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An object $P$ of a category $C$ is a compact projective object if its corepresentable functor $Hom(P,-)\colon C\to Set$ preserves all small sifted colimits.
Equivalently, it is an object that is a compact object ($Hom(P,-)$ preserves all small filtered colimits) and a projective object ($Hom(P,-)$ preserves epimorphisms, which follows from its preservation of coequalizers).
In the category of algebras over an algebraic theory, compact projective objects are retracts of free algebras.
Conversely, if a locally small category has enough compact projective objects (meaning that there is a set of compact projective objects that generates it under small colimits and reflects isomorphisms), then this category is equivalent to the category of algebras over an algebraic theory. Such a category is also known as a locally strongly finitely presentable category
have added hyperlinking to “algebras over”
have added links back to this entry here (i.e. “cross links”) from compact object and from projective object, to give the readers (and us) a chance to know (or remember) that this entry here exists.
I’m not sure that the statement is true in its current form. I’m only aware of the statement that for $C$ an object in a locally small, cocomplete, and Barr-exact category, the functor $Hom(C,-)$ preserves sifted colimits iff it preserves filtered colimits and maps regular epis to surjective functions. This is Corollary 18.3 in the book “Algebraic theories” by Adamek, Rosicky, and Vitale.
In fact I don’t think that $Hom(C,-)$ preserves general epis whenever it preserves sifted colimits even in categories of algebras. A counterexample is that $Ab(\mathbb{Z},-)$ does not preserve the epimorphism $\mathbb{Z}\to\mathbb{Q}$.
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