Theory Category3.Adjunction
chapter Adjunction
theory Adjunction
imports Yoneda
begin
text‹
This theory defines the notions of adjoint functor and adjunction in various
ways and establishes their equivalence.
The notions ``left adjoint functor'' and ``right adjoint functor'' are defined
in terms of universal arrows.
``Meta-adjunctions'' are defined in terms of natural bijections between hom-sets,
where the notion of naturality is axiomatized directly.
``Hom-adjunctions'' formalize the notion of adjunction in terms of natural
isomorphisms of hom-functors.
``Unit-counit adjunctions'' define adjunctions in terms of functors equipped
with unit and counit natural transformations that satisfy the usual
``triangle identities.''
The ‹adjunction› locale is defined as the grand unification of all the
definitions, and includes formulas that connect the data from each of them.
It is shown that each of the definitions induces an interpretation of the
‹adjunction› locale, so that all the definitions are essentially equivalent.
Finally, it is shown that right adjoint functors are unique up to natural
isomorphism.
The reference \<^cite>‹"Wikipedia-Adjoint-Functors"› was useful in constructing this theory.
›
section "Left Adjoint Functor"
text‹
``@{term e} is an arrow from @{term "F x"} to @{term y}.''
›
locale arrow_from_functor =
C: category C +
D: category D +
F: "functor" D C F
for D :: "'d comp" (infixr ‹⋅⇩D› 55)
and C :: "'c comp" (infixr ‹⋅⇩C› 55)
and F :: "'d ⇒ 'c"
and x :: 'd
and y :: 'c
and e :: 'c +
assumes arrow: "D.ide x ∧ C.in_hom e (F x) y"
begin
notation C.in_hom (‹«_ : _ →⇩C _»›)
notation D.in_hom (‹«_ : _ →⇩D _»›)
text‹
``@{term g} is a @{term[source=true] D}-coextension of @{term f} along @{term e}.''
›
definition is_coext :: "'d ⇒ 'c ⇒ 'd ⇒ bool"
where "is_coext x' f g ≡ «g : x' →⇩D x» ∧ f = e ⋅⇩C F g"
end
text‹
``@{term e} is a terminal arrow from @{term "F x"} to @{term y}.''
›
locale terminal_arrow_from_functor =
arrow_from_functor D C F x y e
for D :: "'d comp" (infixr ‹⋅⇩D› 55)
and C :: "'c comp" (infixr ‹⋅⇩C› 55)
and F :: "'d ⇒ 'c"
and x :: 'd
and y :: 'c
and e :: 'c +
assumes is_terminal: "arrow_from_functor D C F x' y f ⟹ (∃!g. is_coext x' f g)"
begin
definition the_coext :: "'d ⇒ 'c ⇒ 'd"
where "the_coext x' f = (THE g. is_coext x' f g)"
lemma the_coext_prop:
assumes "arrow_from_functor D C F x' y f"
shows "«the_coext x' f : x' →⇩D x»" and "f = e ⋅⇩C F (the_coext x' f)"
by (metis assms is_coext_def is_terminal the_coext_def the_equality)+
lemma the_coext_unique:
assumes "arrow_from_functor D C F x' y f" and "is_coext x' f g"
shows "g = the_coext x' f"
using assms is_terminal the_coext_def the_equality by metis
end
text‹
A left adjoint functor is a functor ‹F: D → C›
that enjoys the following universal coextension property: for each object
@{term y} of @{term C} there exists an object @{term x} of @{term D} and an
arrow ‹e ∈ C.hom (F x) y› such that for any arrow
‹f ∈ C.hom (F x') y› there exists a unique ‹g ∈ D.hom x' x›
such that @{term "f = C e (F g)"}.
›
locale left_adjoint_functor =
C: category C +
D: category D +
"functor" D C F
for D :: "'d comp" (infixr ‹⋅⇩D› 55)
and C :: "'c comp" (infixr ‹⋅⇩C› 55)
and F :: "'d ⇒ 'c" +
assumes ex_terminal_arrow: "C.ide y ⟹ (∃x e. terminal_arrow_from_functor D C F x y e)"
begin
notation C.in_hom (‹«_ : _ →⇩C _»›)
notation D.in_hom (‹«_ : _ →⇩D _»›)
end
section "Right Adjoint Functor"
text‹
``@{term e} is an arrow from @{term x} to @{term "G y"}.''
›
locale arrow_to_functor =
C: category C +
D: category D +
G: "functor" C D G
for C :: "'c comp" (infixr ‹⋅⇩C› 55)
and D :: "'d comp" (infixr ‹⋅⇩D› 55)
and G :: "'c ⇒ 'd"
and x :: 'd
and y :: 'c
and e :: 'd +
assumes arrow: "C.ide y ∧ D.in_hom e x (G y)"
begin
notation C.in_hom (‹«_ : _ →⇩C _»›)
notation D.in_hom (‹«_ : _ →⇩D _»›)
text‹
``@{term f} is a @{term[source=true] C}-extension of @{term g} along @{term e}.''
›
definition is_ext :: "'c ⇒ 'd ⇒ 'c ⇒ bool"
where "is_ext y' g f ≡ «f : y →⇩C y'» ∧ g = G f ⋅⇩D e"
end
text‹
``@{term e} is an initial arrow from @{term x} to @{term "G y"}.''
›
locale initial_arrow_to_functor =
arrow_to_functor C D G x y e
for C :: "'c comp" (infixr ‹⋅⇩C› 55)
and D :: "'d comp" (infixr ‹⋅⇩D› 55)
and G :: "'c ⇒ 'd"
and x :: 'd
and y :: 'c
and e :: 'd +
assumes is_initial: "arrow_to_functor C D G x y' g ⟹ (∃!f. is_ext y' g f)"
begin
definition the_ext :: "'c ⇒ 'd ⇒ 'c"
where "the_ext y' g = (THE f. is_ext y' g f)"
lemma the_ext_prop:
assumes "arrow_to_functor C D G x y' g"
shows "«the_ext y' g : y →⇩C y'»" and "g = G (the_ext y' g) ⋅⇩D e"
by (metis assms is_initial is_ext_def the_equality the_ext_def)+
lemma the_ext_unique:
assumes "arrow_to_functor C D G x y' g" and "is_ext y' g f"
shows "f = the_ext y' g"
using assms is_initial the_ext_def the_equality by metis
end
text‹
A right adjoint functor is a functor ‹G: C → D›
that enjoys the following universal extension property:
for each object @{term x} of @{term D} there exists an object @{term y} of @{term C}
and an arrow ‹e ∈ D.hom x (G y)› such that for any arrow
‹g ∈ D.hom x (G y')› there exists a unique ‹f ∈ C.hom y y'›
such that @{term "h = D e (G f)"}.
›
locale right_adjoint_functor =
C: category C +
D: category D +
"functor" C D G
for C :: "'c comp" (infixr ‹⋅⇩C› 55)
and D :: "'d comp" (infixr ‹⋅⇩D› 55)
and G :: "'c ⇒ 'd" +
assumes ex_initial_arrow: "D.ide x ⟹ (∃y e. initial_arrow_to_functor C D G x y e)"
begin
notation C.in_hom (‹«_ : _ →⇩C _»›)
notation D.in_hom (‹«_ : _ →⇩D _»›)
end
section "Various Definitions of Adjunction"
subsection "Meta-Adjunction"
text‹
A ``meta-adjunction'' consists of a functor ‹F: D → C›,
a functor ‹G: C → D›, and for each object @{term x}
of @{term C} and @{term y} of @{term D} a bijection between
‹C.hom (F y) x› to ‹D.hom y (G x)› which is natural in @{term x}
and @{term y}. The naturality is easy to express at the meta-level without having
to resort to the formal baggage of ``set category,'' ``hom-functor,''
and ``natural isomorphism,'' hence the name.
›
locale meta_adjunction =
C: category C +
D: category D +
F: "functor" D C F +
G: "functor" C D G
for C :: "'c comp" (infixr ‹⋅⇩C› 55)
and D :: "'d comp" (infixr ‹⋅⇩D› 55)
and F :: "'d ⇒ 'c"
and G :: "'c ⇒ 'd"
and φ :: "'d ⇒ 'c ⇒ 'd"
and ψ :: "'c ⇒ 'd ⇒ 'c" +
assumes φ_in_hom: "⟦ D.ide y; C.in_hom f (F y) x ⟧ ⟹ D.in_hom (φ y f) y (G x)"
and ψ_in_hom: "⟦ C.ide x; D.in_hom g y (G x) ⟧ ⟹ C.in_hom (ψ x g) (F y) x"
and ψ_φ: "⟦ D.ide y; C.in_hom f (F y) x ⟧ ⟹ ψ x (φ y f) = f"
and φ_ψ: "⟦ C.ide x; D.in_hom g y (G x) ⟧ ⟹ φ y (ψ x g) = g"
and φ_naturality: "⟦ C.in_hom f x x'; D.in_hom g y' y; C.in_hom h (F y) x ⟧ ⟹
φ y' (f ⋅⇩C h ⋅⇩C F g) = G f ⋅⇩D φ y h ⋅⇩D g"
begin
notation C.in_hom (‹«_ : _ →⇩C _»›)
notation D.in_hom (‹«_ : _ →⇩D _»›)
text‹
The naturality of @{term ψ} is a consequence of the naturality of @{term φ}
and the other assumptions.
›
lemma ψ_naturality:
assumes f: "«f : x →⇩C x'»" and g: "«g : y' →⇩D y»" and h: "«h : y →⇩D G x»"
shows "f ⋅⇩C ψ x h ⋅⇩C F g = ψ x' (G f ⋅⇩D h ⋅⇩D g)"
using f g h φ_naturality ψ_in_hom C.ide_dom D.ide_dom D.in_homE φ_ψ ψ_φ
by (metis C.comp_in_homI' F.preserves_hom C.in_homE D.in_homE)
lemma respects_natural_isomorphism:
assumes "natural_isomorphism D C F' F τ" and "natural_isomorphism C D G G' μ"
shows "meta_adjunction C D F' G'
(λy f. μ (C.cod f) ⋅⇩D φ y (f ⋅⇩C inverse_transformation.map D C F τ y))
(λx g. ψ x ((inverse_transformation.map C D G' μ x) ⋅⇩D g) ⋅⇩C τ (D.dom g))"
proof -
interpret τ: natural_isomorphism D C F' F τ
using assms(1) by simp
interpret τ': inverse_transformation D C F' F τ
..
interpret μ: natural_isomorphism C D G G' μ
using assms(2) by simp
interpret μ': inverse_transformation C D G G' μ
..
let ?φ' = "λy f. μ (C.cod f) ⋅⇩D φ y (f ⋅⇩C τ'.map y)"
let ?ψ' = "λx g. ψ x (μ'.map x ⋅⇩D g) ⋅⇩C τ (D.dom g)"
show "meta_adjunction C D F' G' ?φ' ?ψ'"
proof
show "⋀y f x. ⟦D.ide y; «f : F' y →⇩C x»⟧
⟹ «μ (C.cod f) ⋅⇩D φ y (f ⋅⇩C τ'.map y) : y →⇩D G' x»"
proof -
fix x y f
assume y: "D.ide y" and f: "«f : F' y →⇩C x»"
show "«μ (C.cod f) ⋅⇩D φ y (f ⋅⇩C τ'.map y) : y →⇩D G' x»"
proof (intro D.comp_in_homI)
show "«μ (C.cod f) : G x →⇩D G' x»"
using f by fastforce
show "«φ y (f ⋅⇩C τ'.map y) : y →⇩D G x»"
using f y φ_in_hom by auto
qed
qed
show "⋀x g y. ⟦C.ide x; «g : y →⇩D G' x»⟧
⟹ «ψ x (μ'.map x ⋅⇩D g) ⋅⇩C τ (D.dom g) : F' y →⇩C x»"
proof -
fix x y g
assume x: "C.ide x" and g: "«g : y →⇩D G' x»"
show "«ψ x (μ'.map x ⋅⇩D g) ⋅⇩C τ (D.dom g) : F' y →⇩C x»"
proof (intro C.comp_in_homI)
show "«τ (D.dom g) : F' y →⇩C F y»"
using g by fastforce
show "«ψ x (μ'.map x ⋅⇩D g) : F y →⇩C x»"
using x g ψ_in_hom by auto
qed
qed
show "⋀y f x. ⟦D.ide y; «f : F' y →⇩C x»⟧
⟹ ψ x (μ'.map x ⋅⇩D μ (C.cod f) ⋅⇩D φ y (f ⋅⇩C τ'.map y)) ⋅⇩C
τ (D.dom (μ (C.cod f) ⋅⇩D φ y (f ⋅⇩C τ'.map y))) =
f"
proof -
fix x y f
assume y: "D.ide y" and f: "«f : F' y →⇩C x»"
have 1: "«φ y (f ⋅⇩C τ'.map y) : y →⇩D G x»"
using f y φ_in_hom by auto
show "ψ x (μ'.map x ⋅⇩D μ (C.cod f) ⋅⇩D φ y (f ⋅⇩C τ'.map y)) ⋅⇩C
τ (D.dom (μ (C.cod f) ⋅⇩D φ y (f ⋅⇩C τ'.map y))) =
f"
proof -
have "ψ x (μ'.map x ⋅⇩D μ (C.cod f) ⋅⇩D φ y (f ⋅⇩C τ'.map y)) ⋅⇩C
τ (D.dom (μ (C.cod f) ⋅⇩D φ y (f ⋅⇩C τ'.map y))) =
ψ x ((μ'.map x ⋅⇩D μ (C.cod f)) ⋅⇩D φ y (f ⋅⇩C τ'.map y)) ⋅⇩C
τ (D.dom (μ (C.cod f) ⋅⇩D φ y (f ⋅⇩C τ'.map y)))"
using D.comp_assoc by simp
also have "... = ψ x (φ y (f ⋅⇩C τ'.map y)) ⋅⇩C τ y"
by (metis "1" C.arr_cod C.dom_cod C.ide_cod C.in_homE D.comp_ide_arr D.dom_comp
D.ide_compE D.in_homE D.inverse_arrowsE μ'.inverts_components μ.preserves_dom
μ.preserves_reflects_arr category.seqI f meta_adjunction_axioms
meta_adjunction_def)
also have "... = f"
using f y ψ_φ C.comp_assoc τ'.inverts_components [of y] C.comp_arr_dom
by fastforce
finally show ?thesis by blast
qed
qed
show "⋀x g y. ⟦C.ide x; «g : y →⇩D G' x»⟧
⟹ μ (C.cod (ψ x (μ'.map x ⋅⇩D g) ⋅⇩C τ (D.dom g))) ⋅⇩D
φ y ((ψ x (μ'.map x ⋅⇩D g) ⋅⇩C τ (D.dom g)) ⋅⇩C τ'.map y) =
g"
proof -
fix x y g
assume x: "C.ide x" and g: "«g : y →⇩D G' x»"
have 1: "«ψ x (μ'.map x ⋅⇩D g) : F y →⇩C x»"
using x g ψ_in_hom by auto
show "μ (C.cod (ψ x (μ'.map x ⋅⇩D g) ⋅⇩C τ (D.dom g))) ⋅⇩D
φ y ((ψ x (μ'.map x ⋅⇩D g) ⋅⇩C τ (D.dom g)) ⋅⇩C τ'.map y) =
g"
proof -
have "μ (C.cod (ψ x (μ'.map x ⋅⇩D g) ⋅⇩C τ (D.dom g))) ⋅⇩D
φ y ((ψ x (μ'.map x ⋅⇩D g) ⋅⇩C τ (D.dom g)) ⋅⇩C τ'.map y) =
μ (C.cod (ψ x (μ'.map x ⋅⇩D g) ⋅⇩C τ (D.dom g))) ⋅⇩D
φ y (ψ x (μ'.map x ⋅⇩D g) ⋅⇩C τ (D.dom g) ⋅⇩C τ'.map y)"
using C.comp_assoc by simp
also have "... = μ x ⋅⇩D φ y (ψ x (μ'.map x ⋅⇩D g))"
using 1 C.comp_arr_dom C.comp_arr_inv' g by fastforce
also have "... = (μ x ⋅⇩D μ'.map x) ⋅⇩D g"
using x g φ_ψ D.comp_assoc by auto
also have "... = g"
using x g μ'.inverts_components [of x] D.comp_cod_arr by fastforce
finally show ?thesis by blast
qed
qed
show "⋀f x x' g y' y h. ⟦«f : x →⇩C x'»; «g : y' →⇩D y»; «h : F' y →⇩C x»⟧
⟹ μ (C.cod (f ⋅⇩C h ⋅⇩C F' g)) ⋅⇩D φ y' ((f ⋅⇩C h ⋅⇩C F' g) ⋅⇩C τ'.map y') =
G' f ⋅⇩D (μ (C.cod h) ⋅⇩D φ y (h ⋅⇩C τ'.map y)) ⋅⇩D g"
proof -
fix x y x' y' f g h
assume f: "«f : x →⇩C x'»" and g: "«g : y' →⇩D y»" and h: "«h : F' y →⇩C x»"
show "μ (C.cod (f ⋅⇩C h ⋅⇩C F' g)) ⋅⇩D φ y' ((f ⋅⇩C h ⋅⇩C F' g) ⋅⇩C τ'.map y') =
G' f ⋅⇩D (μ (C.cod h) ⋅⇩D φ y (h ⋅⇩C τ'.map y)) ⋅⇩D g"
proof -
have "μ (C.cod (f ⋅⇩C h ⋅⇩C F' g)) ⋅⇩D φ y' ((f ⋅⇩C h ⋅⇩C F' g) ⋅⇩C τ'.map y') =
μ x' ⋅⇩D φ y' ((f ⋅⇩C h ⋅⇩C F' g) ⋅⇩C τ'.map y')"
using f g h by fastforce
also have "... = μ x' ⋅⇩D φ y' (f ⋅⇩C (h ⋅⇩C τ'.map y) ⋅⇩C F g)"
using g τ'.naturality C.comp_assoc by auto
also have "... = (μ x' ⋅⇩D G f) ⋅⇩D φ y (h ⋅⇩C τ'.map y) ⋅⇩D g"
using f g h φ_naturality [of f x x' g y' y "h ⋅⇩C τ'.map y"] D.comp_assoc
by fastforce
also have "... = (G' f ⋅⇩D μ x) ⋅⇩D φ y (h ⋅⇩C τ'.map y) ⋅⇩D g"
using f μ.naturality by auto
also have "... = G' f ⋅⇩D (μ (C.cod h) ⋅⇩D φ y (h ⋅⇩C τ'.map y)) ⋅⇩D g"
using h D.comp_assoc by auto
finally show ?thesis by blast
qed
qed
qed
qed
end
subsection "Hom-Adjunction"
text‹
The bijection between hom-sets that defines an adjunction can be represented
formally as a natural isomorphism of hom-functors. However, stating the definition
this way is more complex than was the case for ‹meta_adjunction›.
One reason is that we need to have a ``set category'' that is suitable as
a target category for the hom-functors, and since the arrows of the categories
@{term C} and @{term D} will in general have distinct types, we need a set category
that simultaneously embeds both. Another reason is that we simply have to formally
construct the various categories and functors required to express the definition.
This is a good place to point out that I have often included more sublocales
in a locale than are strictly required. The main reason for this is the fact that
the locale system in Isabelle only gives one name to each entity introduced by
a locale: the name that it has in the first locale in which it occurs.
This means that entities that make their first appearance deeply nested in sublocales
will have to be referred to by long qualified names that can be difficult to
understand, or even to discover. To counteract this, I have typically introduced
sublocales before the superlocales that contain them to ensure that the entities
in the sublocales can be referred to by short meaningful (and predictable) names.
In my opinion, though, it would be better if the locale system would make entities
that occur in multiple locales accessible by \emph{all} possible qualified names,
so that the most perspicuous name could be used in any particular context.
