Nuprl Lemma : unit-set-presheaf-type
∀[C:SmallCategory]. ∀[I:cat-ob(C)].
({Yoneda(I) ⊢ _} ~ {AF:A:L:cat-ob(C) ⟶ (cat-arrow(C) L I) ⟶ Type × (L:cat-ob(C)
⟶ J:cat-ob(C)
⟶ f:(cat-arrow(C) J L)
⟶ a:(cat-arrow(C) L I)
⟶ (A L a)
⟶ (A J (cat-comp(C) J L I f a)))|
let A,F = AF
in (∀K:cat-ob(C). ∀a:cat-arrow(C) K I. ∀u:A K a. ((F K K (cat-id(C) K) a u) = u ∈ (A K a)))
∧ (∀L,J,K:cat-ob(C). ∀f:cat-arrow(C) J L. ∀g:cat-arrow(C) K J. ∀a:cat-arrow(C) L I. ∀u:A L a.
((F L K (cat-comp(C) K J L g f) a u)
= (F J K g (cat-comp(C) J L I f a) (F L J f a u))
∈ (A K (cat-comp(C) K L I (cat-comp(C) K J L g f) a))))} )
Proof
Definitions occuring in Statement :
presheaf-type: {X ⊢ _},
Yoneda: Yoneda(I),
cat-comp: cat-comp(C),
cat-id: cat-id(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
spread: spread def,
product: x:A × B[x],
universe: Type,
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
presheaf-type: {X ⊢ _},
cat-ob: cat-ob(C),
pi1: fst(t),
I_set: A(I),
functor-ob: ob(F),
Yoneda: Yoneda(I),
cat-arrow: cat-arrow(C),
pi2: snd(t),
psc-restriction: f(s),
cat-comp: cat-comp(C),
all: ∀x:A. B[x]
Lemmas referenced :
cat-ob_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
sqequalAxiom,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
because_Cache
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[I:cat-ob(C)].
(\{Yoneda(I) \mvdash{} \_\} \msim{} \{AF:A:L:cat-ob(C) {}\mrightarrow{} (cat-arrow(C) L I) {}\mrightarrow{} Type \mtimes{} (L:cat-ob(C)
{}\mrightarrow{} J:cat-ob(C)
{}\mrightarrow{} f:(cat-arrow(C) J L)
{}\mrightarrow{} a:(cat-arrow(C) L I)
{}\mrightarrow{} (A L a)
{}\mrightarrow{} (A J
(cat-comp(C) J L I f
a)))|
let A,F = AF
in (\mforall{}K:cat-ob(C). \mforall{}a:cat-arrow(C) K I. \mforall{}u:A K a.
((F K K (cat-id(C) K) a u) = u))
\mwedge{} (\mforall{}L,J,K:cat-ob(C). \mforall{}f:cat-arrow(C) J L. \mforall{}g:cat-arrow(C) K J.
\mforall{}a:cat-arrow(C) L I. \mforall{}u:A L a.
((F L K (cat-comp(C) K J L g f) a u)
= (F J K g (cat-comp(C) J L I f a) (F L J f a u))))\} )
Date html generated:
2018_05_23-AM-08_25_48
Last ObjectModification:
2018_05_20-PM-10_07_24
Theory : presheaf!models!of!type!theory
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