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I am a bit lost understanding some subtleties in various form of epimorphy.

The nLab reports that an effective epimorphism is one that coequalizes its kernel pair. A regular epimorphism is simply one that coequalizes some pair of parallel morphisms.

effective $\rightarrow$ regular

A regular epimorphism with a kernel pair must coequalize it so that

effective $\leftrightarrow$ regular w/ kernel pair

Since kernel pairs are pullbacks (of the epi along itself), in any category with pullbacks effective and regular coincide.

I would then look for an example of a Category with kernel pairs but not necessarily all pullbacks and wonder what can be deduced for epimorphisms in such a category. The point is, I would like to nail down the minimal requirements for regular epimorphisms to be effective and for effective epimorphisms to split an idempotent.

I am aware of the AnnoyingPrecision blog posts, but they still resort to regularity assumptions (which kind of defies the purpose imho).

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    $\begingroup$ Please use MathJax **MathJax**, not $\textbf{TeX trickery}$ $\textbf{TeX trickery}$, for formatting. I edited accordingly. $\endgroup$ Commented Jan 20 at 18:34
  • $\begingroup$ I think "coequalizes" was intended to mean "is the coequalizer of". In that case, a morphism that has a kernel pair need not "coequalize" it. $\endgroup$ Commented Jan 20 at 19:33
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    $\begingroup$ What is your question? As you say, if a morphism has a kernel pair then it is effective epi iff it is regular epi, so in a category with all kernel pairs the two notions coincide. And what exactly is your question about split idempotents? $\endgroup$ Commented Jan 21 at 0:38

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