An object of a category is a compact projective object if its corepresentable functor preserves all small sifted colimits.
In the case of cocomplete Barr-exact categories, it is equivalently an object that is
( preserves all small filtered colimits),
( preserves regular 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
Last revised on May 28, 2022 at 16:19:34. See the history of this page for a list of all contributions to it.