fact
stringlengths 4
3.31k
| type
stringclasses 14
values | library
stringclasses 23
values | imports
listlengths 1
59
| filename
stringlengths 20
105
| symbolic_name
stringlengths 1
89
| docstring
stringlengths 0
1.75k
⌀ |
|---|---|---|---|---|---|---|
transportf_reindex
{C' C : category}
{D : disp_cat C}
{F : C' ⟶ C}
{x y : C'}
{xx : D(F x)} {yy : D(F y)}
{f g : x --> y}
(p : f = g)
(ff : xx -->[# F f] yy)
: transportf
(@mor_disp
C'
(reindex_disp_cat F D)
_ _
xx yy)
p
ff
=
transportf
(@mor_disp
C
D
_ _
xx yy)
(maponpaths (# F) p)
ff.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
transportf_reindex
|
1. Transport lemmas and [idtoiso] in the reindexing
|
transportf_reindex_ob
{C₁ C₂ : category}
{D : disp_cat C₂}
{F : C₁ ⟶ C₂}
{x y : C₁}
(p : x = y)
(xx : D (F x))
: transportf (reindex_disp_cat F D) p xx
=
transportf D (maponpaths (λ z, F z) p) xx.
|
Proposition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
transportf_reindex_ob
| null |
idtoiso_reindex_disp_cat
{C₁ C₂ : category}
{F : C₁ ⟶ C₂}
{D : disp_cat C₂}
{x : C₁}
{xx xx' : D(F x)}
(p : xx = xx')
: pr1 (idtoiso_disp (D := reindex_disp_cat F D) (idpath _) p)
=
transportb
(λ z, _ -->[ z ] _)
(functor_id _ _)
(idtoiso_disp (D := D) (idpath _) p).
|
Proposition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
idtoiso_reindex_disp_cat
| null |
idtoiso_reindex_disp_cat'_path
{C₁ C₂ : category}
{F : C₁ ⟶ C₂}
{D : disp_cat C₂}
{x x' : C₁}
{xx : D(F x)}
{xx' : D(F x')}
{p : x = x'}
(pp : transportf (λ z, D(F z)) p xx = xx')
: transportf D (maponpaths F p) xx = xx'.
|
Proposition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
idtoiso_reindex_disp_cat
| null |
idtoiso_reindex_disp_cat'
{C₁ C₂ : category}
{F : C₁ ⟶ C₂}
{D : disp_cat C₂}
{x x' : C₁}
{xx : D(F x)}
{xx' : D(F x')}
{p : x = x'}
(pp : transportf (λ z, D(F z)) p xx = xx')
: pr1 (idtoiso_disp (D := reindex_disp_cat F D) p pp)
=
transportf
(λ z, _ -->[ z ] _)
(pr1_maponpaths_idtoiso F p)
(idtoiso_disp (D := D) (maponpaths F p) (idtoiso_reindex_disp_cat'_path pp)).