›
locale hom_adjunction =
C: category C +
D: category D +
S: set_category S setp +
Cop: dual_category C +
Dop: dual_category D +
CopxC: product_category Cop.comp C +
DopxD: product_category Dop.comp D +
DopxC: product_category Dop.comp C +
F: "functor" D C F +
G: "functor" C D G +
HomC: hom_functor C S setp φC +
HomD: hom_functor D S setp φD +
Fop: dual_functor Dop.comp Cop.comp F +
FopxC: product_functor Dop.comp C Cop.comp C Fop.map C.map +
DopxG: product_functor Dop.comp C Dop.comp D Dop.map G +
Hom_FopxC: composite_functor DopxC.comp CopxC.comp S FopxC.map HomC.map +
Hom_DopxG: composite_functor DopxC.comp DopxD.comp S DopxG.map HomD.map +
Hom_FopxC: set_valued_functor DopxC.comp S setp Hom_FopxC.map +
Hom_DopxG: set_valued_functor DopxC.comp S setp Hom_DopxG.map +
Φ: set_valued_transformation DopxC.comp S setp Hom_FopxC.map Hom_DopxG.map Φ +
Ψ: set_valued_transformation DopxC.comp S setp Hom_DopxG.map Hom_FopxC.map Ψ +
ΦΨ: inverse_transformations DopxC.comp S Hom_FopxC.map Hom_DopxG.map Φ Ψ
for C :: "'c comp" (infixr ‹⋅⇩C› 55)
and D :: "'d comp" (infixr ‹⋅⇩D› 55)
and S :: "'s comp" (infixr ‹⋅⇩S› 55)
and setp :: "'s set ⇒ bool"
and φC :: "'c * 'c ⇒ 'c ⇒ 's"
and φD :: "'d * 'd ⇒ 'd ⇒ 's"
and F :: "'d ⇒ 'c"
and G :: "'c ⇒ 'd"
and Φ :: "'d * 'c ⇒ 's"
and Ψ :: "'d * 'c ⇒ 's"
begin
notation C.in_hom (‹«_ : _ →⇩C _»›)
notation D.in_hom (‹«_ : _ →⇩D _»›)
abbreviation ψC :: "'c * 'c ⇒ 's ⇒ 'c"
where "ψC ≡ HomC.ψ"
abbreviation ψD :: "'d * 'd ⇒ 's ⇒ 'd"
where "ψD ≡ HomD.ψ"
end
subsection "Unit/Counit Adjunction"
text‹
Expressed in unit/counit terms, an adjunction consists of functors
‹F: D → C› and ‹G: C → D›, equipped with natural transformations
‹η: 1 → GF› and ‹ε: FG → 1› satisfying certain ``triangle identities''.
›
locale unit_counit_adjunction =
C: category C +
D: category D +
F: "functor" D C F +
G: "functor" C D G +
GF: composite_functor D C D F G +
FG: composite_functor C D C G F +
FGF: composite_functor D C C F ‹F o G› +
GFG: composite_functor C D D G ‹G o F› +
η: natural_transformation D D D.map ‹G o F› η +
ε: natural_transformation C C ‹F o G› C.map ε +
Fη: natural_transformation D C F ‹F o G o F› ‹F o η› +
ηG: natural_transformation C D G ‹G o F o G› ‹η o G› +
εF: natural_transformation D C ‹F o G o F› F ‹ε o F› +
Gε: natural_transformation C D ‹G o F o G› G ‹G o ε› +
εFoFη: vertical_composite D C F ‹F o G o F› F ‹F o η› ‹ε o F› +
GεoηG: vertical_composite C D G ‹G o F o G› G ‹η o G› ‹G o ε›
for C :: "'c comp" (infixr ‹⋅⇩C› 55)
and D :: "'d comp" (infixr ‹⋅⇩D› 55)
and F :: "'d ⇒ 'c"
and G :: "'c ⇒ 'd"
and η :: "'d ⇒ 'd"
and ε :: "'c ⇒ 'c" +
assumes triangle_F: "εFoFη.map = F"
and triangle_G: "GεoηG.map = G"
begin
notation C.in_hom (‹«_ : _ →⇩C _»›)
notation D.in_hom (‹«_ : _ →⇩D _»›)
end
lemma unit_determines_counit:
assumes "unit_counit_adjunction C D F G η ε"
and "unit_counit_adjunction C D F G η ε'"
shows "ε = ε'"
proof -
interpret Adj: unit_counit_adjunction C D F G η ε using assms(1) by auto
interpret Adj': unit_counit_adjunction C D F G η ε' using assms(2) by auto
interpret FGFG: composite_functor C D C G ‹F o G o F› ..
interpret FGε: natural_transformation C C ‹(F o G) o (F o G)› ‹F o G› ‹(F o G) o ε›
using Adj.ε.natural_transformation_axioms Adj.FG.as_nat_trans.natural_transformation_axioms
horizontal_composite
by fastforce
interpret FηG: natural_transformation C C ‹F o G› ‹F o G o F o G› ‹F o η o G›
using Adj.η.natural_transformation_axioms Adj.Fη.natural_transformation_axioms
Adj.G.as_nat_trans.natural_transformation_axioms horizontal_composite
by blast
interpret ε'ε: natural_transformation C C ‹F o G o F o G› Adj.C.map ‹ε' o ε›
proof -
have "natural_transformation C C ((F o G) o (F o G)) Adj.C.map (ε' o ε)"
using Adj.ε.natural_transformation_axioms Adj'.ε.natural_transformation_axioms
horizontal_composite Adj.C.is_functor comp_functor_identity
by (metis (no_types, lifting))
thus "natural_transformation C C (F o G o F o G) Adj.C.map (ε' o ε)"
using o_assoc by metis
qed
interpret ε'εoFηG: vertical_composite
C C ‹F o G› ‹F o G o F o G› Adj.C.map ‹F o η o G› ‹ε' o ε› ..
have "ε' = vertical_composite.map C C (F o Adj.GεoηG.map) ε'"
using vcomp_ide_dom [of C C "F o G" Adj.C.map ε'] Adj.triangle_G
by (simp add: Adj'.ε.natural_transformation_axioms)
also have "... = vertical_composite.map C C
(vertical_composite.map C C (F o η o G) (F o G o ε)) ε'"
using whisker_left Adj.F.functor_axioms Adj.Gε.natural_transformation_axioms
Adj.ηG.natural_transformation_axioms o_assoc
by (metis (no_types, lifting))
also have "... = vertical_composite.map C C
(vertical_composite.map C C (F o η o G) (ε' o F o G)) ε"
proof -
have "vertical_composite.map C C
(vertical_composite.map C C (F o η o G) (F o G o ε)) ε'
= vertical_composite.map C C (F o η o G)
(vertical_composite.map C C (F o G o ε) ε')"
using vcomp_assoc
by (metis (no_types, lifting) Adj'.ε.natural_transformation_axioms
FGε.natural_transformation_axioms FηG.natural_transformation_axioms o_assoc)
also have "... = vertical_composite.map C C (F o η o G)
(vertical_composite.map C C (ε' o F o G) ε)"
using Adj'.ε.natural_transformation_axioms Adj.ε.natural_transformation_axioms
interchange_spc [of C C "F o G" Adj.C.map ε C "F o G" Adj.C.map ε']
by (metis hcomp_ide_cod hcomp_ide_dom o_assoc)
also have "... = vertical_composite.map C C
(vertical_composite.map C C (F o η o G) (ε' o F o G)) ε"
using vcomp_assoc
by (metis Adj'.εF.natural_transformation_axioms
Adj.G.as_nat_trans.natural_transformation_axioms
Adj.ε.natural_transformation_axioms FηG.natural_transformation_axioms
horizontal_composite)
finally show ?thesis by simp
qed
also have "... = vertical_composite.map C C
(vertical_composite.map D C (F o η) (ε' o F) o G) ε"
using whisker_right Adj'.εF.natural_transformation_axioms
Adj.Fη.natural_transformation_axioms Adj.G.functor_axioms
by metis
also have "... = ε"
using Adj'.triangle_F vcomp_ide_cod Adj.ε.natural_transformation_axioms by simp
finally show ?thesis by simp
qed
lemma counit_determines_unit:
assumes "unit_counit_adjunction C D F G η ε"
and "unit_counit_adjunction C D F G η' ε"
shows "η = η'"
proof -
interpret Adj: unit_counit_adjunction C D F G η ε using assms(1) by auto
interpret Adj': unit_counit_adjunction C D F G η' ε using assms(2) by auto
interpret GFGF: composite_functor D C D F ‹G o F o G› ..
interpret GFη: natural_transformation D D ‹G o F› ‹(G o F) o (G o F)› ‹(G o F) o η›
using Adj.η.natural_transformation_axioms Adj.GF.functor_axioms
Adj.GF.as_nat_trans.natural_transformation_axioms comp_functor_identity
horizontal_composite
by (metis (no_types, lifting))
interpret η'GF: natural_transformation D D ‹G o F› ‹(G o F) o (G o F)› ‹η' o (G o F)›
using Adj'.η.natural_transformation_axioms Adj.GF.functor_axioms
Adj.GF.as_nat_trans.natural_transformation_axioms comp_identity_functor
horizontal_composite
by (metis (no_types, lifting))
interpret GεF: natural_transformation D D ‹G o F o G o F› ‹G o F› ‹G o ε o F›
using Adj.ε.natural_transformation_axioms Adj.F.as_nat_trans.natural_transformation_axioms
Adj.Gε.natural_transformation_axioms horizontal_composite
by blast
interpret η'η: natural_transformation D D Adj.D.map ‹G o F o G o F› ‹η' o η›
proof -
have "natural_transformation D D Adj.D.map ((G o F) o (G o F)) (η' o η)"
using Adj'.η.natural_transformation_axioms Adj.D.identity_functor_axioms
Adj.η.natural_transformation_axioms horizontal_composite identity_functor.is_functor
by fastforce
thus "natural_transformation D D Adj.D.map (G o F o G o F) (η' o η)"
using o_assoc by metis
qed
interpret GεFoη'η: vertical_composite
D D Adj.D.map ‹G o F o G o F› ‹G o F› ‹η' o η› ‹G o ε o F› ..
have "η' = vertical_composite.map D D η' (G o Adj.εFoFη.map)"
using vcomp_ide_cod [of D D Adj.D.map "G o F" η'] Adj.triangle_F
by (simp add: Adj'.η.natural_transformation_axioms)
also have "... = vertical_composite.map D D η'
(vertical_composite.map D D (G o (F o η)) (G o (ε o F)))"
using whisker_left Adj.Fη.natural_transformation_axioms Adj.G.functor_axioms
Adj.εF.natural_transformation_axioms
by fastforce
also have "... = vertical_composite.map D D
(vertical_composite.map D D η' (G o (F o η))) (G o ε o F)"
using vcomp_assoc Adj'.η.natural_transformation_axioms
GFη.natural_transformation_axioms GεF.natural_transformation_axioms o_assoc
by (metis (no_types, lifting))
also have "... = vertical_composite.map D D
(vertical_composite.map D D η (η' o G o F)) (G o ε o F)"
using interchange_spc [of D D Adj.D.map "G o F" η D Adj.D.map "G o F" η']
Adj.η.natural_transformation_axioms Adj'.η.natural_transformation_axioms
by (metis hcomp_ide_cod hcomp_ide_dom o_assoc)
also have "... = vertical_composite.map D D η
(vertical_composite.map D D (η' o G o F) (G o ε o F))"
using vcomp_assoc
by (metis (no_types, lifting) Adj.η.natural_transformation_axioms
GεF.natural_transformation_axioms η'GF.natural_transformation_axioms o_assoc)
also have "... = vertical_composite.map D D η
(vertical_composite.map C D (η' o G) (G o ε) o F)"
using whisker_right Adj'.ηG.natural_transformation_axioms Adj.F.functor_axioms
Adj.Gε.natural_transformation_axioms
by fastforce
also have "... = η"
using Adj'.triangle_G vcomp_ide_dom Adj.GF.functor_axioms
Adj.η.natural_transformation_axioms
by simp
finally show ?thesis by simp
qed
subsection "Adjunction"
text‹
The grand unification of everything to do with an adjunction.
›
locale adjunction =
C: category C +
D: category D +
S: set_category S setp +
Cop: dual_category C +
Dop: dual_category D +
CopxC: product_category Cop.comp C +
DopxD: product_category Dop.comp D +
DopxC: product_category Dop.comp C +
idDop: identity_functor Dop.comp +
HomC: hom_functor C S setp φC +
HomD: hom_functor D S setp φD +
F: left_adjoint_functor D C F +
G: right_adjoint_functor C D G +
GF: composite_functor D C D F G +
FG: composite_functor C D C G F +
FGF: composite_functor D C C F FG.map +
GFG: composite_functor C D D G GF.map +
Fop: dual_functor Dop.comp Cop.comp F +
FopxC: product_functor Dop.comp C Cop.comp C Fop.map C.map +
DopxG: product_functor Dop.comp C Dop.comp D Dop.map G +
Hom_FopxC: composite_functor DopxC.comp CopxC.comp S FopxC.map HomC.map +
Hom_DopxG: composite_functor DopxC.comp DopxD.comp S DopxG.map HomD.map +
Hom_FopxC: set_valued_functor DopxC.comp S setp Hom_FopxC.map +
Hom_DopxG: set_valued_functor DopxC.comp S setp Hom_DopxG.map +
η: natural_transformation D D D.map GF.map η +
ε: natural_transformation C C FG.map C.map ε +
Fη: natural_transformation D C F ‹F o G o F› ‹F o η› +
ηG: natural_transformation C D G ‹G o F o G› ‹η o G› +
εF: natural_transformation D C ‹F o G o F› F ‹ε o F› +
Gε: natural_transformation C D ‹G o F o G› G ‹G o ε› +
εFoFη: vertical_composite D C F FGF.map F ‹F o η› ‹ε o F› +
GεoηG: vertical_composite C D G GFG.map G ‹η o G› ‹G o ε› +
φψ: meta_adjunction C D F G φ ψ +
ηε: unit_counit_adjunction C D F G η ε +
ΦΨ: hom_adjunction C D S setp φC φD F G Φ Ψ
for C :: "'c comp" (infixr ‹⋅⇩C› 55)
and D :: "'d comp" (infixr ‹⋅⇩D› 55)
and S :: "'s comp" (infixr ‹⋅⇩S› 55)
and setp :: "'s set ⇒ bool"
and φC :: "'c * 'c ⇒ 'c ⇒ 's"
and φD :: "'d * 'd ⇒ 'd ⇒ 's"
and F :: "'d ⇒ 'c"
and G :: "'c ⇒ 'd"
and φ :: "'d ⇒ 'c ⇒ 'd"
and ψ :: "'c ⇒ 'd ⇒ 'c"
and η :: "'d ⇒ 'd"
and ε :: "'c ⇒ 'c"
and Φ :: "'d * 'c ⇒ 's"
and Ψ :: "'d * 'c ⇒ 's" +
assumes φ_in_terms_of_η: "⟦ D.ide y; «f : F y →⇩C x» ⟧ ⟹ φ y f = G f ⋅⇩D η y"
and ψ_in_terms_of_ε: "⟦ C.ide x; «g : y →⇩D G x» ⟧ ⟹ ψ x g = ε x ⋅⇩C F g"
and η_in_terms_of_φ: "D.ide y ⟹ η y = φ y (F y)"
and ε_in_terms_of_ψ: "C.ide x ⟹ ε x = ψ x (G x)"
and φ_in_terms_of_Φ: "⟦ D.ide y; «f : F y →⇩C x» ⟧ ⟹
φ y f = (ΦΨ.ψD (y, G x) o S.Fun (Φ (y, x)) o φC (F y, x)) f"
and ψ_in_terms_of_Ψ: "⟦ C.ide x; «g : y →⇩D G x» ⟧ ⟹
ψ x g = (ΦΨ.ψC (F y, x) o S.Fun (Ψ (y, x)) o φD (y, G x)) g"
and Φ_in_terms_of_φ:
"⟦ C.ide x; D.ide y ⟧ ⟹
Φ (y, x) = S.mkArr (HomC.set (F y, x)) (HomD.set (y, G x))
(φD (y, G x) o φ y o ΦΨ.ψC (F y, x))"
and Ψ_in_terms_of_ψ:
"⟦ C.ide x; D.ide y ⟧ ⟹
Ψ (y, x) = S.mkArr (HomD.set (y, G x)) (HomC.set (F y, x))
(φC (F y, x) o ψ x o ΦΨ.ψD (y, G x))"
section "Meta-Adjunctions Induce Unit/Counit Adjunctions"
context meta_adjunction
begin
interpretation GF: composite_functor D C D F G ..
interpretation FG: composite_functor C D C G F ..
interpretation FGF: composite_functor D C C F FG.map ..
interpretation GFG: composite_functor C D D G GF.map ..
definition ηo :: "'d ⇒ 'd"
where "ηo y = φ y (F y)"
lemma ηo_in_hom:
assumes "D.ide y"
shows "«ηo y : y →⇩D G (F y)»"
using assms D.ide_in_hom ηo_def φ_in_hom by force
lemma φ_in_terms_of_ηo:
assumes "D.ide y" and "«f : F y →⇩C x»"
shows "φ y f = G f ⋅⇩D ηo y"
proof (unfold ηo_def)
have 1: "«F y : F y →⇩C F y»"
using assms(1) D.ide_in_hom by blast
hence "φ y (F y) = φ y (F y) ⋅⇩D y"
by (metis assms(1) D.in_homE φ_in_hom D.comp_arr_dom)
thus "φ y f = G f ⋅⇩D φ y (F y)"
using assms 1 D.ide_in_hom by (metis C.comp_arr_dom C.in_homE φ_naturality)
qed
lemma φ_F_char:
assumes "«g : y' →⇩D y»"
shows "φ y' (F g) = ηo y ⋅⇩D g"
using assms ηo_def φ_in_hom [of y "F y" "F y"]
D.comp_cod_arr [of "D (φ y (F y)) g" "G (F y)"]
φ_naturality [of "F y" "F y" "F y" g y' y "F y"]
by (metis C.ide_in_hom D.arr_cod_iff_arr D.arr_dom D.cod_cod D.cod_dom D.comp_ide_arr
D.comp_ide_self D.ide_cod D.in_homE F.as_nat_trans.is_natural_2 F.functor_axioms
F.preserves_section_retraction φ_in_hom functor.preserves_hom)
interpretation η: transformation_by_components D D D.map GF.map ηo
proof
show "⋀a. D.ide a ⟹ «ηo a : D.map a →⇩D GF.map a»"
using ηo_def φ_in_hom D.ide_in_hom by force
fix f
assume f: "D.arr f"
show "ηo (D.cod f) ⋅⇩D D.map f = GF.map f ⋅⇩D ηo (D.dom f)"
using f φ_F_char [of "D.map f" "D.dom f" "D.cod f"]
φ_in_terms_of_ηo [of "D.dom f" "F f" "F (D.cod f)"]
by force
qed
lemma η_map_simp:
assumes "D.ide y"
shows "η.map y = φ y (F y)"
using assms η.map_simp_ide ηo_def by simp
definition εo :: "'c ⇒ 'c"
where "εo x = ψ x (G x)"
lemma εo_in_hom:
assumes "C.ide x"
shows "«εo x : F (G x) →⇩C x»"
using assms C.ide_in_hom εo_def ψ_in_hom by force
lemma ψ_in_terms_of_εo:
assumes "C.ide x" and "«g : y →⇩D G x»"
shows "ψ x g = εo x ⋅⇩C F g"
proof -
have "εo x ⋅⇩C F g = x ⋅⇩C ψ x (G x) ⋅⇩C F g"
using assms εo_def ψ_in_hom [of x "G x" "G x"]
C.comp_cod_arr [of "ψ x (G x) ⋅⇩C F g" x]
by fastforce
also have "... = ψ x (G x ⋅⇩D G x ⋅⇩D g)"
using assms ψ_naturality [of x x x g y "G x" "G x"] by force
also have "... = ψ x g"
using assms D.comp_cod_arr by fastforce
finally show ?thesis by simp
qed
lemma ψ_G_char:
assumes "«f: x →⇩C x'»"
shows "ψ x' (G f) = f ⋅⇩C εo x"
proof (unfold εo_def)
have 0: "C.ide x ∧ C.ide x'" using assms by auto
thus "ψ x' (G f) = f ⋅⇩C ψ x (G x)"
using 0 assms ψ_naturality ψ_in_hom [of x "G x" "G x"] G.preserves_hom εo_def
ψ_in_terms_of_εo G.as_nat_trans.is_natural_1 C.ide_in_hom
by (metis C.arrI C.in_homE)
qed
interpretation ε: transformation_by_components C C FG.map C.map εo
apply unfold_locales
using εo_in_hom
apply simp
using ψ_G_char ψ_in_terms_of_εo
by (metis C.arr_iff_in_hom C.ide_cod C.map_simp G.preserves_hom comp_apply)
lemma ε_map_simp:
assumes "C.ide x"
shows "ε.map x = ψ x (G x)"
using assms εo_def by simp
interpretation FD: composite_functor D D C D.map F ..
interpretation CF: composite_functor D C C F C.map ..
interpretation GC: composite_functor C C D C.map G ..
interpretation DG: composite_functor C D D G D.map ..
interpretation Fη: natural_transformation D C F ‹F o G o F› ‹F o η.map›
by (metis (no_types, lifting) F.as_nat_trans.natural_transformation_axioms
F.functor_axioms η.natural_transformation_axioms comp_functor_identity
horizontal_composite o_assoc)
interpretation εF: natural_transformation D C ‹F o G o F› F ‹ε.map o F›
using ε.natural_transformation_axioms F.as_nat_trans.natural_transformation_axioms
horizontal_composite
by fastforce
interpretation ηG: natural_transformation C D G ‹G o F o G› ‹η.map o G›
using η.natural_transformation_axioms G.as_nat_trans.natural_transformation_axioms
horizontal_composite
by fastforce
interpretation Gε: natural_transformation C D ‹G o F o G› G ‹G o ε.map›
by (metis (no_types, lifting) G.as_nat_trans.natural_transformation_axioms
G.functor_axioms ε.natural_transformation_axioms comp_functor_identity
horizontal_composite o_assoc)
interpretation εFoFη: vertical_composite D C F ‹F o G o F› F ‹F o η.map› ‹ε.map o F›
..
interpretation GεoηG: vertical_composite C D G ‹G o F o G› G ‹η.map o G› ‹G o ε.map›
..
lemma unit_counit_F:
assumes "D.ide y"
shows "F y = εo (F y) ⋅⇩C F (ηo y)"
using assms ψ_in_terms_of_εo ηo_def ψ_φ ηo_in_hom F.preserves_ide C.ide_in_hom by metis
lemma unit_counit_G:
assumes "C.ide x"
shows "G x = G (εo x) ⋅⇩D ηo (G x)"
using assms φ_in_terms_of_ηo εo_def φ_ψ εo_in_hom G.preserves_ide D.ide_in_hom by metis
lemma induces_unit_counit_adjunction':
shows "unit_counit_adjunction C D F G η.map ε.map"
proof
show "εFoFη.map = F"
using εFoFη.is_natural_transformation εFoFη.map_simp_ide unit_counit_F
F.as_nat_trans.natural_transformation_axioms
by (intro NaturalTransformation.eqI) auto
show "GεoηG.map = G"
using GεoηG.is_natural_transformation GεoηG.map_simp_ide unit_counit_G
G.as_nat_trans.natural_transformation_axioms
by (intro NaturalTransformation.eqI) auto
qed
definition η :: "'d ⇒ 'd" where "η ≡ η.map"
definition ε :: "'c ⇒ 'c" where "ε ≡ ε.map"
theorem induces_unit_counit_adjunction:
shows "unit_counit_adjunction C D F G η ε"
unfolding η_def ε_def
using induces_unit_counit_adjunction' by simp
lemma η_is_natural_transformation:
shows "natural_transformation D D D.map GF.map η"
unfolding η_def ..
lemma ε_is_natural_transformation:
shows "natural_transformation C C FG.map C.map ε"
unfolding ε_def ..
text‹
From the defined @{term η} and @{term ε} we can recover the original @{term φ} and @{term ψ}.
