Nuprl Lemma : mk-presheaf_wf
∀[C:SmallCategory]. ∀[S:cat-ob(C) ⟶ Type]. ∀[morph:I:cat-ob(C) ⟶ J:cat-ob(C) ⟶ f:(cat-arrow(C) J I) ⟶ S[I] ⟶ S[J]].
(Presheaf(Set(I) =S[I];
Morphism(I,J,f,rho) = morph[I;J;f;rho]) ∈ Presheaf(C)) supposing
((∀I,J,K:cat-ob(C). ∀f:cat-arrow(C) J I. ∀g:cat-arrow(C) K J. ∀rho:S[I].
(morph[I;K;cat-comp(C) K J I g f;rho] = morph[J;K;g;morph[I;J;f;rho]] ∈ S[K])) and
(∀I:cat-ob(C). ∀rho:S[I]. (morph[I;I;cat-id(C) I;rho] = rho ∈ S[I])))
Proof
Definitions occuring in Statement :
mk-presheaf: mk-presheaf,
presheaf: Presheaf(C),
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],
so_apply: x[s1;s2;s3;s4],
so_apply: x[s],
all: ∀x:A. B[x],
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
uimplies: b supposing a,
all: ∀x:A. B[x],
top: Top,
type-cat: TypeCat,
so_apply: x[s1;s2;s3;s4],
compose: f o g,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
mk-presheaf: mk-presheaf,
presheaf: Presheaf(C),
so_lambda: λ2x.t[x],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3]
Lemmas referenced :
op-cat_wf,
small-category-subtype,
type-cat_wf,
subtype_rel-equal,
cat-ob_wf,
cat_ob_op_lemma,
op-cat-arrow,
cat_arrow_triple_lemma,
cat-arrow_wf,
op-cat-comp,
cat_comp_tuple_lemma,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
op-cat-id,
cat_id_tuple_lemma,
mk-functor_wf,
all_wf,
cat-comp_wf,
cat-id_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
functionExtensionality,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaEquality,
lambdaFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
independent_functionElimination,
isect_memberFormation,
axiomEquality,
functionEquality,
cumulativity
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[S:cat-ob(C) {}\mrightarrow{} Type]. \mforall{}[morph:I:cat-ob(C)
{}\mrightarrow{} J:cat-ob(C)
{}\mrightarrow{} f:(cat-arrow(C) J I)
{}\mrightarrow{} S[I]
{}\mrightarrow{} S[J]].
(Presheaf(Set(I) =S[I];
Morphism(I,J,f,rho) = morph[I;J;f;rho]) \mmember{} Presheaf(C)) supposing
((\mforall{}I,J,K:cat-ob(C). \mforall{}f:cat-arrow(C) J I. \mforall{}g:cat-arrow(C) K J. \mforall{}rho:S[I].
(morph[I;K;cat-comp(C) K J I g f;rho] = morph[J;K;g;morph[I;J;f;rho]])) and
(\mforall{}I:cat-ob(C). \mforall{}rho:S[I]. (morph[I;I;cat-id(C) I;rho] = rho)))
Date html generated:
2017_10_05-AM-00_47_06
Last ObjectModification:
2017_10_03-PM-01_57_52
Theory : small!categories
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