|
Proposition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
idtoiso_reindex_disp_cat
| null |
z_iso_disp_to_z_iso_disp_reindex
{C₁ C₂ : category}
{F : C₁ ⟶ C₂}
{D : disp_cat C₂}
{x : C₁}
{xx yy : D (F x)}
: z_iso_disp (identity_z_iso (F x)) xx yy
→
@z_iso_disp
_
(reindex_disp_cat F D)
_
_
(identity_z_iso x)
xx
yy.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
z_iso_disp_to_z_iso_disp_reindex
|
2. Characterization of displayed isomorphisms
|
z_iso_disp_reindex_to_z_iso_disp
{C₁ C₂ : category}
{F : C₁ ⟶ C₂}
{D : disp_cat C₂}
{x : C₁}
{xx yy : D (F x)}
: @z_iso_disp
_
(reindex_disp_cat F D)
_
_
(identity_z_iso x)
xx
yy
→
z_iso_disp (identity_z_iso (F x)) xx yy.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
z_iso_disp_reindex_to_z_iso_disp
| null |
z_iso_disp_weq_z_iso_disp_reindex
{C₁ C₂ : category}
{F : C₁ ⟶ C₂}
{D : disp_cat C₂}
{x : C₁}
(xx yy : D (F x))
: z_iso_disp (identity_z_iso (F x)) xx yy
≃
@z_iso_disp
_
(reindex_disp_cat F D)
_
_
(identity_z_iso x)
xx
yy.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
z_iso_disp_weq_z_iso_disp_reindex
| null |
is_z_isomorphism_reindex_disp_cat
{C₁ C₂ : category}
(F : C₁ ⟶ C₂)
(D : disp_cat C₂)
{x y : C₁}
{f : x --> y}
(Hf : is_z_isomorphism f)
{xx : D(F x)}
{yy : D(F y)}
(ff : xx -->[ #F f ] yy)
(Hff : is_z_iso_disp
(make_z_iso' (#F f) (functor_on_is_z_isomorphism F Hf))
ff)
: is_z_iso_disp (D := reindex_disp_cat F D) (make_z_iso' f Hf) ff.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
is_z_isomorphism_reindex_disp_cat
| null |
is_univalent_reindex_disp_cat
{C₁ C₂ : category}
(F : C₁ ⟶ C₂)
(D : disp_cat C₂)
(HD : is_univalent_disp D)
: is_univalent_disp (reindex_disp_cat F D).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
is_univalent_reindex_disp_cat
|
3. The univalence
|
univalent_reindex_cat
{C₁ C₂ : univalent_category}
(F : C₁ ⟶ C₂)
(D : disp_univalent_category C₂)
: univalent_category.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
univalent_reindex_cat
| null |
is_cartesian_in_reindex_disp_cat
{C₁ C₂ : category}
(F : C₁ ⟶ C₂)
(D : disp_cat C₂)
{x y : C₁}
{f : x --> y}
{xx : D (F x)}
{yy : D (F y)}
(ff : xx -->[ #F f ] yy)
(Hff : is_cartesian ff)
: @is_cartesian _ (reindex_disp_cat F D) y x f yy xx ff.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
is_cartesian_in_reindex_disp_cat
|
4. Characterization of cartesian and opcartesian morphisms
|
ℓ : cartesian_lift yy (# F f) := HD (F y) (F x) (#F f) yy.
|
Let
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
ℓ
| null |
m : xx -->[ identity (F x)] pr1 ℓ := cartesian_factorisation
ℓ
(identity _)
(transportb
(λ z, _ -->[ z ] _)
(id_left _)
ff).
|
Let
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
m
| null |
minv' : pr1 ℓ -->[ # F (identity x · f)] yy
:= transportb
_
(maponpaths (λ z, #F z) (id_left _))
(pr12 ℓ).
|
Let
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
minv
| null |
minv : pr1 ℓ -->[ identity (F x)] xx
:= transportf
_
(functor_id _ _)
(cartesian_factorisation Hff _ minv').
|
Let
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
minv
| null |
minv_m
: (minv ;; m)%mor_disp
=
transportb
(λ z, _ -->[ z ] _)
(z_iso_after_z_iso_inv (make_z_iso' _ (identity_is_z_iso (F x))))
(id_disp ℓ).
|
Lemma
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
minv_m
| null |
m_minv
: (m ;; minv)%mor_disp
=
transportb
(λ z, _ -->[ z ] _)
(z_iso_inv_after_z_iso (make_z_iso' _ (identity_is_z_iso (F x))))
(id_disp xx).
