nLab compact projective object


Category theory

Limits and colimits



An object PP of a category CC is a compact projective object if its corepresentable functor Hom(P,):CSetHom(P,-)\colon C\to Set preserves all small sifted colimits.

In the case of cocomplete Barr-exact categories, it is equivalently an object that is

  1. a compact object

    (Hom(P,)Hom(P,-) preserves all small filtered colimits),

  2. a projective object

    (Hom(P,)Hom(P,-) 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.