›
lemma φ_in_terms_of_η:
assumes "D.ide y" and "«f : F y →⇩C x»"
shows "φ y f = G f ⋅⇩D η y"
using assms η_def by (simp add: φ_in_terms_of_ηo)
lemma ψ_in_terms_of_ε:
assumes "C.ide x" and "«g : y →⇩D G x»"
shows "ψ x g = ε x ⋅⇩C F g"
using assms ε_def by (simp add: ψ_in_terms_of_εo)
end
section "Meta-Adjunctions Induce Left and Right Adjoint Functors"
context meta_adjunction
begin
interpretation unit_counit_adjunction C D F G η ε
using induces_unit_counit_adjunction η_def ε_def by auto
lemma has_terminal_arrows_from_functor:
assumes x: "C.ide x"
shows "terminal_arrow_from_functor D C F (G x) x (ε x)"
and "⋀y' f. arrow_from_functor D C F y' x f
⟹ terminal_arrow_from_functor.the_coext D C F (G x) (ε x) y' f = φ y' f"
proof -
interpret εx: arrow_from_functor D C F ‹G x› x ‹ε x›
using x ε.preserves_hom G.preserves_ide by unfold_locales auto
have 1: "⋀y' f. arrow_from_functor D C F y' x f ⟹
εx.is_coext y' f (φ y' f) ∧ (∀g'. εx.is_coext y' f g' ⟶ g' = φ y' f)"
using x
by (metis (full_types) εx.is_coext_def φ_ψ ψ_in_terms_of_ε arrow_from_functor.arrow
φ_in_hom ψ_φ)
interpret εx: terminal_arrow_from_functor D C F ‹G x› x ‹ε x›
using 1 by unfold_locales blast
show "terminal_arrow_from_functor D C F (G x) x (ε x)" ..
show "⋀y' f. arrow_from_functor D C F y' x f ⟹ εx.the_coext y' f = φ y' f"
using 1 εx.the_coext_def by auto
qed
lemma has_left_adjoint_functor:
shows "left_adjoint_functor D C F"
apply unfold_locales using has_terminal_arrows_from_functor by auto
lemma has_initial_arrows_to_functor:
assumes y: "D.ide y"
shows "initial_arrow_to_functor C D G y (F y) (η y)"
and "⋀x' g. arrow_to_functor C D G y x' g ⟹
initial_arrow_to_functor.the_ext C D G (F y) (η y) x' g = ψ x' g"
proof -
interpret ηy: arrow_to_functor C D G y ‹F y› ‹η y›
using y by unfold_locales auto
have 1: "⋀x' g. arrow_to_functor C D G y x' g ⟹
ηy.is_ext x' g (ψ x' g) ∧ (∀f'. ηy.is_ext x' g f' ⟶ f' = ψ x' g)"
using y
by (metis (full_types) ηy.is_ext_def ψ_φ φ_in_terms_of_η arrow_to_functor.arrow
ψ_in_hom φ_ψ)
interpret ηy: initial_arrow_to_functor C D G y ‹F y› ‹η y›
apply unfold_locales using 1 by blast
show "initial_arrow_to_functor C D G y (F y) (η y)" ..
show "⋀x' g. arrow_to_functor C D G y x' g ⟹ ηy.the_ext x' g = ψ x' g"
using 1 ηy.the_ext_def by auto
qed
lemma has_right_adjoint_functor:
shows "right_adjoint_functor C D G"
apply unfold_locales using has_initial_arrows_to_functor by auto
end
section "Unit/Counit Adjunctions Induce Meta-Adjunctions"
context unit_counit_adjunction
begin
definition φ :: "'d ⇒ 'c ⇒ 'd"
where "φ y h = G h ⋅⇩D η y"
definition ψ :: "'c ⇒ 'd ⇒ 'c"
where "ψ x h = ε x ⋅⇩C F h"
interpretation meta_adjunction C D F G φ ψ
proof
fix x :: 'c and y :: 'd and f :: 'c
assume y: "D.ide y" and f: "«f : F y →⇩C x»"
show 0: "«φ y f : y →⇩D G x»"
using f y G.preserves_hom η.preserves_hom φ_def D.ide_in_hom by auto
show "ψ x (φ y f) = f"
proof -
have "ψ x (φ y f) = (ε x ⋅⇩C F (G f)) ⋅⇩C F (η y)"
using y f φ_def ψ_def C.comp_assoc by auto
also have "... = (f ⋅⇩C ε (F y)) ⋅⇩C F (η y)"
using y f ε.naturality by auto
also have "... = f"
using y f εFoFη.map_simp_2 triangle_F C.comp_arr_dom D.ide_in_hom C.comp_assoc
by fastforce
finally show ?thesis by auto
qed
next
fix x :: 'c and y :: 'd and g :: 'd
assume x: "C.ide x" and g: "«g : y →⇩D G x»"
show "«ψ x g : F y →⇩C x»" using g x ψ_def by fastforce
show "φ y (ψ x g) = g"
proof -
have "φ y (ψ x g) = (G (ε x) ⋅⇩D η (G x)) ⋅⇩D g"
using g x φ_def ψ_def η.naturality [of g] D.comp_assoc by auto
also have "... = g"
using x g triangle_G D.comp_ide_arr GεoηG.map_simp_ide by auto
finally show ?thesis by auto
qed
next
fix f :: 'c and g :: 'd and h :: 'c and x :: 'c and x' :: 'c and y :: 'd and y' :: 'd
assume f: "«f : x →⇩C x'»" and g: "«g : y' →⇩D y»" and h: "«h : F y →⇩C x»"
show "φ y' (f ⋅⇩C h ⋅⇩C F g) = G f ⋅⇩D φ y h ⋅⇩D g"
using φ_def f g h η.naturality D.comp_assoc by fastforce
qed
theorem induces_meta_adjunction:
shows "meta_adjunction C D F G φ ψ" ..
text‹
From the defined @{term φ} and @{term ψ} we can recover the original @{term η} and @{term ε}.
›
lemma η_in_terms_of_φ:
assumes "D.ide y"
shows "η y = φ y (F y)"
using assms φ_def D.comp_cod_arr by auto
lemma ε_in_terms_of_ψ:
assumes "C.ide x"
shows "ε x = ψ x (G x)"
using assms ψ_def C.comp_arr_dom by auto
end
section "Left and Right Adjoint Functors Induce Meta-Adjunctions"
text‹
A left adjoint functor induces a meta-adjunction, modulo the choice of a
right adjoint and counit.
›
context left_adjoint_functor
begin
definition Go :: "'c ⇒ 'd"
where "Go a = (SOME b. ∃e. terminal_arrow_from_functor D C F b a e)"
definition εo :: "'c ⇒ 'c"
where "εo a = (SOME e. terminal_arrow_from_functor D C F (Go a) a e)"
lemma Go_εo_terminal:
assumes "∃b e. terminal_arrow_from_functor D C F b a e"
shows "terminal_arrow_from_functor D C F (Go a) a (εo a)"
using assms Go_def εo_def
someI_ex [of "λb. ∃e. terminal_arrow_from_functor D C F b a e"]
someI_ex [of "λe. terminal_arrow_from_functor D C F (Go a) a e"]
by simp
text‹
The right adjoint @{term G} to @{term F} takes each arrow @{term f} of
@{term[source=true] C} to the unique @{term[source=true] D}-coextension of
@{term "C f (εo (C.dom f))"} along @{term "εo (C.cod f)"}.
›
definition G :: "'c ⇒ 'd"
where "G f = (if C.arr f then
terminal_arrow_from_functor.the_coext D C F (Go (C.cod f)) (εo (C.cod f))
(Go (C.dom f)) (f ⋅⇩C εo (C.dom f))
else D.null)"
lemma G_ide:
assumes "C.ide x"
shows "G x = Go x"
proof -
interpret terminal_arrow_from_functor D C F ‹Go x› x ‹εo x›
using assms ex_terminal_arrow Go_εo_terminal by blast
have 1: "arrow_from_functor D C F (Go x) x (εo x)" ..
have "is_coext (Go x) (εo x) (Go x)"
using arrow is_coext_def C.in_homE C.comp_arr_dom by auto
hence "Go x = the_coext (Go x) (εo x)" using 1 the_coext_unique by blast
moreover have "εo x = C x (εo (C.dom x))"
using assms arrow C.comp_ide_arr C.seqI' C.ide_in_hom C.in_homE by metis
ultimately show ?thesis using assms G_def C.cod_dom C.ide_in_hom C.in_homE by metis
qed
lemma G_is_functor:
shows "functor C D G"
proof
fix f :: 'c
assume "¬C.arr f"
thus "G f = D.null" using G_def by auto
next
fix f :: 'c
assume f: "C.arr f"
let ?x = "C.dom f"
let ?x' = "C.cod f"
interpret xε: terminal_arrow_from_functor D C F ‹Go ?x› ‹?x› ‹εo ?x›
using f ex_terminal_arrow Go_εo_terminal by simp
interpret x'ε: terminal_arrow_from_functor D C F ‹Go ?x'› ‹?x'› ‹εo ?x'›
using f ex_terminal_arrow Go_εo_terminal by simp
have 1: "arrow_from_functor D C F (Go ?x) ?x' (C f (εo ?x))"
using f xε.arrow by (unfold_locales, auto)
have "G f = x'ε.the_coext (Go ?x) (C f (εo ?x))" using f G_def by simp
hence Gf: "«G f : Go ?x →⇩D Go ?x'» ∧ f ⋅⇩C εo ?x = εo ?x' ⋅⇩C F (G f)"
using 1 x'ε.the_coext_prop by simp
show "D.arr (G f)" using Gf by auto
show "D.dom (G f) = G ?x" using f Gf G_ide by auto
show "D.cod (G f) = G ?x'" using f Gf G_ide by auto
next
fix f f' :: 'c
assume ff': "C.arr (C f' f)"
have f: "C.arr f" using ff' by auto
let ?x = "C.dom f"
let ?x' = "C.cod f"
let ?x'' = "C.cod f'"
interpret xε: terminal_arrow_from_functor D C F ‹Go ?x› ‹?x› ‹εo ?x›
using f ex_terminal_arrow Go_εo_terminal by simp
interpret x'ε: terminal_arrow_from_functor D C F ‹Go ?x'› ‹?x'› ‹εo ?x'›
using f ex_terminal_arrow Go_εo_terminal by simp
interpret x''ε: terminal_arrow_from_functor D C F ‹Go ?x''› ‹?x''› ‹εo ?x''›
using ff' ex_terminal_arrow Go_εo_terminal by auto
have 1: "arrow_from_functor D C F (Go ?x) ?x' (f ⋅⇩C εo ?x)"
using f xε.arrow by (unfold_locales, auto)
have 2: "arrow_from_functor D C F (Go ?x') ?x'' (f' ⋅⇩C εo ?x')"
using ff' x'ε.arrow by (unfold_locales, auto)
have "G f = x'ε.the_coext (Go ?x) (C f (εo ?x))"
using f G_def by simp
hence Gf: "D.in_hom (G f) (Go ?x) (Go ?x') ∧ f ⋅⇩C εo ?x = εo ?x' ⋅⇩C F (G f)"
using 1 x'ε.the_coext_prop by simp
have "G f' = x''ε.the_coext (Go ?x') (f' ⋅⇩C εo ?x')"
using ff' G_def by auto
hence Gf': "«G f' : Go (C.cod f) →⇩D Go (C.cod f')» ∧ f' ⋅⇩C εo ?x' = εo ?x'' ⋅⇩C F (G f')"
using 2 x''ε.the_coext_prop by simp
show "G (f' ⋅⇩C f) = G f' ⋅⇩D G f"
proof -
have "x''ε.is_coext (Go ?x) ((f' ⋅⇩C f) ⋅⇩C εo ?x) (G f' ⋅⇩D G f)"
proof -
have 3: "«G f' ⋅⇩D G f : Go (C.dom f) →⇩D Go (C.cod f')»" using 1 2 Gf Gf' by auto
moreover have "(f' ⋅⇩C f) ⋅⇩C εo ?x = εo ?x'' ⋅⇩C F (G f' ⋅⇩D G f)"
by (metis 3 C.comp_assoc D.in_homE Gf Gf' preserves_comp)
ultimately show ?thesis using x''ε.is_coext_def by auto
qed
moreover have "arrow_from_functor D C F (Go ?x) ?x'' ((f' ⋅⇩C f) ⋅⇩C εo ?x)"
using ff' xε.arrow by unfold_locales blast
ultimately show ?thesis
using ff' G_def x''ε.the_coext_unique C.seqE C.cod_comp C.dom_comp by auto
qed
qed
interpretation G: "functor" C D G using G_is_functor by auto
lemma G_simp:
assumes "C.arr f"
shows "G f = terminal_arrow_from_functor.the_coext D C F (Go (C.cod f)) (εo (C.cod f))
(Go (C.dom f)) (f ⋅⇩C εo (C.dom f))"
using assms G_def by simp
interpretation idC: identity_functor C ..
interpretation GF: composite_functor C D C G F ..
interpretation ε: transformation_by_components C C GF.map C.map εo
proof
fix x :: 'c
assume x: "C.ide x"
show "«εo x : GF.map x →⇩C C.map x»"
proof -
interpret terminal_arrow_from_functor D C F ‹Go x› x ‹εo x›
using x Go_εo_terminal ex_terminal_arrow by simp
show ?thesis using x G_ide arrow by auto
qed
next
fix f :: 'c
assume f: "C.arr f"
show "εo (C.cod f) ⋅⇩C GF.map f = C.map f ⋅⇩C εo (C.dom f)"
proof -
let ?x = "C.dom f"
let ?x' = "C.cod f"
interpret xε: terminal_arrow_from_functor D C F ‹Go ?x› ?x ‹εo ?x›
using f Go_εo_terminal ex_terminal_arrow by simp
interpret x'ε: terminal_arrow_from_functor D C F ‹Go ?x'› ?x' ‹εo ?x'›
using f Go_εo_terminal ex_terminal_arrow by simp
have 1: "arrow_from_functor D C F (Go ?x) ?x' (C f (εo ?x))"
using f xε.arrow by unfold_locales auto
have "G f = x'ε.the_coext (Go ?x) (f ⋅⇩C εo ?x)"
using f G_simp by blast
hence "x'ε.is_coext (Go ?x) (f ⋅⇩C εo ?x) (G f)"
using 1 x'ε.the_coext_prop x'ε.is_coext_def by auto
thus ?thesis
using f x'ε.is_coext_def by simp
qed
qed
definition ψ
where "ψ x h = C (ε.map x) (F h)"
lemma ψ_in_hom:
assumes "C.ide x" and "«g : y →⇩D G x»"
shows "«ψ x g : F y →⇩C x»"
unfolding ψ_def using assms ε.maps_ide_in_hom by auto
lemma ψ_natural:
assumes f: "«f : x →⇩C x'»" and g: "«g : y' →⇩D y»" and h: "«h : y →⇩D G x»"
shows "f ⋅⇩C ψ x h ⋅⇩C F g = ψ x' ((G f ⋅⇩D h) ⋅⇩D g)"
proof -
have "f ⋅⇩C ψ x h ⋅⇩C F g = f ⋅⇩C (ε.map x ⋅⇩C F h) ⋅⇩C F g"
unfolding ψ_def by auto
also have "... = (f ⋅⇩C ε.map x) ⋅⇩C F h ⋅⇩C F g"
using C.comp_assoc by fastforce
also have "... = (f ⋅⇩C ε.map x) ⋅⇩C F (h ⋅⇩D g)"
using g h by fastforce
also have "... = (ε.map x' ⋅⇩C F (G f)) ⋅⇩C F (h ⋅⇩D g)"
using f ε.naturality by auto
also have "... = ε.map x' ⋅⇩C F ((G f ⋅⇩D h) ⋅⇩D g)"
using f g h C.comp_assoc by fastforce
also have "... = ψ x' ((G f ⋅⇩D h) ⋅⇩D g)"
unfolding ψ_def by auto
finally show ?thesis by auto
qed
lemma ψ_inverts_coext:
assumes x: "C.ide x" and g: "«g : y →⇩D G x»"
shows "arrow_from_functor.is_coext D C F (G x) (ε.map x) y (ψ x g) g"
proof -
interpret xε: arrow_from_functor D C F ‹G x› x ‹ε.map x›
using x ε.maps_ide_in_hom by unfold_locales auto
show "xε.is_coext y (ψ x g) g"
using x g ψ_def xε.is_coext_def G_ide by blast
qed
lemma ψ_invertible:
assumes y: "D.ide y" and f: "«f : F y →⇩C x»"
shows "∃!g. «g : y →⇩D G x» ∧ ψ x g = f"
proof
have x: "C.ide x" using f by auto
interpret xε: terminal_arrow_from_functor D C F ‹Go x› x ‹εo x›
using x ex_terminal_arrow Go_εo_terminal by auto
have 1: "arrow_from_functor D C F y x f"
using y f by (unfold_locales, auto)
let ?g = "xε.the_coext y f"
have "ψ x ?g = f"
using 1 x y ψ_def xε.the_coext_prop G_ide ψ_inverts_coext xε.is_coext_def by simp
thus "«?g : y →⇩D G x» ∧ ψ x ?g = f"
using 1 x xε.the_coext_prop G_ide by simp
show "⋀g'. «g' : y →⇩D G x» ∧ ψ x g' = f ⟹ g' = ?g"
using 1 x y ψ_inverts_coext G_ide xε.the_coext_unique by force
qed
definition φ
where "φ y f = (THE g. «g : y →⇩D G (C.cod f)» ∧ ψ (C.cod f) g = f)"
lemma φ_in_hom:
assumes "D.ide y" and "«f : F y →⇩C x»"
shows "«φ y f : y →⇩D G x»"
using assms ψ_invertible φ_def theI' [of "λg. «g : y →⇩D G x» ∧ ψ x g = f"]
by auto
lemma φ_ψ:
assumes "C.ide x" and "«g : y →⇩D G x»"
shows "φ y (ψ x g) = g"
proof -
have "φ y (ψ x g) = (THE g'. «g' : y →⇩D G x» ∧ ψ x g' = ψ x g)"
proof -
have "C.cod (ψ x g) = x"
using assms ψ_in_hom by auto
thus ?thesis
using φ_def by auto
qed
moreover have "∃!g'. «g' : y →⇩D G x» ∧ ψ x g' = ψ x g"
using assms ψ_in_hom ψ_invertible D.ide_dom by blast
ultimately show "φ y (ψ x g) = g"
using assms(2) by auto
qed
lemma ψ_φ:
assumes "D.ide y" and "«f : F y →⇩C x»"
shows "ψ x (φ y f) = f"
using assms ψ_invertible φ_def theI' [of "λg. «g : y →⇩D G x» ∧ ψ x g = f"]
by auto
lemma φ_natural:
assumes "«f : x →⇩C x'»" and "«g : y' →⇩D y»" and "«h : F y →⇩C x»"
shows "φ y' (f ⋅⇩C h ⋅⇩C F g) = (G f ⋅⇩D φ y h) ⋅⇩D g"
proof -
have "C.ide x' ∧ D.ide y ∧ D.in_hom (φ y h) y (G x)"
using assms φ_in_hom by auto
thus ?thesis
using assms D.comp_in_homI G.preserves_hom ψ_natural [of f x x' g y' y "φ y h"] φ_ψ ψ_φ
by auto
qed
theorem induces_meta_adjunction:
shows "meta_adjunction C D F G φ ψ"
using φ_in_hom ψ_in_hom φ_ψ ψ_φ φ_natural D.comp_assoc
by unfold_locales auto
end
text‹
A right adjoint functor induces a meta-adjunction, modulo the choice of a
left adjoint and unit.
