Nuprl Lemma : mk-adjunction_wf
∀[A,B:SmallCategory]. ∀[F:Functor(A;B)]. ∀[G:Functor(B;A)]. ∀[eps:b:cat-ob(B) ⟶ (cat-arrow(B) (F (G b)) b)].
∀[eta:a:cat-ob(A) ⟶ (cat-arrow(A) a (G (F a)))].
(mk-adjunction(b.eps[b];a.eta[a]) ∈ F -| G) supposing
((∀a1,a2:cat-ob(A). ∀g:cat-arrow(A) a1 a2.
((cat-comp(A) a1 (G (F a1)) (G (F a2)) eta[a1] (G (F a1) (F a2) (F a1 a2 g)))
= (cat-comp(A) a1 a2 (G (F a2)) g eta[a2])
∈ (cat-arrow(A) a1 (G (F a2))))) and
(∀b1,b2:cat-ob(B). ∀g:cat-arrow(B) b1 b2.
((cat-comp(B) (F (G b1)) b1 b2 eps[b1] g)
= (cat-comp(B) (F (G b1)) (F (G b2)) b2 (F (G b1) (G b2) (G b1 b2 g)) eps[b2])
∈ (cat-arrow(B) (F (G b1)) b2))) and
counit-unit-equations(A;B;F;G;λb.eps[b];λa.eta[a]))
Proof
Definitions occuring in Statement :
mk-adjunction: mk-adjunction(b.eps[b];a.eta[a]),
counit-unit-adjunction: F -| G,
counit-unit-equations: counit-unit-equations(D;C;F;G;eps;eta),
functor-arrow: arrow(F),
functor-ob: ob(F),
cat-functor: Functor(C1;C2),
cat-comp: cat-comp(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
member: t ∈ T,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
counit-unit-adjunction: F -| G,
mk-adjunction: mk-adjunction(b.eps[b];a.eta[a]),
pi1: fst(t),
pi2: snd(t),
mk-nat-trans: x |→ T[x],
nat-trans: nat-trans(C;D;F;G),
id_functor: 1,
functor-comp: functor-comp(F;G),
all: ∀x:A. B[x],
top: Top,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
prop: ℙ,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
ob_mk_functor_lemma,
arrow_mk_functor_lemma,
counit-unit-equations_wf,
cat-ob_wf,
cat-arrow_wf,
functor-ob_wf,
pi1_wf_top,
equal_wf,
all_wf,
cat-comp_wf,
functor-arrow_wf,
cat-functor_wf,
small-category_wf,
mk-functor_wf,
squash_wf,
true_wf,
functor-arrow-comp,
iff_weakening_equal,
functor-arrow-id,
cat-id_wf,
mk-nat-trans_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
dependent_set_memberEquality,
sqequalRule,
hypothesis,
productElimination,
thin,
sqequalHypSubstitution,
setElimination,
rename,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
isectElimination,
hypothesisEquality,
independent_pairEquality,
functionExtensionality,
applyEquality,
productEquality,
functionEquality,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
because_Cache,
lambdaEquality,
independent_isectElimination,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[A,B:SmallCategory]. \mforall{}[F:Functor(A;B)]. \mforall{}[G:Functor(B;A)]. \mforall{}[eps:b:cat-ob(B) {}\mrightarrow{} (cat-arrow(B)
(F (G b))
b)].
\mforall{}[eta:a:cat-ob(A) {}\mrightarrow{} (cat-arrow(A) a (G (F a)))].
(mk-adjunction(b.eps[b];a.eta[a]) \mmember{} F -| G) supposing
((\mforall{}a1,a2:cat-ob(A). \mforall{}g:cat-arrow(A) a1 a2.
((cat-comp(A) a1 (G (F a1)) (G (F a2)) eta[a1] (G (F a1) (F a2) (F a1 a2 g)))
= (cat-comp(A) a1 a2 (G (F a2)) g eta[a2]))) and
(\mforall{}b1,b2:cat-ob(B). \mforall{}g:cat-arrow(B) b1 b2.
((cat-comp(B) (F (G b1)) b1 b2 eps[b1] g)
= (cat-comp(B) (F (G b1)) (F (G b2)) b2 (F (G b1) (G b2) (G b1 b2 g)) eps[b2]))) and
counit-unit-equations(A;B;F;G;\mlambda{}b.eps[b];\mlambda{}a.eta[a]))
Date html generated:
2020_05_20-AM-07_58_21
Last ObjectModification:
2017_07_28-AM-09_20_40
Theory : small!categories
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