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Let $(S,f_1,\ldots,f_n)$ be an algebra of some variety and $(T,g_1,\ldots,g_n)$ be another algebra of the same variety. Next let $\varphi:S\to T$ be a homomorphism. I understand well that $\ker\varphi=\{(x,y)\in S^2:\varphi(x)=\varphi(y)\}$, all dandy so far! My question is however for the cokernel, is there a meaningful way to define it in universal algebra. I have tried googling about it and looking in my books but cannot find any definition for it. Does one exist or is it too difficult to define here? I know already in groups it is a bit troublesome.

Zelos Malum
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1 Answers1

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Yes. Your kernel construction is known in category theory as the kernel pair. There is an exactly categorically dual construction called the cokernel pair. (But depending on what you want to do it may not be the right construction to look at.)

Qiaochu Yuan
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  • Would you happen to know the definition for it? – Zelos Malum May 21 '16 at 01:18
  • Isn't that just the kernel again? I was wondering about the cokernel. – Zelos Malum May 21 '16 at 14:08
  • @Zelos: sorry, I misspoke (but no, I described a quotient of $T$ and the kernel is a subset of $S^2$). The cokernel pair is the pushout of $\varphi$ along itself. – Qiaochu Yuan May 21 '16 at 17:10
  • Yeah how do you define it more concretely in universal algebra terms? – Zelos Malum May 21 '16 at 17:15
  • @ZelosMalum I believe this is known as the "free product with amalgamation". It is in general much more complicated than the kernel pair construction (due to the fact that it involves among other things free algebras), so it is probably not exactly "fun" to right this down explicitely. – Stefan Perko May 25 '16 at 16:42