›
context right_adjoint_functor
begin
definition Fo :: "'d ⇒ 'c"
where "Fo y = (SOME x. ∃u. initial_arrow_to_functor C D G y x u)"
definition ηo :: "'d ⇒ 'd"
where "ηo y = (SOME u. initial_arrow_to_functor C D G y (Fo y) u)"
lemma Fo_ηo_initial:
assumes "∃x u. initial_arrow_to_functor C D G y x u"
shows "initial_arrow_to_functor C D G y (Fo y) (ηo y)"
using assms Fo_def ηo_def
someI_ex [of "λx. ∃u. initial_arrow_to_functor C D G y x u"]
someI_ex [of "λu. initial_arrow_to_functor C D G y (Fo y) u"]
by simp
text‹
The left adjoint @{term F} to @{term g} takes each arrow @{term g} of
@{term[source=true] D} to the unique @{term[source=true] C}-extension of
@{term "D (ηo (D.cod g)) g"} along @{term "ηo (D.dom g)"}.
›
definition F :: "'d ⇒ 'c"
where "F g = (if D.arr g then
initial_arrow_to_functor.the_ext C D G (Fo (D.dom g)) (ηo (D.dom g))
(Fo (D.cod g)) (ηo (D.cod g) ⋅⇩D g)
else C.null)"
lemma F_ide:
assumes "D.ide y"
shows "F y = Fo y"
proof -
interpret initial_arrow_to_functor C D G y ‹Fo y› ‹ηo y›
using assms ex_initial_arrow Fo_ηo_initial by blast
have 1: "arrow_to_functor C D G y (Fo y) (ηo y)" ..
have "is_ext (Fo y) (ηo y) (Fo y)"
unfolding is_ext_def using arrow D.comp_ide_arr [of "G (Fo y)" "ηo y"] by force
hence "Fo y = the_ext (Fo y) (ηo y)"
using 1 the_ext_unique by blast
moreover have "ηo y = D (ηo (D.cod y)) y"
using assms arrow D.comp_arr_ide D.comp_arr_dom by auto
ultimately show ?thesis
using assms F_def D.dom_cod D.in_homE D.ide_in_hom by metis
qed
lemma F_is_functor:
shows "functor D C F"
proof
fix g :: 'd
assume "¬D.arr g"
thus "F g = C.null" using F_def by auto
next
fix g :: 'd
assume g: "D.arr g"
let ?y = "D.dom g"
let ?y' = "D.cod g"
interpret yη: initial_arrow_to_functor C D G ?y ‹Fo ?y› ‹ηo ?y›
using g ex_initial_arrow Fo_ηo_initial by simp
interpret y'η: initial_arrow_to_functor C D G ?y' ‹Fo ?y'› ‹ηo ?y'›
using g ex_initial_arrow Fo_ηo_initial by simp
have 1: "arrow_to_functor C D G ?y (Fo ?y') (D (ηo ?y') g)"
using g y'η.arrow by unfold_locales auto
have "F g = yη.the_ext (Fo ?y') (D (ηo ?y') g)"
using g F_def by simp
hence Fg: "«F g : Fo ?y →⇩C Fo ?y'» ∧ ηo ?y' ⋅⇩D g = G (F g) ⋅⇩D ηo ?y"
using 1 yη.the_ext_prop by simp
show "C.arr (F g)" using Fg by auto
show "C.dom (F g) = F ?y" using Fg g F_ide by auto
show "C.cod (F g) = F ?y'" using Fg g F_ide by auto
next
fix g :: 'd
fix g' :: 'd
assume g': "D.arr (D g' g)"
have g: "D.arr g" using g' by auto
let ?y = "D.dom g"
let ?y' = "D.cod g"
let ?y'' = "D.cod g'"
interpret yη: initial_arrow_to_functor C D G ?y ‹Fo ?y› ‹ηo ?y›
using g ex_initial_arrow Fo_ηo_initial by simp
interpret y'η: initial_arrow_to_functor C D G ?y' ‹Fo ?y'› ‹ηo ?y'›
using g ex_initial_arrow Fo_ηo_initial by simp
interpret y''η: initial_arrow_to_functor C D G ?y'' ‹Fo ?y''› ‹ηo ?y''›
using g' ex_initial_arrow Fo_ηo_initial by auto
have 1: "arrow_to_functor C D G ?y (Fo ?y') (ηo ?y' ⋅⇩D g)"
using g y'η.arrow by unfold_locales auto
have "F g = yη.the_ext (Fo ?y') (ηo ?y' ⋅⇩D g)"
using g F_def by simp
hence Fg: "«F g : Fo ?y →⇩C Fo ?y'» ∧ ηo ?y' ⋅⇩D g = G (F g) ⋅⇩D ηo ?y"
using 1 yη.the_ext_prop by simp
have 2: "arrow_to_functor C D G ?y' (Fo ?y'') (ηo ?y'' ⋅⇩D g')"
using g' y''η.arrow by unfold_locales auto
have "F g' = y'η.the_ext (Fo ?y'') (ηo ?y'' ⋅⇩D g')"
using g' F_def by auto
hence Fg': "«F g' : Fo ?y' →⇩C Fo ?y''» ∧ ηo ?y'' ⋅⇩D g' = G (F g') ⋅⇩D ηo ?y'"
using 2 y'η.the_ext_prop by simp
show "F (g' ⋅⇩D g) = F g' ⋅⇩C F g"
proof -
have "yη.is_ext (Fo ?y'') (ηo ?y'' ⋅⇩D g' ⋅⇩D g) (F g' ⋅⇩C F g)"
proof -
have 3: "«F g' ⋅⇩C F g : Fo ?y →⇩C Fo ?y''»" using 1 2 Fg Fg' by auto
moreover have "ηo ?y'' ⋅⇩D g' ⋅⇩D g = G (F g' ⋅⇩C F g) ⋅⇩D ηo ?y"
using Fg Fg' g g' 3 y''η.arrow
by (metis C.arrI D.comp_assoc preserves_comp)
ultimately show ?thesis using yη.is_ext_def by auto
qed
moreover have "arrow_to_functor C D G ?y (Fo ?y'') (ηo ?y'' ⋅⇩D g' ⋅⇩D g)"
using g g' y''η.arrow by unfold_locales auto
ultimately show ?thesis
using g g' F_def yη.the_ext_unique D.dom_comp D.cod_comp by auto
qed
qed
interpretation F: "functor" D C F using F_is_functor by auto
lemma F_simp:
assumes "D.arr g"
shows "F g = initial_arrow_to_functor.the_ext C D G (Fo (D.dom g)) (ηo (D.dom g))
(Fo (D.cod g)) (ηo (D.cod g) ⋅⇩D g)"
using assms F_def by simp
interpretation FG: composite_functor D C D F G ..
interpretation η: transformation_by_components D D D.map FG.map ηo
proof
fix y :: 'd
assume y: "D.ide y"
show "«ηo y : D.map y →⇩D FG.map y»"
proof -
interpret initial_arrow_to_functor C D G y ‹Fo y› ‹ηo y›
using y Fo_ηo_initial ex_initial_arrow by simp
show ?thesis using y F_ide arrow by auto
qed
next
fix g :: 'd
assume g: "D.arr g"
show "ηo (D.cod g) ⋅⇩D D.map g = FG.map g ⋅⇩D ηo (D.dom g)"
proof -
let ?y = "D.dom g"
let ?y' = "D.cod g"
interpret yη: initial_arrow_to_functor C D G ?y ‹Fo ?y› ‹ηo ?y›
using g Fo_ηo_initial ex_initial_arrow by simp
interpret y'η: initial_arrow_to_functor C D G ?y' ‹Fo ?y'› ‹ηo ?y'›
using g Fo_ηo_initial ex_initial_arrow by simp
have "arrow_to_functor C D G ?y (Fo ?y') (ηo ?y' ⋅⇩D g)"
using g y'η.arrow by unfold_locales auto
moreover have "F g = yη.the_ext (Fo ?y') (ηo ?y' ⋅⇩D g)"
using g F_simp by blast
ultimately have "yη.is_ext (Fo ?y') (ηo ?y' ⋅⇩D g) (F g)"
using yη.the_ext_prop yη.is_ext_def by auto
thus ?thesis
using g yη.is_ext_def by simp
qed
qed
definition φ
where "φ y h = D (G h) (η.map y)"
lemma φ_in_hom:
assumes y: "D.ide y" and f: "«f : F y →⇩C x»"
shows "«φ y f : y →⇩D G x»"
unfolding φ_def using assms η.maps_ide_in_hom by auto
lemma φ_natural:
assumes f: "«f : x →⇩C x'»" and g: "«g : y' →⇩D y»" and h: "«h : F y →⇩C x»"
shows "φ y' (f ⋅⇩C h ⋅⇩C F g) = (G f ⋅⇩D φ y h) ⋅⇩D g"
proof -
have "(G f ⋅⇩D φ y h) ⋅⇩D g = (G f ⋅⇩D G h ⋅⇩D η.map y) ⋅⇩D g"
unfolding φ_def by auto
also have "... = (G f ⋅⇩D G h) ⋅⇩D η.map y ⋅⇩D g"
using D.comp_assoc by fastforce
also have "... = G (f ⋅⇩C h) ⋅⇩D G (F g) ⋅⇩D η.map y'"
using f g h η.naturality by fastforce
also have "... = (G (f ⋅⇩C h) ⋅⇩D G (F g)) ⋅⇩D η.map y'"
using D.comp_assoc by fastforce
also have "... = G (f ⋅⇩C h ⋅⇩C F g) ⋅⇩D η.map y'"
using f g h D.comp_assoc by fastforce
also have "... = φ y' (f ⋅⇩C h ⋅⇩C F g)"
unfolding φ_def by auto
finally show ?thesis by auto
qed
lemma φ_inverts_ext:
assumes y: "D.ide y" and f: "«f : F y →⇩C x»"
shows "arrow_to_functor.is_ext C D G (F y) (η.map y) x (φ y f) f"
proof -
interpret yη: arrow_to_functor C D G y ‹F y› ‹η.map y›
using y η.maps_ide_in_hom by unfold_locales auto
show "yη.is_ext x (φ y f) f"
using f y φ_def yη.is_ext_def F_ide by blast
qed
lemma φ_invertible:
assumes x: "C.ide x" and g: "«g : y →⇩D G x»"
shows "∃!f. «f : F y →⇩C x» ∧ φ y f = g"
proof
have y: "D.ide y" using g by auto
interpret yη: initial_arrow_to_functor C D G y ‹Fo y› ‹ηo y›
using y ex_initial_arrow Fo_ηo_initial by auto
have 1: "arrow_to_functor C D G y x g"
using x g by (unfold_locales, auto)
let ?f = "yη.the_ext x g"
have "φ y ?f = g"
using φ_def yη.the_ext_prop 1 F_ide x y φ_inverts_ext yη.is_ext_def by fastforce
moreover have "«?f : F y →⇩C x»"
using 1 y yη.the_ext_prop F_ide by simp
ultimately show "«?f : F y →⇩C x» ∧ φ y ?f = g" by auto
show "⋀f'. «f' : F y →⇩C x» ∧ φ y f' = g ⟹ f' = ?f"
using 1 y φ_inverts_ext yη.the_ext_unique F_ide by force
qed
definition ψ
where "ψ x g = (THE f. «f : F (D.dom g) →⇩C x» ∧ φ (D.dom g) f = g)"
lemma ψ_in_hom:
assumes "C.ide x" and "«g : y →⇩D G x»"
shows "C.in_hom (ψ x g) (F y) x"
using assms φ_invertible ψ_def theI' [of "λf. «f : F y →⇩C x» ∧ φ y f = g"]
by auto
lemma ψ_φ:
assumes "D.ide y" and "«f : F y →⇩C x»"
shows "ψ x (φ y f) = f"
proof -
have "D.dom (φ y f) = y" using assms φ_in_hom by blast
hence "ψ x (φ y f) = (THE f'. «f' : F y →⇩C x» ∧ φ y f' = φ y f)"
using ψ_def by auto
moreover have "∃!f'. «f' : F y →⇩C x» ∧ φ y f' = φ y f"
using assms φ_in_hom φ_invertible C.ide_cod by blast
ultimately show ?thesis using assms(2) by auto
qed
lemma φ_ψ:
assumes "C.ide x" and "«g : y →⇩D G x»"
shows "φ y (ψ x g) = g"
using assms φ_invertible ψ_def theI' [of "λf. «f : F y →⇩C x» ∧ φ y f = g"]
by auto
theorem induces_meta_adjunction:
shows "meta_adjunction C D F G φ ψ"
using φ_in_hom ψ_in_hom φ_ψ ψ_φ φ_natural D.comp_assoc
by (unfold_locales, auto)
end
section "Meta-Adjunctions Induce Hom-Adjunctions"
text‹
To obtain a hom-adjunction from a meta-adjunction, we need to exhibit hom-functors
from @{term C} and @{term D} to a common set category @{term S}, so it is necessary
to apply an actual concrete construction of such a category.
We use the replete set category generated by the disjoint sum
@{typ "('c+'d)"} of the arrow types of @{term C} and @{term D}.
›
context meta_adjunction
begin
interpretation S: replete_setcat ‹TYPE('c+'d)› .
definition inC :: "'c ⇒ ('c+'d) setcat.arr"
where "inC ≡ S.UP o Inl"
definition inD :: "'d ⇒ ('c+'d) setcat.arr"
where "inD ≡ S.UP o Inr"
interpretation S: replete_setcat ‹TYPE('c+'d)› .
interpretation Cop: dual_category C ..
interpretation Dop: dual_category D ..
interpretation CopxC: product_category Cop.comp C ..
interpretation DopxD: product_category Dop.comp D ..
interpretation DopxC: product_category Dop.comp C ..
interpretation HomC: hom_functor C S.comp S.setp ‹λ_. inC›
proof
show "⋀f. C.arr f ⟹ inC f ∈ S.Univ"
unfolding inC_def using S.UP_mapsto by auto
thus "⋀b a. ⟦C.ide b; C.ide a⟧ ⟹ inC ` C.hom b a ⊆ S.Univ"
by blast
show "⋀b a. ⟦C.ide b; C.ide a⟧ ⟹ inj_on inC (C.hom b a)"
unfolding inC_def
using S.inj_UP
by (metis injD inj_Inl inj_compose inj_on_def)
qed
interpretation HomD: hom_functor D S.comp S.setp ‹λ_. inD›
proof
show "⋀f. D.arr f ⟹ inD f ∈ S.Univ"
unfolding inD_def using S.UP_mapsto by auto
thus "⋀b a. ⟦D.ide b; D.ide a⟧ ⟹ inD ` D.hom b a ⊆ S.Univ"
by blast
show "⋀b a. ⟦D.ide b; D.ide a⟧ ⟹ inj_on inD (D.hom b a)"
unfolding inD_def
using S.inj_UP
by (metis injD inj_Inr inj_compose inj_on_def)
qed
interpretation Fop: dual_functor D C F ..
interpretation FopxC: product_functor Dop.comp C Cop.comp C Fop.map C.map ..
interpretation DopxG: product_functor Dop.comp C Dop.comp D Dop.map G ..
interpretation Hom_FopxC: composite_functor DopxC.comp CopxC.comp S.comp
FopxC.map HomC.map ..
interpretation Hom_DopxG: composite_functor DopxC.comp DopxD.comp S.comp
DopxG.map HomD.map ..
