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Let $\mathcal{A}$ be a category. If $\mathcal{A}$ is pointed, i.e. has zero objects, then $f$ monic $\implies \operatorname{ker} f = 0$. If $\mathcal{A}$ is abelian, we have the equivalence $f$ monic $\iff \operatorname{ker} f = 0$. However this is true in $\mathsf{Gp}$, which is not abelian. When in general (or more generally) is this true?

It is true when $\mathcal{A}$ is conormal, or more generally when the coequaliser of any two morphisms is a cokernel, but do we need this?

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    Related: http://math.stackexchange.com/questions/53405/conditions-for-monic-iff-injective2012-06-09
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    Thanks @JuanS that is useful.2012-06-09
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    I think the forward implication is always true, and the reverse is true in normal categories2012-06-09
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    @JuanS do you mean conormal? In a normal category if $f$ is a mono, then indeed it is a kernel, but if I take any $f$ with kernel $0$ it won't necessarily be a mono. However conormality would ensure that the coequaliser of any two morhisms would be a cokernel, which would do it.2012-06-09
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    For categories of algebraic structures, you probably need to have some kind of Mal'cev operation in your structure.2012-06-09
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    @Paul: The forward implication is given in 'Mitchell - Theory of Categories' on page 15. Exercise 9 is to prove that in a normal category with equalizers, a morphism is an epimorphism if and only if its cokernel is 0. Thus it seems I did mean conormal! I should leave the details up to the experts...2012-06-09

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