Processing math: 100%
Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: compact projective object

Bottom of Page

1 to 7 of 7

  1.  
    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 27th 2022

    Created:

    \tableofcontents

    Definition

    An object P of a category C is a compact projective object if its corepresentable functor Hom(P,−):C→Set preserves all small sifted colimits.

    Equivalently, it is an object that is a compact object (Hom(P,−) preserves all small filtered colimits) and a projective object (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 C an object in a locally small, cocomplete, and Barr-exact category, the functor 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,−) preserves general epis whenever it preserves sifted colimits even in categories of algebras. A counterexample is that Ab(ℤ,−) does not preserve the epimorphism ℤ→ℚ.

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 28th 2022

    Corrected the statement.

    diff, v3, current

1 to 7 of 7

  1.