lemma inC_ψ [simp]:
assumes "C.ide b" and "C.ide a" and "x ∈ inC ` C.hom b a"
shows "inC (HomC.ψ (b, a) x) = x"
using assms by auto
lemma ψ_inC [simp]:
assumes "C.arr f"
shows "HomC.ψ (C.dom f, C.cod f) (inC f) = f"
using assms HomC.ψ_φ by blast
lemma inD_ψ [simp]:
assumes "D.ide b" and "D.ide a" and "x ∈ inD ` D.hom b a"
shows "inD (HomD.ψ (b, a) x) = x"
using assms by auto
lemma ψ_inD [simp]:
assumes "D.arr f"
shows "HomD.ψ (D.dom f, D.cod f) (inD f) = f"
using assms HomD.ψ_φ by blast
lemma Hom_FopxC_simp:
assumes "DopxC.arr gf"
shows "Hom_FopxC.map gf =
S.mkArr (HomC.set (F (D.cod (fst gf)), C.dom (snd gf)))
(HomC.set (F (D.dom (fst gf)), C.cod (snd gf)))
(inC ∘ (λh. snd gf ⋅⇩C h ⋅⇩C F (fst gf))
∘ HomC.ψ (F (D.cod (fst gf)), C.dom (snd gf)))"
using assms HomC.map_def by simp
lemma Hom_DopxG_simp:
assumes "DopxC.arr gf"
shows "Hom_DopxG.map gf =
S.mkArr (HomD.set (D.cod (fst gf), G (C.dom (snd gf))))
(HomD.set (D.dom (fst gf), G (C.cod (snd gf))))
(inD ∘ (λh. G (snd gf) ⋅⇩D h ⋅⇩D fst gf)
∘ HomD.ψ (D.cod (fst gf), G (C.dom (snd gf))))"
using assms HomD.map_def by simp
definition Φo
where "Φo yx = S.mkArr (HomC.set (F (fst yx), snd yx))
(HomD.set (fst yx, G (snd yx)))
(inD o φ (fst yx) o HomC.ψ (F (fst yx), snd yx))"
lemma Φo_in_hom:
assumes yx: "DopxC.ide yx"
shows "«Φo yx : Hom_FopxC.map yx →⇩S Hom_DopxG.map yx»"
proof -
have "Hom_FopxC.map yx = S.mkIde (HomC.set (F (fst yx), snd yx))"
using yx HomC.map_ide by auto
moreover have "Hom_DopxG.map yx = S.mkIde (HomD.set (fst yx, G (snd yx)))"
using yx HomD.map_ide by auto
moreover have
"«S.mkArr (HomC.set (F (fst yx), snd yx)) (HomD.set (fst yx, G (snd yx)))
(inD ∘ φ (fst yx) ∘ HomC.ψ (F (fst yx), snd yx)) :
S.mkIde (HomC.set (F (fst yx), snd yx))
→⇩S S.mkIde (HomD.set (fst yx, G (snd yx)))»"
proof (intro S.mkArr_in_hom)
show "HomC.set (F (fst yx), snd yx) ⊆ S.Univ" using yx HomC.set_subset_Univ by simp
show "HomD.set (fst yx, G (snd yx)) ⊆ S.Univ" using yx HomD.set_subset_Univ by simp
show "inD o φ (fst yx) o HomC.ψ (F (fst yx), snd yx)
∈ HomC.set (F (fst yx), snd yx) → HomD.set (fst yx, G (snd yx))"
proof
fix x
assume x: "x ∈ HomC.set (F (fst yx), snd yx)"
show "(inD o φ (fst yx) o HomC.ψ (F (fst yx), snd yx)) x
∈ HomD.set (fst yx, G (snd yx))"
using x yx HomC.ψ_mapsto [of "F (fst yx)" "snd yx"]
φ_in_hom [of "fst yx"] HomD.φ_mapsto [of "fst yx" "G (snd yx)"]
by auto
qed
qed
ultimately show ?thesis using Φo_def by auto
qed
interpretation Φ: transformation_by_components
DopxC.comp S.comp Hom_FopxC.map Hom_DopxG.map Φo
proof
fix yx
assume yx: "DopxC.ide yx"
show "«Φo yx : Hom_FopxC.map yx →⇩S Hom_DopxG.map yx»"
using yx Φo_in_hom by auto
next
fix gf
assume gf: "DopxC.arr gf"
show "S.comp (Φo (DopxC.cod gf)) (Hom_FopxC.map gf)
= S.comp (Hom_DopxG.map gf) (Φo (DopxC.dom gf))"
proof -
let ?g = "fst gf"
let ?f = "snd gf"
let ?x = "C.dom ?f"
let ?x' = "C.cod ?f"
let ?y = "D.cod ?g"
let ?y' = "D.dom ?g"
let ?Fy = "F ?y"
let ?Fy' = "F ?y'"
let ?Fg = "F ?g"
let ?Gx = "G ?x"
let ?Gx' = "G ?x'"
let ?Gf = "G ?f"
have 1: "S.arr (Hom_FopxC.map gf) ∧
Hom_FopxC.map gf = S.mkArr (HomC.set (?Fy, ?x)) (HomC.set (?Fy', ?x'))
(inC o (λh. ?f ⋅⇩C h ⋅⇩C ?Fg) o HomC.ψ (?Fy, ?x))"
using gf Hom_FopxC.preserves_arr Hom_FopxC_simp by blast
have 2: "S.arr (Φo (DopxC.cod gf)) ∧
Φo (DopxC.cod gf) = S.mkArr (HomC.set (?Fy', ?x')) (HomD.set (?y', ?Gx'))
(inD o φ ?y' o HomC.ψ (?Fy', ?x'))"
using gf Φo_in_hom [of "DopxC.cod gf"] Φo_def [of "DopxC.cod gf"] φ_in_hom
by auto
have 3: "S.arr (Φo (DopxC.dom gf)) ∧
Φo (DopxC.dom gf) = S.mkArr (HomC.set (?Fy, ?x)) (HomD.set (?y, ?Gx))
(inD o φ ?y o HomC.ψ (?Fy, ?x))"
using gf Φo_in_hom [of "DopxC.dom gf"] Φo_def [of "DopxC.dom gf"] φ_in_hom
by auto
have 4: "S.arr (Hom_DopxG.map gf) ∧
Hom_DopxG.map gf = S.mkArr (HomD.set (?y, ?Gx)) (HomD.set (?y', ?Gx'))
(inD o (λh. ?Gf ⋅⇩D h ⋅⇩D ?g) o HomD.ψ (?y, ?Gx))"
using gf Hom_DopxG.preserves_arr Hom_DopxG_simp by blast
have 5: "S.seq (Φo (DopxC.cod gf)) (Hom_FopxC.map gf) ∧
S.comp (Φo (DopxC.cod gf)) (Hom_FopxC.map gf)
= S.mkArr (HomC.set (?Fy, ?x)) (HomD.set (?y', ?Gx'))
((inD o φ ?y' o HomC.ψ (?Fy', ?x'))
o (inC o (λh. ?f ⋅⇩C h ⋅⇩C ?Fg) o HomC.ψ (?Fy, ?x)))"
by (metis gf 1 2 DopxC.arr_iff_in_hom DopxC.ide_cod Hom_FopxC.preserves_hom
S.comp_mkArr S.seqI' Φo_in_hom)
have 6: "S.comp (Hom_DopxG.map gf) (Φo (DopxC.dom gf))
= S.mkArr (HomC.set (?Fy, ?x)) (HomD.set (?y', ?Gx'))
((inD o (λh. ?Gf ⋅⇩D h ⋅⇩D ?g) o HomD.ψ (?y, ?Gx))
o (inD o φ ?y o HomC.ψ (?Fy, ?x)))"
by (metis 3 4 S.comp_mkArr)
have 7:
"restrict ((inD o φ ?y' o HomC.ψ (?Fy', ?x'))
o (inC o (λh. ?f ⋅⇩C h ⋅⇩C ?Fg) o HomC.ψ (?Fy, ?x))) (HomC.set (?Fy, ?x))
= restrict ((inD o (λh. ?Gf ⋅⇩D h ⋅⇩D ?g) o HomD.ψ (?y, ?Gx))
o (inD o φ ?y o HomC.ψ (?Fy, ?x))) (HomC.set (?Fy, ?x))"
proof (intro restrict_ext)
show "⋀h. h ∈ HomC.set (?Fy, ?x) ⟹
((inD o φ ?y' o HomC.ψ (?Fy', ?x'))
o (inC o (λh. ?f ⋅⇩C h ⋅⇩C ?Fg) o HomC.ψ (?Fy, ?x))) h
= ((inD o (λh. ?Gf ⋅⇩D h ⋅⇩D ?g) o HomD.ψ (?y, ?Gx))
o (inD o φ ?y o HomC.ψ (?Fy, ?x))) h"
proof -
fix h
assume h: "h ∈ HomC.set (?Fy, ?x)"
have ψh: "«HomC.ψ (?Fy, ?x) h : ?Fy →⇩C ?x»"
using gf h HomC.ψ_mapsto [of ?Fy ?x] CopxC.ide_char by auto
show "((inD o φ ?y' o HomC.ψ (?Fy', ?x'))
o (inC o (λh. ?f ⋅⇩C h ⋅⇩C ?Fg) o HomC.ψ (?Fy, ?x))) h
= ((inD o (λh. ?Gf ⋅⇩D h ⋅⇩D ?g) o HomD.ψ (?y, ?Gx))
o (inD o φ ?y o HomC.ψ (?Fy, ?x))) h"
proof -
have
"((inD o φ ?y' o HomC.ψ (?Fy', ?x'))
o (inC o (λh. ?f ⋅⇩C h ⋅⇩C ?Fg) o HomC.ψ (?Fy, ?x))) h
= inD (φ ?y' (?f ⋅⇩C HomC.ψ (?Fy, ?x) h ⋅⇩C ?Fg))"
using gf ψh HomC.φ_mapsto HomC.ψ_mapsto φ_in_hom
ψ_inC [of "?f ⋅⇩C HomC.ψ (?Fy, ?x) h ⋅⇩C ?Fg"]
by auto
also have "... = inD (D ?Gf (D (φ ?y (HomC.ψ (?Fy, ?x) h)) ?g))"
by (metis (no_types, lifting) C.arr_cod C.arr_dom_iff_arr C.arr_iff_in_hom
C.in_homE D.arr_cod_iff_arr D.arr_iff_in_hom F.preserves_reflects_arr
φ_naturality ψh)
also have "... = ((inD o (λh. ?Gf ⋅⇩D h ⋅⇩D ?g) o HomD.ψ (?y, ?Gx))
o (inD o φ ?y o HomC.ψ (?Fy, ?x))) h"
using gf ψh φ_in_hom by simp
finally show ?thesis by auto
qed
qed
qed
have 8: "S.mkArr (HomC.set (?Fy, ?x)) (HomD.set (?y', ?Gx'))
((inD o φ ?y' o HomC.ψ (?Fy', ?x'))
o (inC o (λh. ?f ⋅⇩C h ⋅⇩C ?Fg) o HomC.ψ (?Fy, ?x)))
= S.mkArr (HomC.set (?Fy, ?x)) (HomD.set (?y', ?Gx'))
((inD o (λh. ?Gf ⋅⇩D h ⋅⇩D ?g) o HomD.ψ (?y, ?Gx))
o (inD o φ ?y o HomC.ψ (?Fy, ?x)))"
using 5 7 by force
show ?thesis using 5 6 8 by auto
qed
qed
lemma Φ_simp:
assumes YX: "DopxC.ide yx"
shows "Φ.map yx =
S.mkArr (HomC.set (F (fst yx), snd yx)) (HomD.set (fst yx, G (snd yx)))
(inD o φ (fst yx) o HomC.ψ (F (fst yx), snd yx))"
using YX Φo_def by simp
abbreviation Ψo
where "Ψo yx ≡ S.mkArr (HomD.set (fst yx, G (snd yx))) (HomC.set (F (fst yx), snd yx))
(inC o ψ (snd yx) o HomD.ψ (fst yx, G (snd yx)))"
lemma Ψo_in_hom:
assumes yx: "DopxC.ide yx"
shows "«Ψo yx : Hom_DopxG.map yx →⇩S Hom_FopxC.map yx»"
proof -
have "Hom_FopxC.map yx = S.mkIde (HomC.set (F (fst yx), snd yx))"
using yx HomC.map_ide by auto
moreover have "Hom_DopxG.map yx = S.mkIde (HomD.set (fst yx, G (snd yx)))"
using yx HomD.map_ide by auto
moreover have "«Ψo yx : S.mkIde (HomD.set (fst yx, G (snd yx)))
→⇩S S.mkIde (HomC.set (F (fst yx), snd yx))»"
proof (intro S.mkArr_in_hom)
show "HomC.set (F (fst yx), snd yx) ⊆ S.Univ" using yx HomC.set_subset_Univ by simp
show "HomD.set (fst yx, G (snd yx)) ⊆ S.Univ" using yx HomD.set_subset_Univ by simp
show "inC o ψ (snd yx) o HomD.ψ (fst yx, G (snd yx))
∈ HomD.set (fst yx, G (snd yx)) → HomC.set (F (fst yx), snd yx)"
proof
fix x
assume x: "x ∈ HomD.set (fst yx, G (snd yx))"
show "(inC o ψ (snd yx) o HomD.ψ (fst yx, G (snd yx))) x
∈ HomC.set (F (fst yx), snd yx)"
using x yx HomD.ψ_mapsto [of "fst yx" "G (snd yx)"] ψ_in_hom [of "snd yx"]
HomC.φ_mapsto [of "F (fst yx)" "snd yx"]
by auto
qed
qed
ultimately show ?thesis by auto
qed
lemma Φ_inv:
assumes yx: "DopxC.ide yx"
shows "S.inverse_arrows (Φ.map yx) (Ψo yx)"
proof -
have 1: "«Φ.map yx : Hom_FopxC.map yx →⇩S Hom_DopxG.map yx»"
using yx Φ.preserves_hom [of yx yx yx] DopxC.ide_in_hom by blast
have 2: "«Ψo yx : Hom_DopxG.map yx →⇩S Hom_FopxC.map yx»"
using yx Ψo_in_hom by simp
have 3: "Φ.map yx = S.mkArr (HomC.set (F (fst yx), snd yx))
(HomD.set (fst yx, G (snd yx)))
(inD o φ (fst yx) o HomC.ψ (F (fst yx), snd yx))"
using yx Φ_simp by blast
have antipar: "S.antipar (Φ.map yx) (Ψo yx)"
using 1 2 by blast
moreover have "S.ide (S.comp (Ψo yx) (Φ.map yx))"
proof -
have "S.comp (Ψo yx) (Φ.map yx) =
S.mkArr (HomC.set (F (fst yx), snd yx)) (HomC.set (F (fst yx), snd yx))
((inC o ψ (snd yx) o HomD.ψ (fst yx, G (snd yx)))
o (inD o φ (fst yx) o HomC.ψ (F (fst yx), snd yx)))"
using 1 2 3 antipar S.comp_mkArr by auto
also have
"... = S.mkArr (HomC.set (F (fst yx), snd yx)) (HomC.set (F (fst yx), snd yx))
(λx. x)"
proof -
have
"S.mkArr (HomC.set (F (fst yx), snd yx)) (HomC.set (F (fst yx), snd yx)) (λx. x)
= ..."
proof
show
"S.arr (S.mkArr (HomC.set (F (fst yx), snd yx)) (HomC.set (F (fst yx), snd yx))
(λx. x))"
using yx HomC.set_subset_Univ by simp
show "⋀x. x ∈ HomC.set (F (fst yx), snd yx) ⟹
x = ((inC o ψ (snd yx) o HomD.ψ (fst yx, G (snd yx)))
o (inD o φ (fst yx) o HomC.ψ (F (fst yx), snd yx))) x"
proof -
fix x
assume x: "x ∈ HomC.set (F (fst yx), snd yx)"
have "((inC o ψ (snd yx) o HomD.ψ (fst yx, G (snd yx)))
o (inD o φ (fst yx) o HomC.ψ (F (fst yx), snd yx))) x
= inC (ψ (snd yx) (HomD.ψ (fst yx, G (snd yx))
(inD (φ (fst yx) (HomC.ψ (F (fst yx), snd yx) x)))))"
by simp
also have "... = inC (ψ (snd yx) (φ (fst yx) (HomC.ψ (F (fst yx), snd yx) x)))"
using x yx HomC.ψ_mapsto [of "F (fst yx)" "snd yx"] φ_in_hom by force
also have "... = inC (HomC.ψ (F (fst yx), snd yx) x)"
using x yx HomC.ψ_mapsto [of "F (fst yx)" "snd yx"] ψ_φ by force
also have "... = x" using x yx inC_ψ by simp
finally show "x = ((inC o ψ (snd yx) o HomD.ψ (fst yx, G (snd yx)))
o (inD o φ (fst yx) o HomC.ψ (F (fst yx), snd yx))) x"
by auto
qed
qed
thus ?thesis by auto
qed
also have "... = S.mkIde (HomC.set (F (fst yx), snd yx))"
using yx S.mkIde_as_mkArr HomC.set_subset_Univ by force
finally have
"S.comp (Ψo yx) (Φ.map yx) = S.mkIde (HomC.set (F (fst yx), snd yx))"
by auto
thus ?thesis using yx HomC.set_subset_Univ S.ide_mkIde by simp
qed
moreover have "S.ide (S.comp (Φ.map yx) (Ψo yx))"
proof -
have "S.comp (Φ.map yx) (Ψo yx) =
S.mkArr (HomD.set (fst yx, G (snd yx))) (HomD.set (fst yx, G (snd yx)))
((inD o φ (fst yx) o HomC.ψ (F (fst yx), snd yx))
o (inC o ψ (snd yx) o HomD.ψ (fst yx, G (snd yx))))"
using 1 2 3 S.comp_mkArr antipar by fastforce
also
have "... = S.mkArr (HomD.set (fst yx, G (snd yx))) (HomD.set (fst yx, G (snd yx)))
(λx. x)"
proof -
have
"S.mkArr (HomD.set (fst yx, G (snd yx))) (HomD.set (fst yx, G (snd yx))) (λx. x)
= ..."
proof
show
"S.arr (S.mkArr (HomD.set (fst yx, G (snd yx))) (HomD.set (fst yx, G (snd yx)))
(λx. x))"
using yx HomD.set_subset_Univ by simp
show "⋀x. x ∈ (HomD.set (fst yx, G (snd yx))) ⟹
x = ((inD o φ (fst yx) o HomC.ψ (F (fst yx), snd yx))
o (inC o ψ (snd yx) o HomD.ψ (fst yx, G (snd yx)))) x"
proof -
fix x
assume x: "x ∈ HomD.set (fst yx, G (snd yx))"
have "((inD o φ (fst yx) o HomC.ψ (F (fst yx), snd yx))
o (inC o ψ (snd yx) o HomD.ψ (fst yx, G (snd yx)))) x
= inD (φ (fst yx) (HomC.ψ (F (fst yx), snd yx)
(inC (ψ (snd yx) (HomD.ψ (fst yx, G (snd yx)) x)))))"
by simp
also have "... = inD (φ (fst yx) (ψ (snd yx) (HomD.ψ (fst yx, G (snd yx)) x)))"
proof -
have "«ψ (snd yx) (HomD.ψ (fst yx, G (snd yx)) x) : F (fst yx) → snd yx»"
using x yx HomD.ψ_mapsto [of "fst yx" "G (snd yx)"] ψ_in_hom by auto
thus ?thesis by simp
qed
also have "... = inD (HomD.ψ (fst yx, G (snd yx)) x)"
using x yx HomD.ψ_mapsto [of "fst yx" "G (snd yx)"] φ_ψ by force
also have "... = x" using x yx inD_ψ by simp
finally show "x = ((inD o φ (fst yx) o HomC.ψ (F (fst yx), snd yx))
o (inC o ψ (snd yx) o HomD.ψ (fst yx, G (snd yx)))) x"
by auto
qed
qed
thus ?thesis by auto
qed
also have "... = S.mkIde (HomD.set (fst yx, G (snd yx)))"
using yx S.mkIde_as_mkArr HomD.set_subset_Univ by force
finally have
"S.comp (Φ.map yx) (Ψo yx) = S.mkIde (HomD.set (fst yx, G (snd yx)))"
by auto
thus ?thesis using yx HomD.set_subset_Univ S.ide_mkIde by simp
qed
ultimately show ?thesis by auto
qed
interpretation Φ: natural_isomorphism DopxC.comp S.comp
Hom_FopxC.map Hom_DopxG.map Φ.map
using Φ_inv by unfold_locales blast
interpretation Ψ: inverse_transformation DopxC.comp S.comp
Hom_FopxC.map Hom_DopxG.map Φ.map ..
interpretation ΦΨ: inverse_transformations DopxC.comp S.comp
Hom_FopxC.map Hom_DopxG.map Φ.map Ψ.map
using Ψ.inverts_components by unfold_locales simp
abbreviation Φ where "Φ ≡ Φ.map"
abbreviation Ψ where "Ψ ≡ Ψ.map"
abbreviation HomC where "HomC ≡ HomC.map"
abbreviation φC where "φC ≡ λ_. inC"
abbreviation HomD where "HomD ≡ HomD.map"
abbreviation φD where "φD ≡ λ_. inD"
theorem induces_hom_adjunction: "hom_adjunction C D S.comp S.setp φC φD F G Φ Ψ" ..
lemma Ψ_simp:
assumes yx: "DopxC.ide yx"
shows "Ψ yx = S.mkArr (HomD.set (fst yx, G (snd yx))) (HomC.set (F (fst yx), snd yx))
(inC o ψ (snd yx) o HomD.ψ (fst yx, G (snd yx)))"
using assms Φo_def Φ_inv S.inverse_unique by simp
text‹
The original @{term φ} and @{term ψ} can be recovered from @{term Φ} and @{term Ψ}.
›
interpretation Φ: set_valued_transformation DopxC.comp S.comp S.setp
Hom_FopxC.map Hom_DopxG.map Φ.map ..
interpretation Ψ: set_valued_transformation DopxC.comp S.comp S.setp
Hom_DopxG.map Hom_FopxC.map Ψ.map ..
