Subobjects in Grothendieck abelian categories #
We study the complete lattice of subjects of X : C
when C is a Grothendieck abelian cateogry. In particular,
for a functor F : J ⥤ MonoOver X from a filtered category,
we relate the colimit of F (computed in C) and the
supremum of the subobjects corresponding to the objects
in the image of F.
If C is a Grothendieck abelian category, X : C, if F : J ⥤ MonoOver X is a
functor from a filtered category J, c is a colimit cocone for the corresponding
functor J ⥤ C, and f : c.pt ⟶ X is induced by the inclusions,
then f is a monomorphism.
If C is a Grothendieck abelian category, X : C, if F : J ⥤ MonoOver X is a
functor from a filtered category J, the colimit of F (computed in C) gives
a subobject of F which is a supremum of the subobjects corresponding to
the objects in the image of the functor F.
Let X : C be an object in a Grothendieck abelian category,
F : J ⥤ MonoOver X a functor from a filtered category, c a cocone for
the composition F ⋙ MonoOver.forget _ : J ⥤ Over X. We assume
that c.pt.hom : c.pt.left ⟶ X is a monomorphism and that the corresponding
subobject of X is the supremum of the subobjects given by (F.obj j).obj.hom,
then c becomes a colimit cocone after the application of
the forget functor Over X ⥤ C. (See also subobjectMk_of_isColimit_eq_iSup.)
Equations
- One or more equations did not get rendered due to their size.
Instances For
If C is a Grothendieck abelian category, X : C, if F : J ⥤ MonoOver X is a
functor from a κ-filtered category J with κ a regular cardinal such
that HasCardinalLT (Subobject X) κ, and if the colimit of F (computed in C)
maps epimorphically onto X, then there exists j : J such that (F.obj j).obj.hom
is an isomorphism.