Timeline for Image, kernel, quotient and first isomorphism theorem, in a category of monoid objects

7 events

when toggle format	what		by	license	comment
Nov 21, 2015 at 17:24	comment	added	Qiaochu Yuan		@Arrow: I mean you don't need to talk about cokernels in order to state the isomorphism theorems in a way that generalizes to other settings (e.g. rings, where you don't have a zero object).
Nov 21, 2015 at 12:50	comment	added	Arrow		@QiaochuYuan but it seems zero objects are very important if you actually want to get (co)kernels out of (co)equalizers. I don't understand what you mean.
Nov 3, 2014 at 21:10	vote	accept	Sebastien Palcoux
Nov 3, 2014 at 18:53	comment	added	Qiaochu Yuan		@Sébastien: zero objects are not important. The first distinguishing feature of abelian categories is the ability to subtract morphisms: this is what allows you to avoid having to learn about kernel and cokernel pairs in favor of defining kernels and cokernels. The counterexample I wrote down above in commutative rings can just as well be done in commutative monoids in $\text{Set}$ (take a localization again).
Nov 3, 2014 at 11:12	comment	added	Sebastien Palcoux		In what your answer changes whether there is a zero object or not in Mon$_{\mathcal{C}}$?
Nov 2, 2014 at 22:25	history	edited	Qiaochu Yuan	CC BY-SA 3.0	edited body
Nov 2, 2014 at 22:18	history	answered	Qiaochu Yuan	CC BY-SA 3.0