|
Lemma
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
m_minv
| null |
is_cartesian_from_reindex_disp_cat
: is_cartesian ff.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
is_cartesian_from_reindex_disp_cat
| null |
is_opcartesian_in_reindex_disp_cat
{C₁ C₂ : category}
(F : C₁ ⟶ C₂)
(D : disp_cat C₂)
{x y : C₁}
{f : x --> y}
{xx : D (F x)}
{yy : D (F y)}
(ff : xx -->[ #F f ] yy)
(Hff : is_opcartesian ff)
: @is_opcartesian _ (reindex_disp_cat F D) x y f xx yy ff.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
is_opcartesian_in_reindex_disp_cat
| null |
ℓ : opcartesian_lift _ xx (# F f) := HD (F x) (F y) xx (#F f).
|
Let
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
ℓ
| null |
m : pr1 ℓ -->[ identity (F y)] yy
:= opcartesian_factorisation
(mor_of_opcartesian_lift_is_opcartesian _ ℓ)
(identity _)
(transportb
(λ z, _ -->[ z ] _)
(id_right _)
ff).
|
Let
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
m
| null |
minv' : xx -->[ # F (f · identity y) ] pr1 ℓ
:= transportb
_
(maponpaths (λ z, #F z) (id_right _))
(pr12 ℓ).
|
Let
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
minv
| null |
minv : yy -->[ identity (F y)] pr1 ℓ
:= transportf
_
(functor_id _ _)
(opcartesian_factorisation Hff _ minv').
|
Let
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
minv
| null |
op_minv_m
: (minv ;; m)%mor_disp
=
transportb
(λ z, _ -->[ z ] _)
(z_iso_after_z_iso_inv (make_z_iso' _ (identity_is_z_iso _)))
(id_disp _).
|
Lemma
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
op_minv_m
| null |
op_m_minv
: (m ;; minv)%mor_disp
=
transportb
(λ z, _ -->[ z ] _)
(z_iso_inv_after_z_iso (make_z_iso' _ (identity_is_z_iso _)))
(id_disp _).
|
Lemma
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
op_m_minv
| null |
is_opcartesian_from_reindex_disp_cat_help
: ff
=
transportf
(λ z, _ -->[ z ] _)
(id_right _)
(opcleaving_mor HD _ _ ;; m)%mor_disp.
|
Lemma
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
is_opcartesian_from_reindex_disp_cat_help
| null |
is_opcartesian_from_reindex_disp_cat
: is_opcartesian ff.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
is_opcartesian_from_reindex_disp_cat
| null |
cleaving_reindex_disp_cat
{C₁ C₂ : category}
(F : C₁ ⟶ C₂)
(D : disp_cat C₂)
(HD : cleaving D)
: cleaving (reindex_disp_cat F D).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
cleaving_reindex_disp_cat
|
5. Cleaving
|
opcleaving_reindex_disp_cat
{C₁ C₂ : category}
(F : C₁ ⟶ C₂)
(D : disp_cat C₂)
(HD : opcleaving D)
: opcleaving (reindex_disp_cat F D).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
opcleaving_reindex_disp_cat
| null |
reindex_disp_cat_disp_functor_data
{C₁ C₂ : category}
(F : C₁ ⟶ C₂)
(D : disp_cat C₂)
: disp_functor_data F (reindex_disp_cat F D) D.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_disp_cat_disp_functor_data
|
6. Functor from reindexing
|
reindex_disp_cat_disp_functor_axioms
{C₁ C₂ : category}
(F : C₁ ⟶ C₂)
(D : disp_cat C₂)
: disp_functor_axioms (reindex_disp_cat_disp_functor_data F D).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_disp_cat_disp_functor_axioms
| null |
reindex_disp_cat_disp_functor
{C₁ C₂ : category}
(F : C₁ ⟶ C₂)
(D : disp_cat C₂)
