Nuprl Lemma : yoneda-embedding_wf
∀[C:SmallCategory]. (yoneda-embedding(C) ∈ Functor(C;FUN(op-cat(C);TypeCat)))
Proof
Definitions occuring in Statement :
yoneda-embedding: yoneda-embedding(C),
type-cat: TypeCat,
op-cat: op-cat(C),
functor-cat: FUN(C1;C2),
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],
yoneda-embedding: yoneda-embedding(C),
member: t ∈ T,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
rep-pre-sheaf: rep-pre-sheaf(C;X),
functor-ob: ob(F),
type-cat: TypeCat,
pi1: fst(t),
functor-arrow: arrow(F),
pi2: snd(t),
compose: f o g,
true: True,
squash: ↓T,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
trans-comp: t1 o t2,
identity-trans: identity-trans(C;D;F)
Lemmas referenced :
small-category_wf,
small-category-subtype,
functor-cat_wf,
op-cat_wf,
type-cat_wf,
functor_cat_ob_lemma,
rep-pre-sheaf_wf,
cat-ob_wf,
functor_cat_arrow_lemma,
mk-nat-trans_wf,
cat-arrow_wf,
functor_cat_comp_lemma,
functor_cat_id_lemma,
mk-functor_wf,
cat_arrow_triple_lemma,
cat-comp_wf,
subtype_rel-equal,
cat_ob_op_lemma,
cat_comp_tuple_lemma,
op-cat-arrow,
equal_wf,
squash_wf,
true_wf,
cat-comp-assoc,
iff_weakening_equal,
ap_mk_nat_trans_lemma,
all_wf,
functor-ob_wf,
functor-arrow_wf,
cat-functor_wf,
cat_id_tuple_lemma,
cat-comp-ident
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
hypothesisEquality,
applyEquality,
sqequalHypSubstitution,
sqequalRule,
thin,
instantiate,
isectElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
lambdaEquality,
independent_isectElimination,
lambdaFormation,
functionExtensionality,
natural_numberEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
functionEquality
Latex:
\mforall{}[C:SmallCategory]. (yoneda-embedding(C) \mmember{} Functor(C;FUN(op-cat(C);TypeCat)))
Date html generated:
2017_10_05-AM-00_47_15
Last ObjectModification:
2017_07_28-AM-09_19_36
Theory : small!categories
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