Nuprl Lemma : adjunction-monad_wf
∀[A,B:SmallCategory]. ∀[F:Functor(A;B)]. ∀[G:Functor(B;A)]. ∀[adj:F -| G]. (adjMonad(adj) ∈ Monad(A))
Proof
Definitions occuring in Statement :
adjunction-monad: adjMonad(adj),
cat-monad: Monad(C),
counit-unit-adjunction: F -| G,
cat-functor: Functor(C1;C2),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
adjunction-monad: adjMonad(adj),
member: t ∈ T,
counit-unit-adjunction: F -| G,
pi2: snd(t),
pi1: fst(t),
uimplies: b supposing a,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
nat-trans: nat-trans(C;D;F;G),
id_functor: 1,
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],
counit-unit-equations: counit-unit-equations(D;C;F;G;eps;eta),
and: P ∧ Q,
true: True,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
mk-monad_wf,
functor-comp_wf,
counit-unit-adjunction_wf,
cat-functor_wf,
small-category_wf,
cat-ob_wf,
mk-nat-trans_wf,
ob_mk_functor_lemma,
arrow_mk_functor_lemma,
functor-arrow_wf,
functor-ob_wf,
cat-arrow_wf,
cat-comp_wf,
equal_wf,
squash_wf,
true_wf,
functor-arrow-comp,
iff_weakening_equal,
ap_mk_nat_trans_lemma,
cat-id_wf,
functor-arrow-id
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
productElimination,
independent_isectElimination,
because_Cache,
dependent_functionElimination,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
applyEquality,
functionExtensionality,
lambdaFormation,
natural_numberEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[A,B:SmallCategory]. \mforall{}[F:Functor(A;B)]. \mforall{}[G:Functor(B;A)]. \mforall{}[adj:F -| G].
(adjMonad(adj) \mmember{} Monad(A))
Date html generated:
2017_10_05-AM-00_52_40
Last ObjectModification:
2017_07_28-AM-09_20_58
Theory : small!categories
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