: disp_functor F (reindex_disp_cat F D) D.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_disp_cat_disp_functor
| null |
is_cartesian_reindex_disp_cat_disp_functor
{C₁ C₂ : category}
(F : C₁ ⟶ C₂)
(D : disp_cat C₂)
(HD : cleaving D)
: is_cartesian_disp_functor (reindex_disp_cat_disp_functor F D).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
is_cartesian_reindex_disp_cat_disp_functor
| null |
is_opcartesian_reindex_disp_cat_disp_functor
{C₁ C₂ : category}
(F : C₁ ⟶ C₂)
(D : disp_cat C₂)
(HD : opcleaving D)
: is_opcartesian_disp_functor (reindex_disp_cat_disp_functor F D).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
is_opcartesian_reindex_disp_cat_disp_functor
| null |
lift_functor_into_reindex_data
{C₁ C₂ C₃ : category}
{D₁ : disp_cat C₁}
{D₃ : disp_cat C₃}
{F₁ : C₁ ⟶ C₂}
{F₂ : C₂ ⟶ C₃}
(FF : disp_functor (F₁ ∙ F₂) D₁ D₃)
: disp_functor_data F₁ D₁ (reindex_disp_cat F₂ D₃).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
lift_functor_into_reindex_data
|
7. Mapping property
|
lift_functor_into_reindex_axioms
{C₁ C₂ C₃ : category}
{D₁ : disp_cat C₁}
{D₃ : disp_cat C₃}
{F₁ : C₁ ⟶ C₂}
{F₂ : C₂ ⟶ C₃}
(FF : disp_functor (F₁ ∙ F₂) D₁ D₃)
: disp_functor_axioms (lift_functor_into_reindex_data FF).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
lift_functor_into_reindex_axioms
| null |
lift_functor_into_reindex
{C₁ C₂ C₃ : category}
{D₁ : disp_cat C₁}
{D₃ : disp_cat C₃}
{F₁ : C₁ ⟶ C₂}
{F₂ : C₂ ⟶ C₃}
(FF : disp_functor (F₁ ∙ F₂) D₁ D₃)
: disp_functor F₁ D₁ (reindex_disp_cat F₂ D₃).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
lift_functor_into_reindex
| null |
is_cartesian_lift_functor_into_reindex
{C₁ C₂ C₃ : category}
{D₁ : disp_cat C₁}
{D₃ : disp_cat C₃}
{F₁ : C₁ ⟶ C₂}
{F₂ : C₂ ⟶ C₃}
{FF : disp_functor (F₁ ∙ F₂) D₁ D₃}
(HFF : is_cartesian_disp_functor FF)
: is_cartesian_disp_functor (lift_functor_into_reindex FF).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
is_cartesian_lift_functor_into_reindex
| null |
is_opcartesian_lift_functor_into_reindex
{C₁ C₂ C₃ : category}
{D₁ : disp_cat C₁}
{D₃ : disp_cat C₃}
{F₁ : C₁ ⟶ C₂}
{F₂ : C₂ ⟶ C₃}
{FF : disp_functor (F₁ ∙ F₂) D₁ D₃}
(HFF : is_opcartesian_disp_functor FF)
: is_opcartesian_disp_functor (lift_functor_into_reindex FF).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
is_opcartesian_lift_functor_into_reindex
| null |
lift_functor_into_reindex_commute_data
{C₁ C₂ C₃ : category}
{D₁ : disp_cat C₁}
{D₃ : disp_cat C₃}
{F₁ : C₁ ⟶ C₂}
{F₂ : C₂ ⟶ C₃}
(FF : disp_functor (F₁ ∙ F₂) D₁ D₃)
: disp_nat_trans_data
(nat_trans_id _)
(disp_functor_composite
(lift_functor_into_reindex FF)
(reindex_disp_cat_disp_functor _ _))
FF
:= λ x xx, id_disp _.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
lift_functor_into_reindex_commute_data
| null |
lift_functor_into_reindex_commute_axioms
{C₁ C₂ C₃ : category}
{D₁ : disp_cat C₁}
{D₃ : disp_cat C₃}
{F₁ : C₁ ⟶ C₂}
{F₂ : C₂ ⟶ C₃}
(FF : disp_functor (F₁ ∙ F₂) D₁ D₃)
: disp_nat_trans_axioms (lift_functor_into_reindex_commute_data FF).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
lift_functor_into_reindex_commute_axioms
| null |
lift_functor_into_reindex_commute
{C₁ C₂ C₃ : category}