lemma φ_in_terms_of_Φ':
assumes y: "D.ide y" and f: "«f: F y →⇩C x»"
shows "φ y f = (HomD.ψ (y, G x) o Φ.FUN (y, x) o inC) f"
proof -
have x: "C.ide x" using f by auto
have "(HomD.ψ (y, G x) o Φ.FUN (y, x) o inC) f =
HomD.ψ (y, G x)
(restrict (inD o φ y o HomC.ψ (F y, x)) (HomC.set (F y, x)) (inC f))"
proof -
have "S.arr (Φ (y, x))" using x y by fastforce
thus ?thesis
using x y Φo_def by simp
qed
also have "... = φ y f"
using x y f HomC.φ_mapsto φ_in_hom HomC.ψ_mapsto C.ide_in_hom D.ide_in_hom
by auto
finally show ?thesis by auto
qed
lemma ψ_in_terms_of_Ψ':
assumes x: "C.ide x" and g: "«g : y →⇩D G x»"
shows "ψ x g = (HomC.ψ (F y, x) o Ψ.FUN (y, x) o inD) g"
proof -
have y: "D.ide y" using g by auto
have "(HomC.ψ (F y, x) o Ψ.FUN (y, x) o inD) g =
HomC.ψ (F y, x)
(restrict (inC o ψ x o HomD.ψ (y, G x)) (HomD.set (y, G x)) (inD g))"
proof -
have "S.arr (Ψ (y, x))"
using x y Ψ.preserves_reflects_arr [of "(y, x)"] by simp
thus ?thesis
using x y Ψ_simp by simp
qed
also have "... = ψ x g"
using x y g HomD.φ_mapsto ψ_in_hom HomD.ψ_mapsto C.ide_in_hom D.ide_in_hom
by auto
finally show ?thesis by auto
qed
end
section "Hom-Adjunctions Induce Meta-Adjunctions"
context hom_adjunction
begin
definition φ :: "'d ⇒ 'c ⇒ 'd"
where
"φ y h = (HomD.ψ (y, G (C.cod h)) o Φ.FUN (y, C.cod h) o φC (F y, C.cod h)) h"
definition ψ :: "'c ⇒ 'd ⇒ 'c"
where
"ψ x h = (HomC.ψ (F (D.dom h), x) o Ψ.FUN (D.dom h, x) o φD (D.dom h, G x)) h"
lemma Hom_FopxC_map_simp:
assumes "DopxC.arr gf"
shows "Hom_FopxC.map gf =
S.mkArr (HomC.set (F (D.cod (fst gf)), C.dom (snd gf)))
(HomC.set (F (D.dom (fst gf)), C.cod (snd gf)))
(φC (F (D.dom (fst gf)), C.cod (snd gf))
o (λh. snd gf ⋅⇩C h ⋅⇩C F (fst gf))
o HomC.ψ (F (D.cod (fst gf)), C.dom (snd gf)))"
using assms HomC.map_def by simp
lemma Hom_DopxG_map_simp:
assumes "DopxC.arr gf"
shows "Hom_DopxG.map gf =
S.mkArr (HomD.set (D.cod (fst gf), G (C.dom (snd gf))))
(HomD.set (D.dom (fst gf), G (C.cod (snd gf))))
(φD (D.dom (fst gf), G (C.cod (snd gf)))
o (λh. G (snd gf) ⋅⇩D h ⋅⇩D fst gf)
o HomD.ψ (D.cod (fst gf), G (C.dom (snd gf))))"
using assms HomD.map_def by simp
lemma Φ_Fun_mapsto:
assumes "D.ide y" and "«f : F y →⇩C x»"
shows "Φ.FUN (y, x) ∈ HomC.set (F y, x) → HomD.set (y, G x)"
proof -
have "S.arr (Φ (y, x)) ∧ Φ.DOM (y, x) = HomC.set (F y, x) ∧
Φ.COD (y, x) = HomD.set (y, G x)"
using assms HomC.set_map HomD.set_map by auto
thus ?thesis using S.Fun_mapsto by blast
qed
lemma φ_mapsto:
assumes y: "D.ide y"
shows "φ y ∈ C.hom (F y) x → D.hom y (G x)"
proof
fix h
assume h: "h ∈ C.hom (F y) x"
hence 1: " «h : F y →⇩C x»" by simp
show "φ y h ∈ D.hom y (G x)"
proof -
have "φC (F y, x) h ∈ HomC.set (F y, x)"
using y h 1 HomC.φ_mapsto [of "F y" x] by fastforce
hence "Φ.FUN (y, x) (φC (F y, x) h) ∈ HomD.set (y, G x)"
using h y Φ_Fun_mapsto by auto
thus ?thesis
using y h 1 φ_def HomC.φ_mapsto HomD.ψ_mapsto [of y "G x"] by fastforce
qed
qed
lemma Φ_simp:
assumes "D.ide y" and "C.ide x"
shows "S.arr (Φ (y, x))"
and "Φ (y, x) = S.mkArr (HomC.set (F y, x)) (HomD.set (y, G x))
(φD (y, G x) o φ y o ψC (F y, x))"
proof -
show 1: "S.arr (Φ (y, x))" using assms by auto
hence "Φ (y, x) = S.mkArr (Φ.DOM (y, x)) (Φ.COD (y, x)) (Φ.FUN (y, x))"
using S.mkArr_Fun by metis
also have "... = S.mkArr (HomC.set (F y, x)) (HomD.set (y, G x)) (Φ.FUN (y, x))"
using assms HomC.set_map HomD.set_map by fastforce
also have "... = S.mkArr (HomC.set (F y, x)) (HomD.set (y, G x))
(φD (y, G x) o φ y o ψC (F y, x))"
proof (intro S.mkArr_eqI')
show 2: "S.arr (S.mkArr (HomC.set (F y, x)) (HomD.set (y, G x)) (Φ.FUN (y, x)))"
using 1 calculation by argo
show "⋀h. h ∈ HomC.set (F y, x) ⟹
Φ.FUN (y, x) h = (φD (y, G x) o φ y o ψC (F y, x)) h"
proof -
fix h
assume h: "h ∈ HomC.set (F y, x)"
have "(φD (y, G x) o φ y o HomC.ψ (F y, x)) h =
φD (y, G x) (ψD (y, G x) (Φ.FUN (y, x) (φC (F y, x) (ψC (F y, x) h))))"
proof -
have "«ψC (F y, x) h : F y →⇩C x»"
using assms h HomC.ψ_mapsto [of "F y" x] by auto
thus ?thesis
using h φ_def by auto
qed
also have "... = φD (y, G x) (ψD (y, G x) (Φ.FUN (y, x) h))"
using assms h HomC.φ_ψ Φ_Fun_mapsto by simp
also have "... = Φ.FUN (y, x) h"
using assms h Φ_Fun_mapsto [of y "ψC (F y, x) h"] HomC.ψ_mapsto
HomD.φ_ψ [of y "G x"] C.ide_in_hom D.ide_in_hom
by blast
finally show "Φ.FUN (y, x) h = (φD (y, G x) o φ y o ψC (F y, x)) h" by auto
qed
qed
finally show "Φ (y, x) = S.mkArr (HomC.set (F y, x)) (HomD.set (y, G x))
(φD (y, G x) o φ y o ψC (F y, x))"
by force
qed
lemma Ψ_Fun_mapsto:
assumes "C.ide x" and "«g : y →⇩D G x»"
shows "Ψ.FUN (y, x) ∈ HomD.set (y, G x) → HomC.set (F y, x)"
proof -
have "S.arr (Ψ (y, x)) ∧ Ψ.COD (y, x) = HomC.set (F y, x) ∧
Ψ.DOM (y, x) = HomD.set (y, G x)"
using assms HomC.set_map HomD.set_map by auto
thus ?thesis using S.Fun_mapsto by fast
qed
lemma ψ_mapsto:
assumes x: "C.ide x"
shows "ψ x ∈ D.hom y (G x) → C.hom (F y) x"
proof
fix h
assume h: "h ∈ D.hom y (G x)"
hence 1: "«h : y →⇩D G x»" by auto
show "ψ x h ∈ C.hom (F y) x"
proof -
have "Ψ.FUN (y, x) (φD (y, G x) h) ∈ HomC.set (F y, x)"
proof -
have "φD (y, G x) h ∈ HomD.set (y, G x)"
using x h 1 HomD.φ_mapsto [of y "G x"] by fastforce
thus ?thesis
using h x Ψ_Fun_mapsto by auto
qed
thus ?thesis
using x h 1 ψ_def HomD.φ_mapsto HomC.ψ_mapsto [of "F y" x] by fastforce
qed
qed
lemma Ψ_simp:
assumes "D.ide y" and "C.ide x"
shows "S.arr (Ψ (y, x))"
and "Ψ (y, x) = S.mkArr (HomD.set (y, G x)) (HomC.set (F y, x))
(φC (F y, x) o ψ x o ψD (y, G x))"
proof -
show 1: "S.arr (Ψ (y, x))" using assms by auto
hence "Ψ (y, x) = S.mkArr (Ψ.DOM (y, x)) (Ψ.COD (y, x)) (Ψ.FUN (y, x))"
using S.mkArr_Fun by metis
also have "... = S.mkArr (HomD.set (y, G x)) (HomC.set (F y, x)) (Ψ.FUN (y, x))"
using assms HomC.set_map HomD.set_map by auto
also have "... = S.mkArr (HomD.set (y, G x)) (HomC.set (F y, x))
(φC (F y, x) o ψ x o ψD (y, G x))"
proof (intro S.mkArr_eqI')
show "S.arr (S.mkArr (HomD.set (y, G x)) (HomC.set (F y, x)) (Ψ.FUN (y, x)))"
using 1 calculation by argo
show "⋀h. h ∈ HomD.set (y, G x) ⟹
Ψ.FUN (y, x) h = (φC (F y, x) o ψ x o ψD (y, G x)) h"
proof -
fix h
assume h: "h ∈ HomD.set (y, G x)"
have "(φC (F y, x) o ψ x o HomD.ψ (y, G x)) h =
φC (F y, x) (ψC (F y, x) (Ψ.FUN (y, x) (φD (y, G x) (ψD (y, G x) h))))"
proof -
have "«ψD (y, G x) h : y →⇩D G x»"
using assms h HomD.ψ_mapsto [of y "G x"] by auto
thus ?thesis
using h ψ_def by auto
qed
also have "... = φC (F y, x) (ψC (F y, x) (Ψ.FUN (y, x) h))"
using assms h HomD.φ_ψ Ψ_Fun_mapsto by simp
also have "... = Ψ.FUN (y, x) h"
using assms h Ψ_Fun_mapsto HomD.ψ_mapsto [of y "G x"] HomC.φ_ψ [of "F y" x]
C.ide_in_hom D.ide_in_hom
by blast
finally show "Ψ.FUN (y, x) h = (φC (F y, x) o ψ x o HomD.ψ (y, G x)) h" by auto
qed
qed
finally show "Ψ (y, x) = S.mkArr (HomD.set (y, G x)) (HomC.set (F y, x))
(φC (F y, x) o ψ x o ψD (y, G x))"
by force
qed
text‹
The length of the next proof stems from having to use properties of composition
of arrows in @{term[source=true] S} to infer properties of the composition of the
corresponding functions.
›
interpretation φψ: meta_adjunction C D F G φ ψ
proof
fix y :: 'd and x :: 'c and h :: 'c
assume y: "D.ide y" and h: "«h : F y →⇩C x»"
have x: "C.ide x" using h by auto
show "«φ y h : y →⇩D G x»"
proof -
have "Φ.FUN (y, x) ∈ HomC.set (F y, x) → HomD.set (y, G x)"
using y h Φ_Fun_mapsto by blast
thus ?thesis
using x y h φ_def HomD.ψ_mapsto [of y "G x"] HomC.φ_mapsto [of "F y" x] by auto
qed
show "ψ x (φ y h) = h"
proof -
have 0: "restrict (λh. h) (HomC.set (F y, x))
= restrict (φC (F y, x) o (ψ x o φ y) o ψC (F y, x)) (HomC.set (F y, x))"
proof -
have 1: "S.ide (Ψ (y, x) ⋅⇩S Φ (y, x))"
using x y ΦΨ.inv [of "(y, x)"] by auto
hence 6: "S.seq (Ψ (y, x)) (Φ (y, x))" by auto
have 2: "Φ (y, x) = S.mkArr (HomC.set (F y, x)) (HomD.set (y, G x))
(φD (y, G x) o φ y o ψC (F y, x)) ∧
Ψ (y, x) = S.mkArr (HomD.set (y, G x)) (HomC.set (F y, x))
(φC (F y, x) o ψ x o ψD (y, G x))"
using x y Φ_simp Ψ_simp by force
have 3: "S (Ψ (y, x)) (Φ (y, x))
= S.mkArr (HomC.set (F y, x)) (HomC.set (F y, x))
(φC (F y, x) o (ψ x o φ y) o ψC (F y, x))"
proof -
have 4: "S.arr (Ψ (y, x) ⋅⇩S Φ (y, x))" using 1 by auto
hence "S (Ψ (y, x)) (Φ (y, x))
= S.mkArr (HomC.set (F y, x)) (HomC.set (F y, x))
((φC (F y, x) o ψ x o ψD (y, G x))
o (φD (y, G x) o φ y o ψC (F y, x)))"
using 1 2 S.ide_in_hom S.comp_mkArr by fastforce
also have "... = S.mkArr (HomC.set (F y, x)) (HomC.set (F y, x))
(φC (F y, x) o (ψ x o φ y) o ψC (F y, x))"
proof (intro S.mkArr_eqI')
show "S.arr (S.mkArr (HomC.set (F y, x)) (HomC.set (F y, x))
((φC (F y, x) o ψ x o ψD (y, G x))
o (φD (y, G x) o φ y o ψC (F y, x))))"
using 4 calculation by simp
show "⋀h. h ∈ HomC.set (F y, x) ⟹
((φC (F y, x) o ψ x o ψD (y, G x))
o (φD (y, G x) o φ y o ψC (F y, x))) h =
(φC (F y, x) o (ψ x o φ y) o ψC (F y, x)) h"
proof -
fix h
assume h: "h ∈ HomC.set (F y, x)"
hence "«φ y (ψC (F y, x) h) : y →⇩D G x»"
using x y h HomC.ψ_mapsto [of "F y" x] φ_mapsto by auto
thus "((φC (F y, x) o ψ x o ψD (y, G x))
o (φD (y, G x) o φ y o ψC (F y, x))) h =
(φC (F y, x) o (ψ x o φ y) o ψC (F y, x)) h"
using x y 1 φ_mapsto HomD.ψ_φ by simp
qed
qed
finally show ?thesis by simp
qed
moreover have "Ψ (y, x) ⋅⇩S Φ (y, x)
= S.mkArr (HomC.set (F y, x)) (HomC.set (F y, x)) (λh. h)"
using 1 2 6 calculation S.mkIde_as_mkArr S.arr_mkArr S.dom_mkArr S.ideD(2)
by metis
ultimately have 4: "S.mkArr (HomC.set (F y, x)) (HomC.set (F y, x))
(φC (F y, x) o (ψ x o φ y) o ψC (F y, x))
= S.mkArr (HomC.set (F y, x)) (HomC.set (F y, x)) (λh. h)"
by auto
have 5: "S.arr (S.mkArr (HomC.set (F y, x)) (HomC.set (F y, x))
(φC (F y, x) o (ψ x o φ y) o ψC (F y, x)))"
using 1 3 6 by presburger
hence "restrict (φC (F y, x) o (ψ x o φ y) o ψC (F y, x)) (HomC.set (F y, x))
= S.Fun (S.mkArr (HomC.set (F y, x)) (HomC.set (F y, x))
(φC (F y, x) o (ψ x o φ y) o ψC (F y, x)))"
by auto
also have "... = restrict (λh. h) (HomC.set (F y, x))"
using 4 5 by auto
finally show ?thesis by auto
qed
moreover have "φC (F y, x) h ∈ HomC.set (F y, x)"
using x y h HomC.φ_mapsto [of "F y" x] by auto
ultimately have
"φC (F y, x) h = (φC (F y, x) o (ψ x o φ y) o ψC (F y, x)) (φC (F y, x) h)"
using x y h HomC.φ_mapsto [of "F y" x] by fast
hence "ψC (F y, x) (φC (F y, x) h) =
ψC (F y, x) ((φC (F y, x) o (ψ x o φ y) o ψC (F y, x)) (φC (F y, x) h))"
by simp
hence "h = ψC (F y, x) (φC (F y, x) (ψ x (φ y (ψC (F y, x) (φC (F y, x) h)))))"
using x y h HomC.ψ_φ [of "F y" x] by simp
also have "... = ψ x (φ y h)"
using x y h HomC.ψ_φ HomC.ψ_φ φ_mapsto ψ_mapsto
by (metis PiE mem_Collect_eq)
finally show ?thesis by auto
qed
next
fix x :: 'c and h :: 'd and y :: 'd
assume x: "C.ide x" and h: "«h : y →⇩D G x»"
have y: "D.ide y" using h by auto
show "«ψ x h : F y →⇩C x»" using x y h ψ_mapsto [of x y] by auto
show "φ y (ψ x h) = h"
proof -
have 0: "restrict (λh. h) (HomD.set (y, G x))
= restrict (φD (y, G x) o (φ y o ψ x) o ψD (y, G x)) (HomD.set (y, G x))"
proof -
have 1: "S.ide (S (Φ (y, x)) (Ψ (y, x)))"
using x y ΦΨ.inv by force
hence 6: "S.seq (Φ (y, x)) (Ψ (y, x))" by auto
have 2: "Φ (y, x) = S.mkArr (HomC.set (F y, x)) (HomD.set (y, G x))
(φD (y, G x) o φ y o ψC (F y, x)) ∧
Ψ (y, x) = S.mkArr (HomD.set (y, G x)) (HomC.set (F y, x))
(φC (F y, x) o ψ x o ψD (y, G x))"
using x h Φ_simp Ψ_simp by auto
have 3: "S (Φ (y, x)) (Ψ (y, x))
= S.mkArr (HomD.set (y, G x)) (HomD.set (y, G x))
(φD (y, G x) o (φ y o ψ x) o ψD (y, G x))"
proof -
have 4: "S.seq (Φ (y, x)) (Ψ (y, x))" using 1 by auto
hence "S (Φ (y, x)) (Ψ (y, x))
= S.mkArr (HomD.set (y, G x)) (HomD.set (y, G x))
((φD (y, G x) o φ y o ψC (F y, x))
o (φC (F y, x) o ψ x o ψD (y, G x)))"
using 1 2 6 S.ide_in_hom S.comp_mkArr by fastforce
also have "... = S.mkArr (HomD.set (y, G x)) (HomD.set (y, G x))
(φD (y, G x) o (φ y o ψ x) o ψD (y, G x))"
proof
show "S.arr (S.mkArr (HomD.set (y, G x)) (HomD.set (y, G x))
((φD (y, G x) o φ y o ψC (F y, x))
o (φC (F y, x) o ψ x o ψD (y, G x))))"
using 4 calculation by simp
show "⋀h. h ∈ HomD.set (y, G x) ⟹
((φD (y, G x) o φ y o ψC (F y, x))
o (φC (F y, x) o ψ x o ψD (y, G x))) h =
(φD (y, G x) o (φ y o ψ x) o ψD (y, G x)) h"
proof -
fix h
assume h: "h ∈ HomD.set (y, G x)"
hence "«ψ x (ψD (y, G x) h) : F y →⇩C x»"
using x y HomD.ψ_mapsto [of y "G x"] ψ_mapsto by auto
thus "((φD (y, G x) o φ y o ψC (F y, x))
o (φC (F y, x) o ψ x o ψD (y, G x))) h =
(φD (y, G x) o (φ y o ψ x) o ψD (y, G x)) h"
using x y HomC.ψ_φ by simp
qed
qed
finally show ?thesis by auto
qed
moreover have "Φ (y, x) ⋅⇩S Ψ (y, x) =
S.mkArr (HomD.set (y, G x)) (HomD.set (y, G x)) (λh. h)"
using 1 2 6 calculation
by (metis S.arr_mkArr S.cod_mkArr S.ide_in_hom S.mkIde_as_mkArr S.in_homE)
ultimately have 4: "S.mkArr (HomD.set (y, G x)) (HomD.set (y, G x))
(φD (y, G x) o (φ y o ψ x) o ψD (y, G x))
= S.mkArr (HomD.set (y, G x)) (HomD.set (y, G x)) (λh. h)"
by auto
have 5: "S.arr (S.mkArr (HomD.set (y, G x)) (HomD.set (y, G x))
(φD (y, G x) o (φ y o ψ x) o ψD (y, G x)))"
using 1 3 by fastforce
hence "restrict (φD (y, G x) o (φ y o ψ x) o ψD (y, G x)) (HomD.set (y, G x))
= S.Fun (S.mkArr (HomD.set (y, G x)) (HomD.set (y, G x))
(φD (y, G x) o (φ y o ψ x) o ψD (y, G x)))"
by auto
also have "... = restrict (λh. h) (HomD.set (y, G x))"
using 4 5 by auto
finally show ?thesis by auto
qed
moreover have "φD (y, G x) h ∈ HomD.set (y, G x)"
using x y h HomD.φ_mapsto [of y "G x"] by auto
ultimately have
"φD (y, G x) h = (φD (y, G x) o (φ y o ψ x) o ψD (y, G x)) (φD (y, G x) h)"
by fast
hence "ψD (y, G x) (φD (y, G x) h) =
ψD (y, G x) ((φD (y, G x) o (φ y o ψ x) o ψD (y, G x)) (φD (y, G x) h))"
by simp
hence "h = ψD (y, G x) (φD (y, G x) (φ y (ψ x (ψD (y, G x) (φD (y, G x) h)))))"
using x y h HomD.ψ_φ by simp
also have "... = φ y (ψ x h)"
using x y h HomD.ψ_φ HomD.ψ_φ [of "φ y (ψ x h)" y "G x"] φ_mapsto ψ_mapsto
by fastforce
finally show ?thesis by auto
qed
next
fix x :: 'c and x' :: 'c and y :: 'd and y' :: 'd
and f :: 'c and g :: 'd and h :: 'c
assume f: "«f : x →⇩C x'»" and g: "«g : y' →⇩D y»" and h: "«h : F y →⇩C x»"
have x: "C.ide x" using f by auto
have y: "D.ide y" using g by auto
have x': "C.ide x'" using f by auto
have y': "D.ide y'" using g by auto
show "φ y' (f ⋅⇩C h ⋅⇩C F g) = G f ⋅⇩D φ y h ⋅⇩D g"
proof -
have 0: "restrict ((φD (y', G x') o (λh. G f ⋅⇩D h ⋅⇩D g) o ψD (y, G x))
o (φD (y, G x) o φ y o ψC (F y, x)))
(HomC.set (F y, x))
= restrict ((φD (y', G x') o φ y' o ψC (F y', x'))
o (φC (F y', x') o (λh. f ⋅⇩C h ⋅⇩C F g)) o ψC (F y, x))
(HomC.set (F y, x))"
proof -
have 1: "S.arr (Φ (y, x)) ∧
Φ (y, x) = S.mkArr (HomC.set (F y, x)) (HomD.set (y, G x))
(φD (y, G x) o φ y o ψC (F y, x))"
using x y Φ_simp [of y x] by auto
have 2: "S.arr (Φ (y', x')) ∧
Φ (y', x') = S.mkArr (HomC.set (F y', x')) (HomD.set (y', G x'))
(φD (y', G x') o φ y' o ψC (F y', x'))"
using x' y' Φ_simp [of y' x'] by auto
have 3: "S.arr (S.mkArr (HomC.set (F y, x)) (HomD.set (y', G x'))
((φD (y', G x') o (λh. G f ⋅⇩D h ⋅⇩D g) o ψD (y, G x))
o (φD (y, G x) o φ y o ψC (F y, x))))
∧ S.mkArr (HomC.set (F y, x)) (HomD.set (y', G x'))
