Nuprl Lemma : mk-monad_wf
∀[C:SmallCategory]. ∀[T:Functor(C;C)]. ∀[u:nat-trans(C;C;1;T)]. ∀[m:nat-trans(C;C;functor-comp(T;T);T)].
(mk-monad(T;u;m) ∈ Monad(C)) supposing
((∀X:cat-ob(C)
((cat-comp(C) (ob(T) X) (ob(T) (ob(T) X)) (ob(T) X) (u (ob(T) X)) (m X))
= (cat-id(C) (ob(T) X))
∈ (cat-arrow(C) (ob(T) X) (ob(T) X)))) and
(∀X:cat-ob(C)
((cat-comp(C) (ob(T) X) (ob(T) (ob(T) X)) (ob(T) X) (arrow(T) X (ob(T) X) (u X)) (m X))
= (cat-id(C) (ob(T) X))
∈ (cat-arrow(C) (ob(T) X) (ob(T) X)))) and
(∀X:cat-ob(C)
((cat-comp(C) (ob(T) (ob(T) (ob(T) X))) (ob(T) (ob(T) X)) (ob(T) X) (m (ob(T) X)) (m X))
= (cat-comp(C) (ob(T) (ob(T) (ob(T) X))) (ob(T) (ob(T) X)) (ob(T) X)
(arrow(T) (ob(T) (ob(T) X)) (ob(T) X) (m X))
(m X))
∈ (cat-arrow(C) (ob(T) (ob(T) (ob(T) X))) (ob(T) X)))))
Proof
Definitions occuring in Statement :
mk-monad: mk-monad(T;u;m),
cat-monad: Monad(C),
id_functor: 1,
functor-comp: functor-comp(F;G),
nat-trans: nat-trans(C;D;F;G),
functor-arrow: arrow(F),
functor-ob: ob(F),
cat-functor: Functor(C1;C2),
cat-comp: cat-comp(C),
cat-id: cat-id(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
cand: A c∧ B,
and: P ∧ Q,
spreadn: spread3,
prop: ℙ,
mk-monad: mk-monad(T;u;m),
cat-monad: Monad(C),
uimplies: b supposing a,
id_functor: 1,
so_apply: x[s],
so_lambda: λ2x.t[x],
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
member: t ∈ T,
all: ∀x:A. B[x],
nat-trans: nat-trans(C;D;F;G),
functor-comp: functor-comp(F;G),
uall: ∀[x:A]. B[x]
Lemmas referenced :
small-category_wf,
cat-functor_wf,
id_functor_wf,
functor-comp_wf,
nat-trans_wf,
cat-id_wf,
functor-ob_wf,
cat-arrow_wf,
cat-ob_wf,
functor-arrow_wf,
cat-comp_wf,
equal_wf,
all_wf,
arrow_mk_functor_lemma,
ob_mk_functor_lemma
Rules used in proof :
productElimination,
independent_pairFormation,
functionEquality,
setEquality,
productEquality,
functionExtensionality,
applyEquality,
lambdaEquality,
isectElimination,
because_Cache,
rename,
setElimination,
independent_pairEquality,
hypothesisEquality,
dependent_pairEquality,
dependent_set_memberEquality,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[T:Functor(C;C)]. \mforall{}[u:nat-trans(C;C;1;T)].
\mforall{}[m:nat-trans(C;C;functor-comp(T;T);T)].
(mk-monad(T;u;m) \mmember{} Monad(C)) supposing
((\mforall{}X:cat-ob(C)
((cat-comp(C) (ob(T) X) (ob(T) (ob(T) X)) (ob(T) X) (u (ob(T) X)) (m X))
= (cat-id(C) (ob(T) X)))) and
(\mforall{}X:cat-ob(C)
((cat-comp(C) (ob(T) X) (ob(T) (ob(T) X)) (ob(T) X) (arrow(T) X (ob(T) X) (u X)) (m X))
= (cat-id(C) (ob(T) X)))) and
(\mforall{}X:cat-ob(C)
((cat-comp(C) (ob(T) (ob(T) (ob(T) X))) (ob(T) (ob(T) X)) (ob(T) X) (m (ob(T) X)) (m X))
= (cat-comp(C) (ob(T) (ob(T) (ob(T) X))) (ob(T) (ob(T) X)) (ob(T) X)
(arrow(T) (ob(T) (ob(T) X)) (ob(T) X) (m X))
(m X)))))
Date html generated:
2017_01_19-PM-02_57_51
Last ObjectModification:
2017_01_16-PM-08_02_54
Theory : small!categories
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