nLab
compact projective object
Context
Category theory
Limits and colimits
limits and colimits

1-Categorical
limit and colimit

limits and colimits by example

commutativity of limits and colimits

small limit

filtered colimit

sifted colimit

connected limit , wide pullback

preserved limit , reflected limit , created limit

product , fiber product , base change , coproduct , pullback , pushout , cobase change , equalizer , coequalizer , join , meet , terminal object , initial object , direct product , direct sum

finite limit

Kan extension

weighted limit

end and coend

fibered limit

2-Categorical
(∞,1)-Categorical
Model-categorical
Contents
Definition
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 .

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

a compact object

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

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

Last revised on May 28, 2022 at 16:19:34.
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