Timeline for Reference request for category theory works which quickly prove the theorem which generalises the 1st isomorphism theorem for groups/rings/...
Current License: CC BY-SA 2.5
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 2, 2011 at 1:38 | vote | accept | teil | ||
| May 24, 2010 at 15:33 | answer | added | castal | timeline score: 0 | |
| May 23, 2010 at 20:59 | answer | added | Buschi Sergio | timeline score: 3 | |
| May 16, 2010 at 22:43 | answer | added | Martin Brandenburg | timeline score: 2 | |
| May 15, 2010 at 20:09 | answer | added | S. Carnahan♦ | timeline score: 6 | |
| May 15, 2010 at 18:43 | comment | added | Tim Perutz | A variant of Pete's comment: the "isomorphism theorem" for sets is the observation that a map $f:A\to B$ induces a bijection from a quotient of $A$ (where elements are identified if they map to the same element of $B$) to $im(f)$. If you're interested in some category of structured sets and structure-preserving maps (groups, topological spaces, etc.) you just have to ask yourself: can you form the quotient and image in the category? Is the bijection a morphism when $f$ is? Is its inverse a morphism? | |
| May 15, 2010 at 17:09 | comment | added | Sean Rostami | like Xandi Tuni said, in category theory the isomorphism theorem is just the axiom that says "the kernel of the cokernel is the cokernel of the kernel" | |
| May 15, 2010 at 16:11 | comment | added | user2734 | Perhaps instead of category theory, you should look at some basic book on universal algebra, for example, you can try part 3 of Cohn’s algebra. | |
| May 15, 2010 at 15:14 | comment | added | Pete L. Clark | I'm having trouble understanding your motivation. You can simultaneously prove the first isomorphism theorem for groups, rings and modules just by understanding the proof in any one of the cases: the arguments are all the same. | |
| May 15, 2010 at 15:06 | comment | added | S. Carnahan♦ | What specific category theory work reaches the first isomorphism theorem after hundreds of pages? | |
| May 15, 2010 at 14:36 | comment | added | Xandi Tuni | What is the first isomorphism theorem for fields? Also, I don't think categories are a framework to prove such things, from my point of view the isomorphism theorems are usually taken as axioms for a certain type of categories, say abelian categories, and then one proves that e.g. vector spaces form an abelian category. | |
| May 15, 2010 at 14:33 | comment | added | teil | As I don't know the theory, it's very difficult for me to tell what to skip, I don't know which results depend on which. | |
| May 15, 2010 at 14:23 | comment | added | Regenbogen | But you don't have to read all these leading hundreds of pages to read the particular proof you want. . | |
| May 15, 2010 at 14:20 | history | asked | teil | CC BY-SA 2.5 |