{D₁ : disp_cat C₁}
{D₃ : disp_cat C₃}
{F₁ : C₁ ⟶ C₂}
{F₂ : C₂ ⟶ C₃}
(FF : disp_functor (F₁ ∙ F₂) D₁ D₃)
: disp_nat_trans
(nat_trans_id _)
(disp_functor_composite
(lift_functor_into_reindex FF)
(reindex_disp_cat_disp_functor _ _))
FF.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
lift_functor_into_reindex_commute
| null |
lift_functor_into_reindex_disp_nat_trans_data
{C₁ C₂ C₃ : category}
{D₁ : disp_cat C₁}
{D₃ : disp_cat C₃}
{F₁ F₁' : C₁ ⟶ C₂}
{α : F₁ ⟹ F₁'}
{F₂ : C₂ ⟶ C₃}
{FF₁ : disp_functor (F₁ ∙ F₂) D₁ D₃}
{FF₂ : disp_functor (F₁' ∙ F₂) D₁ D₃}
(αα : disp_nat_trans (post_whisker α _) FF₁ FF₂)
: disp_nat_trans_data
α
(lift_functor_into_reindex_data FF₁)
(lift_functor_into_reindex_data FF₂)
:= λ x xx, αα x xx.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
lift_functor_into_reindex_disp_nat_trans_data
| null |
lift_functor_into_reindex_disp_nat_trans_axioms
{C₁ C₂ C₃ : category}
{D₁ : disp_cat C₁}
{D₃ : disp_cat C₃}
{F₁ F₁' : C₁ ⟶ C₂}
{α : F₁ ⟹ F₁'}
{F₂ : C₂ ⟶ C₃}
{FF₁ : disp_functor (F₁ ∙ F₂) D₁ D₃}
{FF₂ : disp_functor (F₁' ∙ F₂) D₁ D₃}
(αα : disp_nat_trans (post_whisker α _) FF₁ FF₂)
: disp_nat_trans_axioms (lift_functor_into_reindex_disp_nat_trans_data αα).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
lift_functor_into_reindex_disp_nat_trans_axioms
| null |
lift_functor_into_reindex_disp_nat_trans
{C₁ C₂ C₃ : category}
{D₁ : disp_cat C₁}
{D₃ : disp_cat C₃}
{F₁ F₁' : C₁ ⟶ C₂}
{α : F₁ ⟹ F₁'}
{F₂ : C₂ ⟶ C₃}
{FF₁ : disp_functor (F₁ ∙ F₂) D₁ D₃}
{FF₂ : disp_functor (F₁' ∙ F₂) D₁ D₃}
(αα : disp_nat_trans (post_whisker α _) FF₁ FF₂)
: disp_nat_trans
α
(lift_functor_into_reindex FF₁)
(lift_functor_into_reindex FF₂).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
lift_functor_into_reindex_disp_nat_trans
| null |
reindex_pb_ump_1_data
: functor_data
C₀
(total_category (reindex_disp_cat F D₂)).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_pb_ump_1_data
| null |
reindex_pb_ump_1_is_functor
: is_functor reindex_pb_ump_1_data.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_pb_ump_1_is_functor
| null |
reindex_pb_ump_1
: C₀ ⟶ total_category (reindex_disp_cat F D₂).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_pb_ump_1
| null |
reindex_pb_ump_1_pr1
: reindex_pb_ump_1 ∙ pr1_category _ ⟹ H.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_pb_ump_1_pr1
| null |
reindex_pb_ump_1_pr1_nat_iso
: nat_z_iso
(reindex_pb_ump_1 ∙ pr1_category _)
H.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_pb_ump_1_pr1_nat_iso
| null |
reindex_pb_ump_1_pr2_nat_z_iso_data
: nat_trans_data
(reindex_pb_ump_1 ∙ total_functor (reindex_disp_cat_disp_functor F D₂))
G
:= λ x, α x ,, pr12 (HD₂ _ _ (nat_z_iso_pointwise_z_iso α x) (pr2 (G x))).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_pb_ump_1_pr2_nat_z_iso_data
| null |
reindex_pb_ump_1_pr2_is_nat_trans
: is_nat_trans
_ _
reindex_pb_ump_1_pr2_nat_z_iso_data.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_pb_ump_1_pr2_is_nat_trans
| null |
reindex_pb_ump_1_pr2
: reindex_pb_ump_1 ∙ total_functor (reindex_disp_cat_disp_functor F D₂) ⟹ G.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_pb_ump_1_pr2
| null |
reindex_pb_ump_1_pr2_nat_z_iso
: nat_z_iso
