Functor categories are enriched #
If C
is a V
-enriched ordinary category, then J ⥤ C
is also
both a V
-enriched ordinary category and a J ⥤ V
-enriched
ordinary category, provided C
has suitable limits.
We first define the V
-enriched structure on J ⥤ C
by saying
that if F₁
and F₂
are in J ⥤ C
, then enrichedHom V F₁ F₂ : V
is a suitable limit involving F₁.obj j ⟶[V] F₂.obj j
for all j : C
.
The J ⥤ V
object of morphisms functorEnrichedHom V F₁ F₂ : J ⥤ V
is defined by sending j : J
to the previously defined enrichedHom
for the "restriction" of F₁
and F₂
to the category Under j
.
The definition isLimitConeFunctorEnrichedHom
shows that
enriched V F₁ F₂
is the limit of the functor functorEnrichedHom V F₁ F₂
.
Given two functors F₁
and F₂
from a category J
to a V
-enriched
ordinary category C
, this is the diagram Jᵒᵖ ⥤ J ⥤ V
whose end shall be
the V
-morphisms in J ⥤ V
from F₁
to F₂
.
Equations
- CategoryTheory.Enriched.FunctorCategory.diagram V F₁ F₂ = F₁.op.comp ((CategoryTheory.eHomFunctor V C).comp ((CategoryTheory.whiskeringLeft J C V).obj F₂))
Instances For
The condition that the end diagram V F₁ F₂
exists, see enrichedHom
.
Equations
Instances For
The V
-enriched hom from F₁
to F₂
when F₁
and F₂
are functors J ⥤ C
and C
is a V
-enriched category.
Equations
Instances For
The projection enrichedHom V F₁ F₂ ⟶ F₁.obj j ⟶[V] F₂.obj j
in the category V
for any j : J
when F₁
and F₂
are functors J ⥤ C
and C
is a V
-enriched category.
Equations
Instances For
Given functors F₁
and F₂
in J ⥤ C
, where C
is a V
-enriched ordinary category,
this is the bijection (F₁ ⟶ F₂) ≃ (𝟙_ V ⟶ enrichedHom V F₁ F₂)
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The identity for the V
-enrichment of the category J ⥤ C
.
Equations
Instances For
The composition for the V
-enrichment of the category J ⥤ C
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
If C
is a V
-enriched ordinary category, and C
has suitable limits,
then J ⥤ C
is also a V
-enriched ordinary category.
Equations
- One or more equations did not get rendered due to their size.
Instances For
If F₁
and F₂
are functors J ⥤ C
, G : K ⥤ J
, and
F₁'
and F₂'
are functors K ⥤ C
that are respectively
isomorphic to G ⋙ F₁
and G ⋙ F₂
, then this is the
induced morphism enrichedHom V F₁ F₂ ⟶ enrichedHom V F₁' F₂'
in V
when C
is a category enriched in V
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
If F₁
and F₂
are functors J ⥤ C
, and G : K ⥤ J
,
then this is the induced morphism
enrichedHom V F₁ F₂ ⟶ enrichedHom V (G ⋙ F₁) (G ⋙ F₂)
in V
when C
is a category enriched in V
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Given functors F₁
and F₂
in J ⥤ C
, where C
is a category enriched in V
,
this condition allows the definition of functorEnrichedHom V F₁ F₂ : J ⥤ V
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Given functors F₁
and F₂
in J ⥤ C
, where C
is a category enriched in V
,
this is the enriched hom functor from F₁
to F₂
in J ⥤ V
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The (limit) cone expressing that the limit of functorEnrichedHom V F₁ F₂
is enrichedHom V F₁ F₂
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Auxiliary definition for Enriched.FunctorCategory.isLimitConeFunctorEnrichedHom
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The limit of functorEnrichedHom V F₁ F₂
is enrichedHom V F₁ F₂
.
Equations
- CategoryTheory.Enriched.FunctorCategory.isLimitConeFunctorEnrichedHom V F₁ F₂ = { lift := CategoryTheory.Enriched.FunctorCategory.isLimitConeFunctorEnrichedHom.lift, fac := ⋯, uniq := ⋯ }
Instances For
The identity for the J ⥤ V
-enrichment of the category J ⥤ C
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The composition for the J ⥤ V
-enrichment of the category J ⥤ C
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
If C
is a V
-enriched ordinary category, and C
has suitable limits,
then J ⥤ C
is also a J ⥤ V
-enriched ordinary category.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Given functors F₁
and F₂
in J ⥤ C
, where C
is a V
-enriched ordinary category,
this is the bijection (F₁ ⟶ F₂) ≃ (𝟙_ (J ⥤ V) ⟶ functorEnrichedHom V F₁ F₂)
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
If C
is a V
-enriched ordinary category, and C
has suitable limits,
then J ⥤ C
is also a J ⥤ V
-enriched ordinary category.
Equations
- One or more equations did not get rendered due to their size.