nLab compact projective object

Definition

An object $P$ of a category $C$ is a compact projective object, also known as strongly of finite presentation, if its corepresentable functor $Hom(P,-)\colon C\to Set$ preserves all small sifted colimits.

Thus $Hom(P,-)$ preserves all small filtered colimits (i.e., it is a compact object) and all reflexive coequalizers. The converse holds if $C$ is finitely cocomplete, see Adamek, Rosicky, Vitale, Theorem 2.1.

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

1. a compact object

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

2. a projective object

   ($Hom(P,-)$ preserves regular epimorphisms, which follows from its preservation of coequalizers).

Examples

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

References

• Kęstutis Česnavičius and Peter Scholze, Purity for Flat Cohomology

• Jiri Adamek, Jiri Rosicky, On sifted colimits and generalized varieties, TAC 8 (2001) pp 33–53. (web)

Related concepts

• sifted colimit

• algebraic theory

• locally strongly finitely presentable category