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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 27th 2022

    Created:

    \tableofcontents

    Definition

    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.

    Equivalently, it is an object that is a compact object (Hom(P,)Hom(P,-) preserves all small filtered colimits) and a projective object (Hom(P,)Hom(P,-) preserves 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

    Related concepts

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 28th 2022

    have added hyperlinking to “algebras over

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 28th 2022
    • (edited May 28th 2022)

    turned page name into singular form

    diff, v2, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 28th 2022

    I found the sentence starting with

    Equivalently, it is an

    hard to read. Have broken it up into a little numbered list, to make it easier on the eye.

    diff, v2, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 28th 2022

    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.

    diff, v2, current

    • CommentRowNumber6.
    • CommentAuthorJonasFrey
    • CommentTimeMay 28th 2022
    • (edited May 28th 2022)

    I’m not sure that the statement is true in its current form. I’m only aware of the statement that for CC an object in a locally small, cocomplete, and Barr-exact category, the functor Hom(C,)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,)Hom(C,-) preserves general epis whenever it preserves sifted colimits even in categories of algebras. A counterexample is that Ab(,)Ab(\mathbb{Z},-) does not preserve the epimorphism \mathbb{Z}\to\mathbb{Q}.

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 28th 2022

    Corrected the statement.

    diff, v3, current

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