Timeline for Are generators defined in Tohoku paper equivalent to that defined in Wikipedia (Which I believe is a more widely used definition)
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 7, 2021 at 15:14 | vote | accept | jy z | ||
| Oct 27, 2020 at 13:26 | comment | added | jy z | Thank uou for all your advice, I have found the way to prove proposition 2. | |
| Oct 27, 2020 at 9:02 | comment | added | Peter LeFanu Lumsdaine | @jyz: To show the “balanced” implication, I recommend rephrasing the “extremal generator” condition to the contrapositive of Grothendick’s version: show it as “given a subobject $B ≤ A$, if all maps from objects $U_i$ into $A$ factor through $B$, then $B = A$”. This now looks similar to the definition of balancedness: “given a subobject $B≤A$, if its map is epi, then $B = A$”. | |
| Oct 26, 2020 at 23:55 | comment | added | Alexander Campbell | That’s not quite right. A morphism $f \colon A \to B$ in a category is an epimorphism if the pre-composition function $Hom(B,C) \to Hom(A,C)$ is injective for every object $C$. The condition you describe on $f$ says that it is a split epimorphism, I.e. that it has a right inverse. | |
| Oct 26, 2020 at 23:26 | comment | added | jy z | Just to check that, in a balanced category, if $f \colon A\rightarrow B$ induces a surjection(in the sense of set theory) from $Hom(C,A)$ to $Hom(C,B)$ for any object $C$, then $f$ is surjective. I believe that it will give me the desired result.(Hope that I am on the right trend) | |
| Oct 26, 2020 at 22:34 | comment | added | Alexander Campbell | The idea is that in a balanced category, if a subobject inclusion $B \to A$ is an epimorphism, then that subobject is not proper. | |
| Oct 26, 2020 at 22:30 | comment | added | jy z | Thank you for your reference, for proposition 1, I proved it successfully, but I was now stuck at proposition 2, I nearly have no idea on how to relate the definition of extremal generators with generators via the definition of balanced category | |
| Oct 26, 2020 at 19:55 | comment | added | Alexander Campbell | I don’t know a reference for these particular statements, but you might find them or something similar in Chapter 4 of Borceux’s Handbook of Categorical Algebra Vol. 1. The proofs of the two propositions I gave are very simple, and I would encourage you to try to prove them for yourself. | |
| Oct 26, 2020 at 16:57 | comment | added | jy z | Thank you for your answer, I would like to know that is there any further reference and proof for the statement? | |
| Oct 26, 2020 at 12:27 | history | answered | Alexander Campbell | CC BY-SA 4.0 |