(reindex_pb_ump_1 ∙ total_functor (reindex_disp_cat_disp_functor F D₂))
G.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_pb_ump_1_pr2_nat_z_iso
| null |
reindex_pb_ump_2_data
: nat_trans_data Φ₁ Φ₂.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_pb_ump_2_data
| null |
reindex_pb_ump_2_is_nat_trans
: is_nat_trans Φ₁ Φ₂ reindex_pb_ump_2_data.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_pb_ump_2_is_nat_trans
| null |
reindex_pb_ump_2
: Φ₁ ⟹ Φ₂.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_pb_ump_2
| null |
reindex_pb_ump_2_pr1
: post_whisker
reindex_pb_ump_2
(pr1_category _)
=
τ₁.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_pb_ump_2_pr1
| null |
reindex_pb_ump_2_pr2
: post_whisker
reindex_pb_ump_2
(total_functor (reindex_disp_cat_disp_functor F D₂))
=
τ₂.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_pb_ump_2_pr2
| null |
reindex_pb_ump_eq
: n₁ = n₂.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_pb_ump_eq
| null |
reindex_of_disp_functor_data
: disp_functor_data
(functor_identity _)
(reindex_disp_cat F D₁)
(reindex_disp_cat F D₂).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_of_disp_functor_data
| null |
reindex_of_disp_functor_axioms
: disp_functor_axioms reindex_of_disp_functor_data.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_of_disp_functor_axioms
| null |
reindex_of_disp_functor
: disp_functor
(functor_identity _)
(reindex_disp_cat F D₁)
(reindex_disp_cat F D₂).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_of_disp_functor
| null |
reindex_of_disp_functor_is_cartesian_disp_functor
(HD₁ : cleaving D₁)
(HG : is_cartesian_disp_functor G)
: is_cartesian_disp_functor reindex_of_disp_functor.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_of_disp_functor_is_cartesian_disp_functor
| null |
reindex_of_disp_functor_is_opcartesian_disp_functor
(HD₁ : opcleaving D₁)
(HG : is_opcartesian_disp_functor G)
: is_opcartesian_disp_functor reindex_of_disp_functor.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_of_disp_functor_is_opcartesian_disp_functor
| null |
reindex_of_cartesian_disp_functor
{C₁ C₂ : category}
(F : C₁ ⟶ C₂)
{D₁ D₂ : disp_cat C₂}
(G : cartesian_disp_functor (functor_identity _) D₁ D₂)
(HD₁ : cleaving D₁)
: cartesian_disp_functor
(functor_identity _)
(reindex_disp_cat F D₁)
(reindex_disp_cat F D₂)
:= reindex_of_disp_functor F G
,,
reindex_of_disp_functor_is_cartesian_disp_functor F G HD₁ (pr2 G).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_of_cartesian_disp_functor
| null |
reindex_of_opcartesian_disp_functor
{C₁ C₂ : category}
(F : C₁ ⟶ C₂)
{D₁ D₂ : disp_cat C₂}
(G : opcartesian_disp_functor (functor_identity _) D₁ D₂)
(HD₁ : opcleaving D₁)
: opcartesian_disp_functor
(functor_identity _)
(reindex_disp_cat F D₁)
(reindex_disp_cat F D₂)
:= reindex_of_disp_functor F G
,,
reindex_of_disp_functor_is_opcartesian_disp_functor F G HD₁ (pr2 G).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_of_opcartesian_disp_functor
| null |
reindex_of_disp_nat_trans_data
: disp_nat_trans_data
(nat_trans_id _)
(reindex_of_disp_functor F G₁)
(reindex_of_disp_functor F G₂)
:= λ x xx,
transportb
(λ z, _ -->[ z ] _)
(functor_id F x)
(α (F x) xx).
