Nuprl Lemma : monad-of-Kleisli-adjunction
∀[C:SmallCategory]. ∀[M:Monad(C)]. (adjMonad(Kl(C;M)) = M ∈ Monad(C))
Proof
Definitions occuring in Statement :
Kleisli-adjunction: Kl(C;M),
Kleisli-right: KlG(C;M),
Kleisli-left: KlF(C;M),
Kleisli-cat: Kl(C;M),
adjunction-monad: adjMonad(adj),
cat-monad: Monad(C),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
monad-unit: monad-unit(M;x),
pi1: fst(t),
pi2: snd(t),
adjunction-monad: adjMonad(adj),
mk-monad: mk-monad(T;u;m),
Kleisli-adjunction: Kl(C;M),
mk-adjunction: mk-adjunction(b.eps[b];a.eta[a]),
mk-nat-trans: x |→ T[x],
monad-functor: monad-functor(M),
Kleisli-right: KlG(C;M),
Kleisli-left: KlF(C;M),
functor-comp: functor-comp(F;G),
top: Top,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
so_lambda: λ2x.t[x],
so_apply: x[s],
monad-fun: M(x),
cat_comp: g o f,
monad-extend: monad-extend(C;M;x;y;f),
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
monad-op: monad-op(M;x)
Lemmas referenced :
equal-monads,
adjunction-monad_wf,
Kleisli-cat_wf,
Kleisli-left_wf,
Kleisli-right_wf,
Kleisli-adjunction_wf,
monad-unit_wf,
cat-monad_wf,
small-category_wf,
monad-functor_wf,
cat-ob_wf,
cat-arrow_wf,
equal-functors,
ob_mk_functor_lemma,
monad-fun_wf,
arrow_mk_functor_lemma,
cat-comp_wf,
monad-op_wf,
functor-arrow_wf,
functor-ob_wf,
equal_wf,
squash_wf,
true_wf,
functor-arrow-comp,
iff_weakening_equal,
monad-equations,
cat_comp_assoc,
cat_comp_wf,
cat-comp-ident2,
ap_mk_nat_trans_lemma,
cat-comp-ident,
functor-arrow-id
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
hypothesis,
independent_isectElimination,
lambdaFormation,
sqequalRule,
because_Cache,
applyEquality,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaEquality,
natural_numberEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[M:Monad(C)]. (adjMonad(Kl(C;M)) = M)
Date html generated:
2017_10_05-AM-00_53_01
Last ObjectModification:
2017_07_28-AM-09_21_13
Theory : small!categories
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