A note on effectively projective objects (Q5931465)

An object \(P\) in the category \(\mathcal K\) is said to be an effective projective if \(\mathcal K(P,-)\) preserves coequalizers of reflexive graphs. The following conditions are equivalent for a category \(\mathcal K\): (i) \(\mathcal K\) is equivalent to a (finitary, multi-sorted) variety; (ii) \(\mathcal K\) has finite limits, there is a strong generating set \(\mathcal P\) consisting of finitely presentable objects, \(\mathcal K\) has all sums of objects of \(\mathcal P\), effective equivalence relations and the objects in \(\mathcal P\) are regular projectives; (iii) \(\mathcal K\) has finite limits, there is a strong generating set \(\mathcal P\) consisting of finitely presentable objects, \(\mathcal K\) has all sums of objects of \(\mathcal P\), coequalizers of equivalence relations and the objects in \(\mathcal P\) are effective projective (theorem 3). If \(\mathcal K\) is a locally finitely presentable category with \(\mathcal C\) the full subcategory of finitely presentables, then \(\mathcal K\) is a variety if and only if \(\mathcal C\) has enough effective projectives (theorem 4).