|
Let
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_of_disp_nat_trans_data
| null |
reindex_of_disp_nat_trans_axioms
: disp_nat_trans_axioms reindex_of_disp_nat_trans_data.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_of_disp_nat_trans_axioms
| null |
reindex_of_disp_nat_trans
: disp_nat_trans
(nat_trans_id _)
(reindex_of_disp_functor F G₁)
(reindex_of_disp_functor F G₂).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_of_disp_nat_trans
| null |
reindex_of_disp_functor_identity_data
: disp_nat_trans_data
(nat_trans_id _)
(disp_functor_identity _)
(reindex_of_disp_functor_data F (disp_functor_identity D))
:= λ x xx,
transportb
(λ z, _ -->[ z ] _)
(functor_id F x)
(id_disp _).
|
Let
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_of_disp_functor_identity_data
| null |
reindex_of_disp_functor_identity_axioms
: disp_nat_trans_axioms reindex_of_disp_functor_identity_data.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_of_disp_functor_identity_axioms
| null |
reindex_of_disp_functor_identity
: disp_nat_trans
(nat_trans_id _)
(disp_functor_identity _)
(reindex_of_disp_functor F (disp_functor_identity D)).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_of_disp_functor_identity
| null |
reindex_of_disp_functor_composite_data
: disp_nat_trans_data
(nat_trans_id _)
(disp_functor_over_id_composite
(reindex_of_disp_functor F G₁)
(reindex_of_disp_functor F G₂))
(reindex_of_disp_functor
F
(disp_functor_over_id_composite G₁ G₂))
:= λ x xx,
transportb
(λ z, _ -->[ z ] _)
(functor_id F x)
(id_disp _).
|
Let
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_of_disp_functor_composite_data
| null |
reindex_of_disp_functor_composite_axioms
: disp_nat_trans_axioms reindex_of_disp_functor_composite_data.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_of_disp_functor_composite_axioms
| null |
reindex_of_disp_functor_composite
: disp_nat_trans
(nat_trans_id _)
(disp_functor_over_id_composite
(reindex_of_disp_functor F G₁)
(reindex_of_disp_functor F G₂))
(reindex_of_disp_functor
F
(disp_functor_over_id_composite G₁ G₂)).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/Reindexing.v
|
reindex_of_disp_functor_composite
| null |
total_set_groupoid
{G : setgroupoid}
(D : disp_cat_isofib G)
: setgroupoid.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/SetGroupoidComprehension.v
|
total_set_groupoid
|
* 1. The total category of an isofibration
|
total_functor_commute_eq
{C₁ C₂ : category}
{F : C₁ ⟶ C₂}
{D₁ : disp_cat C₁}
{D₂ : disp_cat C₂}
(FF : disp_functor F D₁ D₂)
: total_functor FF ∙ pr1_category D₂
=
pr1_category D₁ ∙ F.
