Nuprl Lemma : yoneda-lemma
∀C:SmallCategory. ff-functor(C;FUN(op-cat(C);TypeCat);yoneda-embedding(C))
Proof
Definitions occuring in Statement :
yoneda-embedding: yoneda-embedding(C),
type-cat: TypeCat,
op-cat: op-cat(C),
functor-cat: FUN(C1;C2),
full-faithful-functor: ff-functor(C;D;F),
small-category: SmallCategory,
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
full-faithful-functor: ff-functor(C;D;F),
biject: Bij(A;B;f),
and: P ∧ Q,
inject: Inj(A;B;f),
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
surject: Surj(A;B;f),
yoneda-embedding: yoneda-embedding(C),
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],
nat-trans: nat-trans(C;D;F;G),
squash: ↓T,
uimplies: b supposing a,
cat-arrow: cat-arrow(C),
pi1: fst(t),
pi2: snd(t),
type-cat: TypeCat,
functor-ob: ob(F),
rep-pre-sheaf: rep-pre-sheaf(C;X),
true: True,
guard: {T},
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x],
mk-functor: mk-functor,
functor-cat: FUN(C1;C2),
functor-arrow: arrow(F),
compose: f o g
Lemmas referenced :
equal_wf,
cat-arrow_wf,
functor-cat_wf,
op-cat_wf,
small-category-subtype,
type-cat_wf,
functor-ob_wf,
yoneda-embedding_wf,
functor-arrow_wf,
cat-ob_wf,
small-category_wf,
functor_cat_arrow_lemma,
ob_mk_functor_lemma,
arrow_mk_functor_lemma,
subtype_rel-equal,
cat_ob_op_lemma,
subtype_rel_self,
rep-pre-sheaf_wf,
cat-id_wf,
ap_mk_nat_trans_lemma,
squash_wf,
true_wf,
cat-comp-ident1,
iff_weakening_equal,
functor_cat_ob_lemma,
cat_arrow_triple_lemma,
nat-trans-equal,
op-cat-arrow,
cat_comp_tuple_lemma,
cat-comp_wf,
cat-comp-ident2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
hypothesis,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
applyEquality,
hypothesisEquality,
sqequalRule,
because_Cache,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
applyLambdaEquality,
setElimination,
rename,
imageMemberEquality,
baseClosed,
imageElimination,
functionExtensionality,
independent_isectElimination,
functionEquality,
natural_numberEquality,
lambdaEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
productElimination,
independent_functionElimination,
dependent_pairFormation
Latex:
\mforall{}C:SmallCategory. ff-functor(C;FUN(op-cat(C);TypeCat);yoneda-embedding(C))
Date html generated:
2017_10_05-AM-00_47_20
Last ObjectModification:
2017_07_28-AM-09_19_38
Theory : small!categories
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