Relative Yoneda preserves certain colimits #
In this file we turn the statement yonedaYonedaColimit
from
CategoryTheory.Limits.Preserves.Yoneda
from a functor F : J ⥤ Cᵒᵖ ⥤ Type v
into a statement
about families of presheaves over A
, i.e., functors F : J ⥤ Over A
.
noncomputable def
CategoryTheory.CostructuredArrow.toOverCompYonedaColimit
{C : Type u}
[CategoryTheory.Category.{v, u} C]
{J : Type v}
[CategoryTheory.SmallCategory J]
{A : CategoryTheory.Functor Cᵒᵖ (Type v)}
(F : CategoryTheory.Functor J (CategoryTheory.Over A))
:
(CategoryTheory.CostructuredArrow.toOver CategoryTheory.yoneda A).op.comp
(CategoryTheory.yoneda.obj (CategoryTheory.Limits.colimit F)) ≅ (CategoryTheory.CostructuredArrow.toOver CategoryTheory.yoneda A).op.comp
(CategoryTheory.Limits.colimit (F.comp CategoryTheory.yoneda))
Naturally in X
, we have Hom(YX, colim_i Fi) ≅ colim_i Hom(YX, Fi)
, where Y
is the
"Yoneda embedding" CostructuredArrow.toOver yoneda A
. This is a relative version of
yonedaYonedaColimit
.
Equations
- One or more equations did not get rendered due to their size.