Timeline for Does there exist a full and faithful embedding of $\mathsf{Poset}$ in $\mathsf{Set}$?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 23, 2018 at 0:48 | comment | added | Jesse Elliott | I thought @JulianRosen's answer was neat, too! A more philosophical question: why do so many people still seem to think that all mathematical objects are sets? The category of topological spaces with morphisms as homotopy classes of continuous functions is not even concretizable. | |
| Dec 21, 2018 at 12:21 | answer | added | Julian Rosen | timeline score: 27 | |
| Dec 21, 2018 at 8:36 | comment | added | Todd Trimble | A philosophy is that any full and faithful embedding $C \to D$ realizes the objects of $C$ as special types of $D$-objects ($D$-objects satisfying a certain property); see ncatlab.org/nlab/show/stuff,+structure,+property. Thus posets would have to be certain types of sets. This leads to considerations like the nice answers by Julian and Dylan. | |
| Dec 21, 2018 at 7:00 | comment | added | Dylan Wilson | Julian’s answer is better, but also: there are infinitely many isomorphism classes of posets with a unique automorphism, but only two such sets. | |
| Dec 21, 2018 at 4:13 | comment | added | David Roberts♦ | @JulianRosen you should add this as an answer, it's neat :-) | |
| Dec 21, 2018 at 3:03 | comment | added | Julian Rosen | The poset $\{0,1\}$ with $0<1$ has three endomorphisms. There is no set with exactly three endomorphisms. | |
| Dec 21, 2018 at 2:59 | history | asked | Jesse Elliott | CC BY-SA 4.0 |