Projective objects in abelian categories #
In an abelian category, an object P
is projective iff the functor
preadditiveCoyonedaObj (op P)
preserves finite colimits.
noncomputable def
CategoryTheory.preservesFiniteColimitsPreadditiveCoyonedaObjOfProjective
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.Abelian C]
(P : C)
[hP : CategoryTheory.Projective P]
:
The preadditive Co-Yoneda functor on P
preserves finite colimits if P
is projective.
Equations
- CategoryTheory.preservesFiniteColimitsPreadditiveCoyonedaObjOfProjective P = (CategoryTheory.preadditiveCoyonedaObj { unop := P }).preservesFiniteColimitsOfPreservesEpisAndKernels
Instances For
theorem
CategoryTheory.projective_of_preservesFiniteColimits_preadditiveCoyonedaObj
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.Abelian C]
(P : C)
[hP : CategoryTheory.Limits.PreservesFiniteColimits (CategoryTheory.preadditiveCoyonedaObj { unop := P })]
:
An object is projective if its preadditive Co-Yoneda functor preserves finite colimits.