrad | OK some guy, but how does this relate to the question seeking a concept that was abstractly defined chronologically earliest by a universal mapping property? I'd think an answer to the original question would be an example with a lot more going on than a supremum on a poset, i.e., an example where the universal mapping property point of view provides an essential way to understand what the concept "really means". | |
| May 4, 2010 at 20:32 | comment | added | some guy on the street | eep! I should say uniqueness, existence would mean $X$ is a supremum, and in any poset suprema might not exist... this is why colimits and limits "are" limits, btw. | |
| May 4, 2010 at 20:31 | comment | added | some guy on the street | The idea --- and it's pretty standard in introducing Categories --- is that any partial order is a (free) category with at most one morphism between any two objects, called $\leq$. The supremum of a set $S$, if it exists, is an object $X$ with $s\leq X$ (a specified structure map $s\to X$) for $s\in S$ such that for any given object $Y$ and maps $s\leq Y$ there exists a unique map $X\leq Y$ such that $s\leq X\leq Y$ is $s\leq Y$. In this poset case, existence and uniqueness both follow from the at-most-one property. | |
| May 4, 2010 at 18:56 | comment | added | KConrad | What does that mean? There is no need to conceive of a supremum by a universal mapping property, whatever that might be, to think about and work with it. | |
| May 4, 2010 at 18:08 | history | answered | some guy on the street | CC BY-SA 2.5 |