((φD (y', G x') o (λh. G f ⋅⇩D h ⋅⇩D g) o ψD (y, G x))
o (φD (y, G x) o φ y o ψC (F y, x)))
= S (S.mkArr (HomD.set (y, G x)) (HomD.set (y', G x'))
(φD (y', G x') o (λh. G f ⋅⇩D h ⋅⇩D g) o ψD (y, G x)))
(S.mkArr (HomC.set (F y, x)) (HomD.set (y, G x))
(φD (y, G x) o φ y o ψC (F y, x)))"
proof -
have 1: "S.seq (S.mkArr (HomD.set (y, G x)) (HomD.set (y', G x'))
(φD (y', G x') o (λh. G f ⋅⇩D h ⋅⇩D g) o ψD (y, G x)))
(S.mkArr (HomC.set (F y, x)) (HomD.set (y, G x))
(φD (y, G x) o φ y o ψC (F y, x)))"
proof -
have "S.arr (Hom_DopxG.map (g, f)) ∧
Hom_DopxG.map (g, f)
= S.mkArr (HomD.set (y, G x)) (HomD.set (y', G x'))
(φD (y', G x') o (λh. G f ⋅⇩D h ⋅⇩D g) o ψD (y, G x))"
using f g Hom_DopxG.preserves_arr Hom_DopxG_map_simp by fastforce
thus ?thesis
using 1 S.cod_mkArr S.dom_mkArr S.seqI by metis
qed
have "S.seq (S.mkArr (HomD.set (y, G x)) (HomD.set (y', G x'))
(φD (y', G x') o (λh. G f ⋅⇩D h ⋅⇩D g) o ψD (y, G x)))
(S.mkArr (HomC.set (F y, x)) (HomD.set (y, G x))
(φD (y, G x) o φ y o ψC (F y, x)))"
using 1 by (intro S.seqI', auto)
moreover have "S.mkArr (HomC.set (F y, x)) (HomD.set (y', G x'))
((φD (y', G x') o (λh. G f ⋅⇩D h ⋅⇩D g) o ψD (y, G x))
o (φD (y, G x) o φ y o ψC (F y, x)))
= S (S.mkArr (HomD.set (y, G x)) (HomD.set (y', G x'))
(φD (y', G x') o (λh. G f ⋅⇩D h ⋅⇩D g) o ψD (y, G x)))
(S.mkArr (HomC.set (F y, x)) (HomD.set (y, G x))
(φD (y, G x) o φ y o ψC (F y, x)))"
using 1 S.comp_mkArr by fastforce
ultimately show ?thesis by auto
qed
moreover have
4: "S.arr (S.mkArr (HomC.set (F y, x)) (HomD.set (y', G x'))
((φD (y', G x') o φ y' o ψC (F y', x'))
o (φC (F y', x') o (λh. f ⋅⇩C h ⋅⇩C F g) o ψC (F y, x))))
∧ S.mkArr (HomC.set (F y, x)) (HomD.set (y', G x'))
((φD (y', G x') o φ y' o ψC (F y', x'))
o (φC (F y', x') o (λh. f ⋅⇩C h ⋅⇩C F g) o ψC (F y, x)))
= S (S.mkArr (HomC.set (F y', x')) (HomD.set (y', G x'))
(φD (y', G x') o φ y' o ψC (F y', x')))
(S.mkArr (HomC.set (F y, x)) (HomC.set (F y', x'))
(φC (F y', x') o (λh. f ⋅⇩C h ⋅⇩C F g) o ψC (F y, x)))"
proof -
have 5: "S.seq (S.mkArr (HomC.set (F y', x')) (HomD.set (y', G x'))
(φD (y', G x') o φ y' o ψC (F y', x')))
(S.mkArr (HomC.set (F y, x)) (HomC.set (F y', x'))
(φC (F y', x') o (λh. f ⋅⇩C h ⋅⇩C F g) o ψC (F y, x)))"
proof -
have "S.arr (Hom_FopxC.map (g, f)) ∧
Hom_FopxC.map (g, f)
= S.mkArr (HomC.set (F y, x)) (HomC.set (F y', x'))
(φC (F y', x') o (λh. f ⋅⇩C h ⋅⇩C F g) o ψC (F y, x))"
using f g Hom_FopxC.preserves_arr Hom_FopxC_map_simp by fastforce
thus ?thesis using 2 S.cod_mkArr S.dom_mkArr S.seqI by metis
qed
have "S.seq (S.mkArr (HomC.set (F y', x')) (HomD.set (y', G x'))
(φD (y', G x') o φ y' o ψC (F y', x')))
(S.mkArr (HomC.set (F y, x)) (HomC.set (F y', x'))
(φC (F y', x') o (λh. f ⋅⇩C h ⋅⇩C F g) o ψC (F y, x)))"
using 5 by (intro S.seqI', auto)
moreover have "S.mkArr (HomC.set (F y, x)) (HomD.set (y', G x'))
((φD (y', G x') o φ y' o ψC (F y', x'))
o (φC (F y', x') o (λh. f ⋅⇩C h ⋅⇩C F g) o ψC (F y, x)))
= S (S.mkArr (HomC.set (F y', x')) (HomD.set (y', G x'))
(φD (y', G x') o φ y' o ψC (F y', x')))
(S.mkArr (HomC.set (F y, x)) (HomC.set (F y', x'))
(φC (F y', x') o (λh. f ⋅⇩C h ⋅⇩C F g) o ψC (F y, x)))"
using 5 S.comp_mkArr by fastforce
ultimately show ?thesis by argo
qed
moreover have 2:
"S.mkArr (HomC.set (F y, x)) (HomD.set (y', G x'))
((φD (y', G x') o (λh. G f ⋅⇩D h ⋅⇩D g) o ψD (y, G x))
o (φD (y, G x) o φ y o ψC (F y, x)))
= S.mkArr (HomC.set (F y, x)) (HomD.set (y', G x'))
((φD (y', G x') o φ y' o ψC (F y', x'))
o (φC (F y', x') o (λh. f ⋅⇩C h ⋅⇩C F g) o ψC (F y, x)))"
proof -
have
"S (Hom_DopxG.map (g, f)) (Φ (y, x)) = S (Φ (y', x')) (Hom_FopxC.map (g, f))"
using f g Φ.is_natural_1 Φ.is_natural_2 by fastforce
moreover have "Hom_DopxG.map (g, f)
= S.mkArr (HomD.set (y, G x)) (HomD.set (y', G x'))
(φD (y', G x') o (λh. G f ⋅⇩D h ⋅⇩D g) o ψD (y, G x))"
using f g Hom_DopxG_map_simp [of "(g, f)"] by fastforce
moreover have "Hom_FopxC.map (g, f)
= S.mkArr (HomC.set (F y, x)) (HomC.set (F y', x'))
(φC (F y', x') o (λh. f ⋅⇩C h ⋅⇩C F g) o ψC (F y, x))"
using f g Hom_FopxC_map_simp [of "(g, f)"] by fastforce
ultimately show ?thesis using 1 2 3 4 by simp
qed
ultimately have 6: "S.arr (S.mkArr (HomC.set (F y, x)) (HomD.set (y', G x'))
((φD (y', G x') o (λh. G f ⋅⇩D h ⋅⇩D g) o ψD (y, G x))
o (φD (y, G x) o φ y o ψC (F y, x))))"
by fast
hence "restrict ((φD (y', G x') o (λh. D (G f) (D h g)) o ψD (y, G x))
o (φD (y, G x) o φ y o ψC (F y, x)))
(HomC.set (F y, x))
= S.Fun (S.mkArr (HomC.set (F y, x)) (HomD.set (y', G x'))
((φD (y', G x') o (λh. G f ⋅⇩D h ⋅⇩D g) o ψD (y, G x))
o (φD (y, G x) o φ y o ψC (F y, x))))"
by simp
also have "... = S.Fun (S.mkArr (HomC.set (F y, x)) (HomD.set (y', G x'))
((φD (y', G x') o φ y' o ψC (F y', x'))
o (φC (F y', x') o (λh. f ⋅⇩C h ⋅⇩C F g) o ψC (F y, x))))"
using 2 by argo
also have "... = restrict ((φD (y', G x') o φ y' o ψC (F y', x'))
o (φC (F y', x') o (λh. f ⋅⇩C h ⋅⇩C F g) o ψC (F y, x)))
(HomC.set (F y, x))"
using 4 S.Fun_mkArr by meson
finally show ?thesis by auto
qed
hence 5: "((φD (y', G x') ∘ (λh. G f ⋅⇩D h ⋅⇩D g) ∘ ψD (y, G x))
∘ (φD (y, G x) ∘ φ y ∘ ψC (F y, x))) (φC (F y, x) h) =
(φD (y', G x') ∘ φ y' ∘ ψC (F y', x')
∘ (φC (F y', x') ∘ (λh. f ⋅⇩C h ⋅⇩C F g)) ∘ ψC (F y, x)) (φC (F y, x) h)"
proof -
have "φC (F y, x) h ∈ HomC.set (F y, x)"
using x y h HomC.φ_mapsto [of "F y" x] by auto
thus ?thesis
using 0 h restr_eqE [of "(φD (y', G x') ∘ (λh. G f ⋅⇩D h ⋅⇩D g) ∘ ψD (y, G x))
∘ (φD (y, G x) ∘ φ y ∘ ψC (F y, x))"
"HomC.set (F y, x)"
"(φD (y', G x') ∘ φ y' ∘ ψC (F y', x'))
∘ (φC (F y', x') ∘ (λh. f ⋅⇩C h ⋅⇩C F g) o ψC (F y, x))"]
by fast
qed
show ?thesis
proof -
have "φ y' (C f (C h (F g))) =
ψD (y', G x') (φD (y', G x') (φ y' (ψC (F y', x') (φC (F y', x')
(C f (C (ψC (F y, x) (φC (F y, x) h)) (F g)))))))"
proof -
have "ψD (y', G x') (φD (y', G x') (φ y' (ψC (F y', x') (φC (F y', x')
(C f (C (ψC (F y, x) (φC (F y, x) h)) (F g)))))))
= ψD (y', G x') (φD (y', G x') (φ y' (ψC (F y', x') (φC (F y', x')
(C f (C h (F g)))))))"
using x y h HomC.ψ_φ by simp
also have "... = ψD (y', G x') (φD (y', G x') (φ y' (C f (C h (F g)))))"
using f g h HomC.ψ_φ [of "C f (C h (F g))"] by fastforce
also have "... = φ y' (C f (C h (F g)))"
proof -
have "«φ y' (f ⋅⇩C h ⋅⇩C F g) : y' →⇩D G x'»"
using f g h y' x' φ_mapsto [of y' x'] by auto
thus ?thesis by simp
qed
finally show ?thesis by auto
qed
also have
"... = ψD (y', G x')
(φD (y', G x')
(G f ⋅⇩D ψD (y, G x) (φD (y, G x) (φ y (ψC (F y, x) (φC (F y, x) h))))
⋅⇩D g))"
using 5 by force
also have "... = D (G f) (D (φ y h) g)"
proof -
have φyh: "«φ y h : y →⇩D G x»"
using x y h φ_mapsto by auto
have "ψD (y', G x')
(φD (y', G x')
(G f ⋅⇩D ψD (y, G x) (φD (y, G x) (φ y (ψC (F y, x) (φC (F y, x) h))))
⋅⇩D g)) =
ψD (y', G x') (φD (y', G x') (G f ⋅⇩D ψD (y, G x) (φD (y, G x) (φ y h)) ⋅⇩D g))"
using x y f g h by auto
also have "... = ψD (y', G x') (φD (y', G x') (G f ⋅⇩D φ y h ⋅⇩D g))"
using φyh x' y' f g by simp
also have "... = G f ⋅⇩D φ y h ⋅⇩D g"
using φyh f g by fastforce
finally show ?thesis by auto
qed
finally show ?thesis by auto
qed
qed
qed
theorem induces_meta_adjunction:
shows "meta_adjunction C D F G φ ψ" ..
end
section "Putting it All Together"
text‹
Combining the above results, an interpretation of any one of the locales:
‹left_adjoint_functor›, ‹right_adjoint_functor›, ‹meta_adjunction›,
‹hom_adjunction›, and ‹unit_counit_adjunction› extends to an interpretation
of ‹adjunction›.
›
context meta_adjunction
begin
interpretation S: replete_setcat .
interpretation F: left_adjoint_functor D C F using has_left_adjoint_functor by auto
interpretation G: right_adjoint_functor C D G using has_right_adjoint_functor by auto
interpretation ηε: unit_counit_adjunction C D F G η ε
using induces_unit_counit_adjunction η_def ε_def by auto
interpretation ΦΨ: hom_adjunction C D S.comp S.setp φC φD F G Φ Ψ
using induces_hom_adjunction by auto
theorem induces_adjunction:
shows "adjunction C D S.comp S.setp φC φD F G φ ψ η ε Φ Ψ"
using ε_map_simp η_map_simp φ_in_terms_of_η φ_in_terms_of_Φ' ψ_in_terms_of_ε
ψ_in_terms_of_Ψ' Φ_simp Ψ_simp η_def ε_def
by unfold_locales auto
end
context unit_counit_adjunction
begin
interpretation φψ: meta_adjunction C D F G φ ψ using induces_meta_adjunction by auto
interpretation S: replete_setcat .
interpretation F: left_adjoint_functor D C F using φψ.has_left_adjoint_functor by auto
interpretation G: right_adjoint_functor C D G using φψ.has_right_adjoint_functor by auto
interpretation ΦΨ: hom_adjunction C D S.comp S.setp
φψ.φC φψ.φD F G φψ.Φ φψ.Ψ
using φψ.induces_hom_adjunction by auto
theorem induces_adjunction:
shows "adjunction C D S.comp S.setp φψ.φC φψ.φD F G φ ψ η ε φψ.Φ φψ.Ψ"
using ε_in_terms_of_ψ η_in_terms_of_φ φψ.φ_in_terms_of_Φ' ψ_def φψ.ψ_in_terms_of_Ψ'
φψ.Φ_simp φψ.Ψ_simp φ_def
by unfold_locales auto
end
context hom_adjunction
begin
interpretation φψ: meta_adjunction C D F G φ ψ
using induces_meta_adjunction by auto
interpretation F: left_adjoint_functor D C F using φψ.has_left_adjoint_functor by auto
interpretation G: right_adjoint_functor C D G using φψ.has_right_adjoint_functor by auto
interpretation ηε: unit_counit_adjunction C D F G φψ.η φψ.ε
using φψ.induces_unit_counit_adjunction φψ.η_def φψ.ε_def by auto
theorem induces_adjunction:
shows "adjunction C D S setp φC φD F G φ ψ φψ.η φψ.ε Φ Ψ"
proof
fix x
assume "C.ide x"
thus "φψ.ε x = ψ x (G x)"
using φψ.ε_map_simp φψ.ε_def by simp
next
fix y
assume "D.ide y"
thus "φψ.η y = φ y (F y)"
using φψ.η_map_simp φψ.η_def by simp
fix x y f
assume y: "D.ide y" and f: "«f : F y →⇩C x»"
show "φ y f = G f ⋅⇩D φψ.η y"
using y f φψ.φ_in_terms_of_η φψ.η_def by simp
show "φ y f = (ψD (y, G x) ∘ Φ.FUN (y, x) ∘ φC (F y, x)) f"
using y f φ_def by auto
next
fix x y g
assume x: "C.ide x" and g: "«g : y →⇩D G x»"
show "ψ x g = φψ.ε x ⋅⇩C F g"
using x g φψ.ψ_in_terms_of_ε φψ.ε_def by simp
show "ψ x g = (ψC (F y, x) ∘ Ψ.FUN (y, x) ∘ φD (y, G x)) g"
using x g ψ_def by fast
next
fix x y
assume x: "C.ide x" and y: "D.ide y"
show "Φ (y, x) = S.mkArr (HomC.set (F y, x)) (HomD.set (y, G x))
(φD (y, G x) o φ y o ψC (F y, x))"
using x y Φ_simp by simp
show "Ψ (y, x) = S.mkArr (HomD.set (y, G x)) (HomC.set (F y, x))
(φC (F y, x) o ψ x o ψD (y, G x))"
using x y Ψ_simp by simp
qed
end
context left_adjoint_functor
begin
interpretation φψ: meta_adjunction C D F G φ ψ
using induces_meta_adjunction by auto
interpretation S: replete_setcat .
theorem induces_adjunction:
shows "adjunction C D S.comp S.setp φψ.φC φψ.φD F G φ ψ φψ.η φψ.ε φψ.Φ φψ.Ψ"
using φψ.induces_adjunction by auto
end
context right_adjoint_functor
begin
interpretation φψ: meta_adjunction C D F G φ ψ
using induces_meta_adjunction by auto
interpretation S: replete_setcat .
theorem induces_adjunction:
shows "adjunction C D S.comp S.setp φψ.φC φψ.φD F G φ ψ φψ.η φψ.ε φψ.Φ φψ.Ψ"
using φψ.induces_adjunction by auto
end
definition adjoint_functors
where "adjoint_functors C D F G = (∃φ ψ. meta_adjunction C D F G φ ψ)"
lemma adjoint_functors_respects_naturally_isomorphic:
assumes "adjoint_functors C D F G"
and "naturally_isomorphic D C F' F" and "naturally_isomorphic C D G G'"
shows "adjoint_functors C D F' G'"
proof -
obtain φ ψ where φψ: "meta_adjunction C D F G φ ψ"
using assms(1) adjoint_functors_def by blast
interpret φψ: meta_adjunction C D F G φ ψ
using φψ by simp
obtain τ where τ: "natural_isomorphism D C F' F τ"
using assms(2) naturally_isomorphic_def by blast
obtain μ where μ: "natural_isomorphism C D G G' μ"
using assms(3) naturally_isomorphic_def by blast
show ?thesis
using adjoint_functors_def τ μ φψ.respects_natural_isomorphism by blast
qed
lemma left_adjoint_functor_respects_naturally_isomorphic:
assumes "left_adjoint_functor D C F"
and "naturally_isomorphic D C F F'"
shows "left_adjoint_functor D C F'"
proof -
interpret F: left_adjoint_functor D C F
using assms(1) by simp
have 1: "meta_adjunction C D F F.G F.φ F.ψ"
using F.induces_meta_adjunction by simp
interpret φψ: meta_adjunction C D F F.G F.φ F.ψ
using 1 by simp
have "adjoint_functors C D F F.G"
using 1 adjoint_functors_def by blast
hence 2: "adjoint_functors C D F' F.G"
using assms(2) adjoint_functors_respects_naturally_isomorphic [of C D F F.G F' F.G]
naturally_isomorphic_reflexive naturally_isomorphic_symmetric
φψ.G.functor_axioms
by blast
obtain φ' ψ' where φ'ψ': "meta_adjunction C D F' F.G φ' ψ'"
using 2 adjoint_functors_def by blast
interpret φ'ψ': meta_adjunction C D F' F.G φ' ψ'
using φ'ψ' by simp
show ?thesis
using φ'ψ'.has_left_adjoint_functor by simp
qed
lemma right_adjoint_functor_respects_naturally_isomorphic:
assumes "right_adjoint_functor C D G"
and "naturally_isomorphic C D G G'"
shows "right_adjoint_functor C D G'"
proof -
interpret G: right_adjoint_functor C D G
using assms(1) by simp
have 1: "meta_adjunction C D G.F G G.φ G.ψ"
using G.induces_meta_adjunction by simp
interpret φψ: meta_adjunction C D G.F G G.φ G.ψ
using 1 by simp
have "adjoint_functors C D G.F G"
using 1 adjoint_functors_def by blast
hence 2: "adjoint_functors C D G.F G'"
using assms(2) adjoint_functors_respects_naturally_isomorphic
naturally_isomorphic_reflexive naturally_isomorphic_symmetric
φψ.F.functor_axioms
by blast
obtain φ' ψ' where φ'ψ': "meta_adjunction C D G.F G' φ' ψ'"
using 2 adjoint_functors_def by blast
interpret φ'ψ': meta_adjunction C D G.F G' φ' ψ'
using φ'ψ' by simp
show ?thesis
using φ'ψ'.has_right_adjoint_functor by simp
qed
section "Inverse Functors are Adjoints"
lemma inverse_functors_induce_meta_adjunction:
assumes "inverse_functors C D F G"
shows "meta_adjunction C D F G (λx. G) (λy. F)"
proof -
interpret inverse_functors C D F G using assms by auto
interpret meta_adjunction C D F G ‹λx. G› ‹λy. F›
proof -
have 1: "⋀y. B.arr y ⟹ G (F y) = y"
by (metis B.map_simp comp_apply inv)
have 2: "⋀x. A.arr x ⟹ F (G x) = x"
by (metis A.map_simp comp_apply inv')
show "meta_adjunction C D F G (λx. G) (λy. F)"
proof
fix y f x
assume y: "B.ide y" and f: "«f : F y →⇩A x»"
show "«G f : y →⇩B G x»"
using y f 1 G.preserves_hom by (elim A.in_homE, auto)
show "F (G f) = f"
using f 2 by auto
next
fix x g y
assume x: "A.ide x" and g: "«g : y →⇩B G x»"
show "«F g : F y →⇩A x»"
using x g 2 F.preserves_hom by (elim B.in_homE, auto)
show "G (F g) = g" using g 1 A.map_def by blast
next
fix f x x' g y' y h
assume f: "«f : x →⇩A x'»" and g: "«g : y' →⇩B y»" and h: "«h : F y →⇩A x»"
show "G (C f (C h (F g))) = D (G f) (D (G h) g)"
using f g h 1 2 inv inv' A.map_def B.map_def by (elim A.in_homE B.in_homE, auto)
qed
qed
show ?thesis ..
qed
lemma inverse_functors_are_adjoints:
assumes "inverse_functors A B F G"
shows "adjoint_functors A B F G"
using assms inverse_functors_induce_meta_adjunction adjoint_functors_def by fast
context inverse_functors
begin
lemma η_char:
shows "meta_adjunction.η B F (λx. G) = identity_functor.map B"
proof (intro eqI)
interpret meta_adjunction A B F G ‹λy. G› ‹λx. F›
using inverse_functors_induce_meta_adjunction inverse_functors_axioms by auto
interpret S: replete_setcat .
