Timeline for On a corollary in Mitchell's book
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 8, 2012 at 23:53 | vote | accept | Ralph | ||
| Mar 10, 2011 at 9:38 | comment | added | Steve Lack | OK, thanks Ralph, I've now given a fix. | |
| Mar 10, 2011 at 9:38 | history | edited | Steve Lack | CC BY-SA 2.5 |
added 650 characters in body
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| Mar 10, 2011 at 8:45 | comment | added | Ralph | No, this isn't part of the definition. Informally, Mitchell's exact category is just a plain framework to work reasonable with exact sequences, i.e. such that $A \to B \to 0$ is exact iff $A \to B$ is epi, $0 \to A \to B$ is exact iff $A \to B$ is mono, etc. For instance, an abelian category is an exact additive category. | |
| Mar 10, 2011 at 8:09 | comment | added | Steve Lack | Ralph: yes, of course, you're quite right, this argument needs the category to be enriched over abelian groups, not just over pointed sets. Is this part of the Mitchell definition? | |
| Mar 10, 2011 at 7:10 | comment | added | Ralph | For the sake of completeness: A category is exact (in the sense of Mitchell) if (i) kernels and cokernels exist, and (ii) each monomorphism is a kernel and dually each epimorphism is a cokernel, and (iii) each morphism can be written as a composition of an epimorphism followed by a monomorphism. | |
| Mar 10, 2011 at 6:59 | vote | accept | Ralph | ||
| Mar 10, 2011 at 6:59 | |||||
| Mar 10, 2011 at 6:59 | comment | added | Ralph | Steve, thanks for the argument, like it. In fact, it shows: If the category is Ab (Hom's are abelian groups with bilinear composition), $p$ is the cokernel of $i$ and $q$ the cokernel of $j$ then the statement is true. However, if there is no additive structure on the Hom's your argument doesn't seem to work. For, then we start with $fqi = f^'qi$ and the universal property of $p$ couldn't be used in the same way as above. | |
| Mar 10, 2011 at 2:25 | history | answered | Steve Lack | CC BY-SA 2.5 |