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RS17, Proposition 9.10. #149

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RS17, Proposition 9.10. #149

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RS17, Proposition 9.10
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215 changes: 215 additions & 0 deletions src/simplicial-hott/09-yoneda.rzk.md
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Original file line number Diff line number Diff line change
Expand Up @@ -892,6 +892,28 @@ types are contractible.
:= (x : A) → is-contr (hom A a x)
```

Initial objects map to initial objects by equivalences.

```rzk
#def is-inital-equiv-is-initial uses (extext)
( A B : U)
( e : Equiv A B)
( a : A)
( is-initial-a : is-initial A a)
:
is-initial B (first e a)
:=
\ b →
ind-has-section-equiv A B e
( \ b → is-contr (hom B (first e a) b))
( \ a' → is-contr-equiv-is-contr (hom A a a')
( hom B (first e a) (first e a'))
( ap-hom A B (first e) a a'
, is-equiv-ap-hom-is-equiv extext A B (first e) (second e) a a')
( is-initial-a a'))
( b)
```

Initial objects satisfy an induction principle relative to covariant families.

```rzk
Expand Down Expand Up @@ -1114,6 +1136,28 @@ types are contractible.
:= (x : A) → is-contr (hom A x a)
```

Final objects map to final objects by equivalences.

```rzk
#def is-final-equiv-is-final uses (extext)
( A B : U)
( e : Equiv A B)
( a : A)
( is-final-a : is-final A a)
:
is-final B (first e a)
:=
\ b →
ind-has-section-equiv A B e
( \ b → is-contr (hom B b (first e a)))
( \ a' → is-contr-equiv-is-contr (hom A a' a)
( hom B (first e a') (first e a))
( ap-hom A B (first e) a' a
, is-equiv-ap-hom-is-equiv extext A B (first e) (second e) a' a)
( is-final-a a'))
( b)
```

Final objects satisfy an induction principle relative to contravariant families.

```rzk
Expand Down Expand Up @@ -1364,4 +1408,175 @@ condition a name.
: U
:= Σ ((a , u) : Σ (x : A) , (C x))
, is-initial (Σ (x : A) , (C x)) (a , u)

#def has-initial-tot-is-representable-family-is-segal uses (extext)
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( A : U)
( is-segal-A : is-segal A)
( C : A → U)
( is-rep-C : is-representable-family A C)
: has-initial-tot A C
:=
( ( first is-rep-C
, evid
( A)
( first is-rep-C)
( C)
( equiv-for-is-representable-family A C is-rep-C))
, is-inital-equiv-is-initial
( coslice A (first is-rep-C))
( Σ ( x : A) , (C x))
( total-equiv-family-of-equiv
( A)
( \ x → hom A (first is-rep-C) x)
( C)
( second is-rep-C))
( ( first is-rep-C , id-hom A (first is-rep-C)))
( is-initial-id-hom-is-segal A is-segal-A (first is-rep-C)))

#def is-representable-family-has-initial-tot
( A : U)
( is-segal-A : is-segal A)
( C : A → U)
( is-covariant-C : is-covariant A C)
( has-initial-tot-A : has-initial-tot A C)
: ( is-representable-family A C)
:=
( first(first has-initial-tot-A)
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, \ b →
( ( yon
( A)
( is-segal-A)
( first(first has-initial-tot-A))
( C)
( is-covariant-C)
( second(first has-initial-tot-A))
( b))
, is-equiv-is-contr-map
( hom A (first(first has-initial-tot-A)) b)
( C b)
( yon
( A)
( is-segal-A)
( first(first has-initial-tot-A))
( C)
( is-covariant-C)
( second(first has-initial-tot-A))
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( b))
( \ v →
is-contr-equiv-is-contr
( hom (Σ (a : A) , (C a)) (first has-initial-tot-A) (b , v))
( fib
( hom A (first(first has-initial-tot-A)) b)
( C b)
( yon
( A)
( is-segal-A)
( first(first has-initial-tot-A))
( C)
( is-covariant-C)
( second(first has-initial-tot-A))
( b))
( v))
( equiv-comp
( hom (Σ (a : A) , (C a)) (first has-initial-tot-A) (b , v))
( Σ ( f : hom A (first(first has-initial-tot-A)) b)
, dhom
( A)
( first(first has-initial-tot-A))
( b)
( f)
( C)
( second(first has-initial-tot-A))
( v))
( fib
( hom A (first(first has-initial-tot-A)) b)
( C b)
( yon
( A)
( is-segal-A)
( first(first has-initial-tot-A))
( C)
( is-covariant-C)
( second(first has-initial-tot-A))
( b))
( v))
( axiom-choice
( 2)
( Δ¹)
( ∂Δ¹)
( \ _ → A)
( \ t x → C x)
( \ t →
recOR
( t ≡ 0₂ ↦ first(first has-initial-tot-A)
, t ≡ 1₂ ↦ b))
( \ t →
recOR
( t ≡ 0₂ ↦ second(first has-initial-tot-A)
, t ≡ 1₂ ↦ v)))
( total-equiv-family-of-equiv
( hom A (first(first has-initial-tot-A)) b)
( \ f →
dhom
( A)
( first(first has-initial-tot-A))
( b)
( f)
( C)
( second(first has-initial-tot-A))
( v))
( \ f →
covariant-transport
( A)
( first(first has-initial-tot-A))
( b)
( f)
( C)
( is-covariant-C)
( second(first has-initial-tot-A))
= v)
( \ f →
( covariant-uniqueness-curried
( A)
( first(first has-initial-tot-A))
( b)
( f)
( C)
( is-covariant-C)
( second(first has-initial-tot-A))
( v)
, is-equiv-covariant-uniqueness-curried
( A)
( first(first has-initial-tot-A))
( b)
( f)
( C)
( is-covariant-C)
( second(first has-initial-tot-A))
( v)))))
( second has-initial-tot-A (b , v)))))
```

```rzk title="RS17, Proposition 9.10"
#def is-representable-iff-has-initial-tot uses (extext)
( A : U)
( is-segal-A : is-segal A)
( C : A → U)
( is-covariant-C : is-covariant A C)
: iff (is-representable-family A C) (has-initial-tot A C)
:=
( \ is-rep-C →
has-initial-tot-is-representable-family-is-segal
( A)
( is-segal-A)
( C)
( is-rep-C)
, \ has-initial-tot-A →
is-representable-family-has-initial-tot
( A)
( is-segal-A)
( C)
( is-covariant-C)
( has-initial-tot-A))
```

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