interpret adjunction A B S.comp S.setp φC φD F G ‹λy. G› ‹λx. F› η ε Φ Ψ
using induces_adjunction by force
show "natural_transformation B B B.map GF.map η"
using η.natural_transformation_axioms by auto
show "natural_transformation B B B.map GF.map B.map"
by (simp add: B.as_nat_trans.natural_transformation_axioms inv)
show "⋀b. B.ide b ⟹ η b = B.map b"
using η_in_terms_of_φ ηo_def ηo_in_hom by fastforce
qed
lemma ε_char:
shows "meta_adjunction.ε A F G (λy. F) = identity_functor.map A"
proof (intro eqI)
interpret meta_adjunction A B F G ‹λy. G› ‹λx. F›
using inverse_functors_induce_meta_adjunction inverse_functors_axioms by auto
interpret S: replete_setcat .
interpret adjunction A B S.comp S.setp φC φD F G ‹λy. G› ‹λx. F› η ε Φ Ψ
using induces_adjunction by force
show "natural_transformation A A FG.map A.map ε"
using ε.natural_transformation_axioms by auto
show "natural_transformation A A FG.map A.map A.map"
by (simp add: A.as_nat_trans.natural_transformation_axioms inv')
show "⋀a. A.ide a ⟹ ε a = A.map a"
using ε_in_terms_of_ψ εo_def εo_in_hom by fastforce
qed
end
section "Composition of Adjunctions"
locale composite_adjunction =
A: category A +
B: category B +
C: category C +
F: "functor" B A F +
G: "functor" A B G +
F': "functor" C B F' +
G': "functor" B C G' +
FG: meta_adjunction A B F G φ ψ +
F'G': meta_adjunction B C F' G' φ' ψ'
for A :: "'a comp" (infixr ‹⋅⇩A› 55)
and B :: "'b comp" (infixr ‹⋅⇩B› 55)
and C :: "'c comp" (infixr ‹⋅⇩C› 55)
and F :: "'b ⇒ 'a"
and G :: "'a ⇒ 'b"
and F' :: "'c ⇒ 'b"
and G' :: "'b ⇒ 'c"
and φ :: "'b ⇒ 'a ⇒ 'b"
and ψ :: "'a ⇒ 'b ⇒ 'a"
and φ' :: "'c ⇒ 'b ⇒ 'c"
and ψ' :: "'b ⇒ 'c ⇒ 'b"
begin
interpretation S: replete_setcat .
interpretation FG: adjunction A B S.comp S.setp
FG.φC FG.φD F G φ ψ FG.η FG.ε FG.Φ FG.Ψ
using FG.induces_adjunction by simp
interpretation F'G': adjunction B C S.comp S.setp F'G'.φC F'G'.φD F' G' φ' ψ'
F'G'.η F'G'.ε F'G'.Φ F'G'.Ψ
using F'G'.induces_adjunction by simp
lemma is_meta_adjunction:
shows "meta_adjunction A C (F o F') (G' o G) (λz. φ' z o φ (F' z)) (λx. ψ x o ψ' (G x))"
proof -
interpret G'oG: composite_functor A B C G G' ..
interpret FoF': composite_functor C B A F' F ..
show ?thesis
proof
fix y f x
assume y: "C.ide y" and f: "«f : FoF'.map y →⇩A x»"
show "«(φ' y ∘ φ (F' y)) f : y →⇩C G'oG.map x»"
using y f FG.φ_in_hom F'G'.φ_in_hom by simp
show "(ψ x ∘ ψ' (G x)) ((φ' y ∘ φ (F' y)) f) = f"
using y f FG.φ_in_hom F'G'.φ_in_hom FG.ψ_φ F'G'.ψ_φ by simp
next
fix x g y
assume x: "A.ide x" and g: "«g : y →⇩C G'oG.map x»"
show "«(ψ x ∘ ψ' (G x)) g : FoF'.map y →⇩A x»"
using x g FG.ψ_in_hom F'G'.ψ_in_hom by auto
show "(φ' y ∘ φ (F' y)) ((ψ x ∘ ψ' (G x)) g) = g"
using x g FG.ψ_in_hom F'G'.ψ_in_hom FG.φ_ψ F'G'.φ_ψ by simp
next
fix f x x' g y' y h
assume f: "«f : x →⇩A x'»" and g: "«g : y' →⇩C y»" and h: "«h : FoF'.map y →⇩A x»"
show "(φ' y' ∘ φ (F' y')) (f ⋅⇩A h ⋅⇩A FoF'.map g) =
G'oG.map f ⋅⇩C (φ' y ∘ φ (F' y)) h ⋅⇩C g"
using f g h FG.φ_naturality [of f x x' "F' g" "F' y'" "F' y" h]
F'G'.φ_naturality [of "G f" "G x" "G x'" g y' y "φ (F' y) h"]
FG.φ_in_hom
by fastforce
qed
qed
interpretation KηH: natural_transformation C C ‹G' o F'› ‹G' o G o F o F'›
‹G' o FG.η o F'›
proof -
interpret ηF': natural_transformation C B F' ‹(G o F) o F'› ‹FG.η o F'›
using FG.η_is_natural_transformation F'.as_nat_trans.natural_transformation_axioms
horizontal_composite
by fastforce
interpret G'ηF': natural_transformation C C ‹G' o F'› ‹G' o (G o F o F')›
‹G' o (FG.η o F')›
using ηF'.natural_transformation_axioms G'.as_nat_trans.natural_transformation_axioms
horizontal_composite
by blast
show "natural_transformation C C (G' o F') (G' o G o F o F') (G' o FG.η o F')"
using G'ηF'.natural_transformation_axioms o_assoc by metis
qed
interpretation G'ηF'oη': vertical_composite C C C.map ‹G' o F'› ‹G' o G o F o F'›
F'G'.η ‹G' o FG.η o F'› ..
interpretation FεG: natural_transformation A A ‹F o F' o G' o G› ‹F o G›
‹F o F'G'.ε o G›
proof -
interpret Fε': natural_transformation B A ‹F o (F' o G')› F ‹F o F'G'.ε›
using F'G'.ε.natural_transformation_axioms F.as_nat_trans.natural_transformation_axioms
horizontal_composite
by fastforce
interpret Fε'G: natural_transformation A A ‹F o (F' o G') o G› ‹F o G› ‹F o F'G'.ε o G›
using Fε'.natural_transformation_axioms G.as_nat_trans.natural_transformation_axioms
horizontal_composite
by blast
show "natural_transformation A A (F o F' o G' o G) (F o G) (F o F'G'.ε o G)"
using Fε'G.natural_transformation_axioms o_assoc by metis
qed
interpretation εoFε'G: vertical_composite A A ‹F ∘ F' ∘ G' ∘ G› ‹F o G› A.map
‹F o F'G'.ε o G› FG.ε ..
interpretation meta_adjunction A C ‹F o F'› ‹G' o G›
‹λz. φ' z o φ (F' z)› ‹λx. ψ x o ψ' (G x)›
using is_meta_adjunction by auto
interpretation S: replete_setcat .
interpretation adjunction A C S.comp S.setp φC φD ‹F ∘ F'› ‹G' ∘ G›
‹λz. φ' z ∘ φ (F' z)› ‹λx. ψ x ∘ ψ' (G x)› η ε Φ Ψ
using induces_adjunction by simp
lemma η_char:
shows "η = G'ηF'oη'.map"
proof (intro NaturalTransformation.eqI)
show "natural_transformation C C C.map (G' o G o F o F') G'ηF'oη'.map" ..
show "natural_transformation C C C.map (G' o G o F o F') η"
by (metis (no_types, lifting) η_is_natural_transformation o_assoc)
fix a
assume a: "C.ide a"
show "η a = G'ηF'oη'.map a"
unfolding η_def
using a G'ηF'oη'.map_def FG.η.preserves_hom [of "F' a" "F' a" "F' a"]
F'G'.φ_in_terms_of_η FG.η_map_simp η_map_simp [of a] C.ide_in_hom
F'G'.η_def FG.η_def
by auto
qed
lemma ε_char:
shows "ε = εoFε'G.map"
proof (intro NaturalTransformation.eqI)
show "natural_transformation A A (F o F' o G' o G) A.map ε"
by (metis (no_types, lifting) ε_is_natural_transformation o_assoc)
show "natural_transformation A A (F ∘ F' ∘ G' ∘ G) A.map εoFε'G.map" ..
fix a
assume a: "A.ide a"
show "ε a = εoFε'G.map a"
proof -
have "ε a = ψ a (ψ' (G a) (G' (G a)))"
using a ε_in_terms_of_ψ by simp
also have "... = FG.ε a ⋅⇩A F (F'G'.ε (G a) ⋅⇩B F' (G' (G a)))"
by (metis F'G'.ε_in_terms_of_ψ F'G'.εo_def F'G'.εo_in_hom F'G'.ηε.ε_in_terms_of_ψ
F'G'.ηε.ψ_def FG.Gε.natural_transformation_axioms FG.ψ_in_terms_of_ε a
functor.preserves_ide natural_transformation_def)
also have "... = εoFε'G.map a"
using a B.comp_arr_dom εoFε'G.map_def by simp
finally show ?thesis by blast
qed
qed
end
section "Right Adjoints are Unique up to Natural Isomorphism"
text‹
As an example of the use of the of the foregoing development, we show that two right adjoints
to the same functor are naturally isomorphic.
›
theorem two_right_adjoints_naturally_isomorphic:
assumes "adjoint_functors C D F G" and "adjoint_functors C D F G'"
shows "naturally_isomorphic C D G G'"
proof -
text‹
For any object @{term x} of @{term C}, we have that ‹ε x ∈ C.hom (F (G x)) x›
is a terminal arrow from @{term F} to @{term x}, and similarly for ‹ε' x›.
We may therefore obtain the unique coextension ‹τ x ∈ D.hom (G x) (G' x)›
of ‹ε x› along ‹ε' x›.
An explicit formula for ‹τ x› is ‹D (G' (ε x)) (η' (G x))›.
Similarly, we obtain ‹τ' x = D (G (ε' x)) (η (G' x)) ∈ D.hom (G' x) (G x)›.
We show these are the components of inverse natural transformations between
@{term G} and @{term G'}.
›
obtain φ ψ where φψ: "meta_adjunction C D F G φ ψ"
using assms adjoint_functors_def by blast
obtain φ' ψ' where φ'ψ': "meta_adjunction C D F G' φ' ψ'"
using assms adjoint_functors_def by blast
interpret Adj: meta_adjunction C D F G φ ψ using φψ by auto
interpret S: replete_setcat .
interpret Adj: adjunction C D S.comp S.setp Adj.φC Adj.φD
F G φ ψ Adj.η Adj.ε Adj.Φ Adj.Ψ
using Adj.induces_adjunction by auto
interpret Adj': meta_adjunction C D F G' φ' ψ' using φ'ψ' by auto
interpret Adj': adjunction C D S.comp S.setp Adj'.φC Adj'.φD
F G' φ' ψ' Adj'.η Adj'.ε Adj'.Φ Adj'.Ψ
using Adj'.induces_adjunction by auto
write C (infixr ‹⋅⇩C› 55)
write D (infixr ‹⋅⇩D› 55)
write Adj.C.in_hom (‹«_ : _ →⇩C _»›)
write Adj.D.in_hom (‹«_ : _ →⇩D _»›)
let ?τo = "λa. G' (Adj.ε a) ⋅⇩D Adj'.η (G a)"
interpret τ: transformation_by_components C D G G' ?τo
proof
show "⋀a. Adj.C.ide a ⟹ «G' (Adj.ε a) ⋅⇩D Adj'.η (G a) : G a →⇩D G' a»"
by fastforce
show "⋀f. Adj.C.arr f ⟹
(G' (Adj.ε (Adj.C.cod f)) ⋅⇩D Adj'.η (G (Adj.C.cod f))) ⋅⇩D G f =
G' f ⋅⇩D G' (Adj.ε (Adj.C.dom f)) ⋅⇩D Adj'.η (G (Adj.C.dom f))"
proof -
fix f
assume f: "Adj.C.arr f"
let ?x = "Adj.C.dom f"
let ?x' = "Adj.C.cod f"
have "(G' (Adj.ε (Adj.C.cod f)) ⋅⇩D Adj'.η (G (Adj.C.cod f))) ⋅⇩D G f =
G' (Adj.ε (Adj.C.cod f) ⋅⇩C F (G f)) ⋅⇩D Adj'.η (G (Adj.C.dom f))"
using f Adj'.η.naturality [of "G f"] Adj.D.comp_assoc by simp
also have "... = G' (f ⋅⇩C Adj.ε (Adj.C.dom f)) ⋅⇩D Adj'.η (G (Adj.C.dom f))"
using f Adj.ε.naturality by simp
also have "... = G' f ⋅⇩D G' (Adj.ε (Adj.C.dom f)) ⋅⇩D Adj'.η (G (Adj.C.dom f))"
using f Adj.D.comp_assoc by simp
finally show "(G' (Adj.ε (Adj.C.cod f)) ⋅⇩D Adj'.η (G (Adj.C.cod f))) ⋅⇩D G f =
G' f ⋅⇩D G' (Adj.ε (Adj.C.dom f)) ⋅⇩D Adj'.η (G (Adj.C.dom f))"
by auto
qed
qed
interpret natural_isomorphism C D G G' τ.map
proof
fix a
assume a: "Adj.C.ide a"
show "Adj.D.iso (τ.map a)"
proof
show "Adj.D.inverse_arrows (τ.map a) (φ (G' a) (Adj'.ε a))"
proof
text‹
The proof that the two composites are identities is a modest diagram chase.
This is a good example of the inference rules for the ‹category›,
‹functor›, and ‹natural_transformation› locales in action.
Isabelle is able to use the single hypothesis that ‹a› is an identity to
implicitly fill in all the details that the various quantities are in fact arrows
and that the indicated composites are all well-defined, as well as to apply
associativity of composition. In most cases, this is done by auto or simp without
even mentioning any of the rules that are used.
$$\xymatrix{
{G' a} \ar[dd]_{\eta'(G'a)} \ar[rr]^{\tau' a} \ar[dr]_{\eta(G'a)}
&& {G a} \ar[rr]^{\tau a} \ar[dr]_{\eta'(Ga)} && {G' a} \\
& {GFG'a} \rrtwocell\omit{\omit(2)} \ar[ur]_{G(\epsilon' a)} \ar[dr]_{\eta'(GFG'a)}
&& {G'FGa} \drtwocell\omit{\omit(3)} \ar[ur]_{G'(\epsilon a)} & \\
{G'FG'a} \urtwocell\omit{\omit(1)} \ar[rr]_{G'F\eta(G'a)} \ar@/_8ex/[rrrr]_{G'FG'a}
&& {G'FGFG'a} \dtwocell\omit{\omit(4)} \ar[ru]_{G'FG(\epsilon' a)} \ar[rr]_{G'(\epsilon(FG'a))}
&& {G'FG'a} \ar[uu]_{G'(\epsilon' a)} \\
&&&&
}$$
›
show "Adj.D.ide (τ.map a ⋅⇩D φ (G' a) (Adj'.ε a))"
proof -
have "τ.map a ⋅⇩D φ (G' a) (Adj'.ε a) = G' a"
proof -
have "τ.map a ⋅⇩D φ (G' a) (Adj'.ε a) =
G' (Adj.ε a) ⋅⇩D (Adj'.η (G a) ⋅⇩D G (Adj'.ε a)) ⋅⇩D Adj.η (G' a)"
using a τ.map_simp_ide Adj.φ_in_terms_of_η Adj'.φ_in_terms_of_η
Adj'.ε.preserves_hom [of a a a] Adj.C.ide_in_hom Adj.D.comp_assoc
Adj.ε_def Adj.η_def
by simp
also have "... = G' (Adj.ε a) ⋅⇩D (G' (F (G (Adj'.ε a))) ⋅⇩D Adj'.η (G (F (G' a)))) ⋅⇩D
Adj.η (G' a)"
using a Adj'.η.naturality [of "G (Adj'.ε a)"] by auto
also have "... = (G' (Adj.ε a) ⋅⇩D G' (F (G (Adj'.ε a)))) ⋅⇩D G' (F (Adj.η (G' a))) ⋅⇩D
Adj'.η (G' a)"
using a Adj'.η.naturality [of "Adj.η (G' a)"] Adj.D.comp_assoc by auto
also have
"... = G' (Adj'.ε a) ⋅⇩D (G' (Adj.ε (F (G' a))) ⋅⇩D G' (F (Adj.η (G' a)))) ⋅⇩D
Adj'.η (G' a)"
proof -
have
"G' (Adj.ε a) ⋅⇩D G' (F (G (Adj'.ε a))) = G' (Adj'.ε a) ⋅⇩D G' (Adj.ε (F (G' a)))"
proof -
have "G' (Adj.ε a ⋅⇩C F (G (Adj'.ε a))) = G' (Adj'.ε a ⋅⇩C Adj.ε (F (G' a)))"
using a Adj.ε.naturality [of "Adj'.ε a"] by auto
thus ?thesis using a by force
qed
thus ?thesis using Adj.D.comp_assoc by auto
qed
also have "... = G' (Adj'.ε a) ⋅⇩D Adj'.η (G' a)"
proof -
have "G' (Adj.ε (F (G' a))) ⋅⇩D G' (F (Adj.η (G' a))) = G' (F (G' a))"
proof -
have
"G' (Adj.ε (F (G' a))) ⋅⇩D G' (F (Adj.η (G' a))) = G' (Adj.εFoFη.map (G' a))"
using a Adj.εFoFη.map_simp_1 by auto
moreover have "Adj.εFoFη.map (G' a) = F (G' a)"
using a by (simp add: Adj.ηε.triangle_F)
ultimately show ?thesis by auto
qed
thus ?thesis
using a Adj.D.comp_cod_arr [of "Adj'.η (G' a)"] by auto
qed
also have "... = G' a"
using a Adj'.ηε.triangle_G Adj'.GεoηG.map_simp_1 [of a] by auto
finally show ?thesis by auto
qed
thus ?thesis using a by simp
qed
show "Adj.D.ide (φ (G' a) (Adj'.ε a) ⋅⇩D τ.map a)"
proof -
have "φ (G' a) (Adj'.ε a) ⋅⇩D τ.map a = G a"
proof -
have "φ (G' a) (Adj'.ε a) ⋅⇩D τ.map a =
G (Adj'.ε a) ⋅⇩D (Adj.η (G' a) ⋅⇩D G' (Adj.ε a)) ⋅⇩D Adj'.η (G a)"
using a τ.map_simp_ide Adj.φ_in_terms_of_η Adj'.ε.preserves_hom [of a a a]
Adj.C.ide_in_hom Adj.D.comp_assoc Adj.η_def
by auto
also have
"... = G (Adj'.ε a) ⋅⇩D (G (F (G' (Adj.ε a))) ⋅⇩D Adj.η (G' (F (G a)))) ⋅⇩D
Adj'.η (G a)"
using a Adj.η.naturality [of "G' (Adj.ε a)"] by auto
also have
"... = (G (Adj'.ε a) ⋅⇩D G (F (G' (Adj.ε a)))) ⋅⇩D G (F (Adj'.η (G a))) ⋅⇩D
Adj.η (G a)"
using a Adj.η.naturality [of "Adj'.η (G a)"] Adj.D.comp_assoc by auto
also have
"... = G (Adj.ε a) ⋅⇩D (G (Adj'.ε (F (G a))) ⋅⇩D G (F (Adj'.η (G a)))) ⋅⇩D
Adj.η (G a)"
proof -
have "G (Adj'.ε a) ⋅⇩D G (F (G' (Adj.ε a))) = G (Adj.ε a) ⋅⇩D G (Adj'.ε (F (G a)))"
proof -
have "G (Adj'.ε a ⋅⇩C F (G' (Adj.ε a))) = G (Adj.ε a ⋅⇩C Adj'.ε (F (G a)))"
using a Adj'.ε.naturality [of "Adj.ε a"] by auto
thus ?thesis using a by force
qed
thus ?thesis using Adj.D.comp_assoc by auto
qed
also have "... = G (Adj.ε a) ⋅⇩D Adj.η (G a)"
proof -
have "G (Adj'.ε (F (G a))) ⋅⇩D G (F (Adj'.η (G a))) = G (F (G a))"
proof -
have
"G (Adj'.ε (F (G a))) ⋅⇩D G (F (Adj'.η (G a))) = G (Adj'.εFoFη.map (G a))"
using a Adj'.εFoFη.map_simp_1 [of "G a"] by auto
moreover have "Adj'.εFoFη.map (G a) = F (G a)"
using a by (simp add: Adj'.ηε.triangle_F)
ultimately show ?thesis by auto
qed
thus ?thesis
using a Adj.D.comp_cod_arr by auto
qed
also have "... = G a"
using a Adj.ηε.triangle_G Adj.GεoηG.map_simp_1 [of a] by auto
finally show ?thesis by auto
qed
thus ?thesis using a by auto
qed
qed
qed
qed
have "natural_isomorphism C D G G' τ.map" ..
thus "naturally_isomorphic C D G G'"
using naturally_isomorphic_def by blast
qed
end