|
Proposition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/SetGroupoidComprehension.v
|
total_functor_commute_eq
| null |
disp_cat_isofib_comprehension_data
: disp_functor_data
(functor_identity _)
disp_cat_isofib
(disp_codomain _).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/SetGroupoidComprehension.v
|
disp_cat_isofib_comprehension_data
|
* 2. The comprehension functor
|
disp_cat_isofib_comprehension_axioms
: disp_functor_axioms disp_cat_isofib_comprehension_data.
|
Proposition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/SetGroupoidComprehension.v
|
disp_cat_isofib_comprehension_axioms
| null |
disp_cat_isofib_comprehension
: disp_functor
(functor_identity _)
disp_cat_isofib
(disp_codomain _).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/SetGroupoidComprehension.v
|
disp_cat_isofib_comprehension
| null |
transportf_functor_disp_ob
{C₁ C₂ : category}
{D : disp_cat C₂}
{F₁ F₂ : C₁ ⟶ C₂}
(p : F₁ = F₂)
{x : C₁}
(xx : D (F₁ x))
: D (F₂ x)
:= transportf D (maponpaths (λ (F : _ ⟶ _), F x) p) xx.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/SetGroupoidComprehension.v
|
transportf_functor_disp_ob
|
* 3. Some transport lemmas that we need
|
transportf_functor_disp_ob_eq_b
{C₁ C₂ : category}
{D : disp_cat C₂}
{F₁ F₂ : C₁ ⟶ C₂}
(p : F₁ = F₂)
{x : C₁}
(xx : D (F₁ x))
: transportb D (path_functor_ob p _) (transportf_functor_disp_ob p xx)
=
xx.
|
Proposition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/SetGroupoidComprehension.v
|
transportf_functor_disp_ob_eq_b
| null |
transportf_functor_disp_ob_eq_f
{C₁ C₂ : category}
{D : disp_cat C₂}
{F₁ F₂ : C₁ ⟶ C₂}
(p : F₁ = F₂)
{x : C₁}
(xx : D (F₁ x))
: transportf D (path_functor_ob p _) xx
=
transportf_functor_disp_ob p xx.
|
Proposition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/SetGroupoidComprehension.v
|
transportf_functor_disp_ob_eq_f
| null |
transportf_functor_disp_mor
{C₁ C₂ : category}
{D : disp_cat C₂}
{F₁ F₂ : C₁ ⟶ C₂}
(p : F₁ = F₂)
{x y : C₁}
{f : x --> y}
{xx : D (F₁ x)}
{yy : D (F₁ y)}
(ff : xx -->[ #F₁ f ] yy)
: transportf_functor_disp_ob p xx
-->[ #F₂ f ]
transportf_functor_disp_ob p yy.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/SetGroupoidComprehension.v
|
transportf_functor_disp_mor
| null |
transportf_functor_disp_mor_eq
{C₁ C₂ : category}
{D : disp_cat C₂}
{F₁ F₂ : C₁ ⟶ C₂}
(p : F₁ = F₂)
{x y : C₁}
{f : x --> y}
{xx : D (F₁ x)}
{yy : D (F₁ y)}
(ff : xx -->[ #F₁ f ] yy)
: transportf_functor_disp_mor p ff
=
transportf
(λ z, _ -->[ z ] _)
(path_functor_mor_left p f)
(idtoiso_disp _ (transportf_functor_disp_ob_eq_b p xx)
;; ff
;; idtoiso_disp _ (transportf_functor_disp_ob_eq_f p yy))%mor_disp.
|
Proposition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/SetGroupoidComprehension.v
|
transportf_functor_disp_mor_eq
| null |
transportf_functor_disp_mor_id
{C₁ C₂ : category}
{D : disp_cat C₂}
{F₁ F₂ : C₁ ⟶ C₂}
(p : F₁ = F₂)
{x : C₁}
(xx : D (F₁ x))
: transportf_functor_disp_mor
p
(transportb (λ z, _ -->[ z ] _) (functor_id _ _) (id_disp xx))
=
transportb
(λ z, _ -->[ z ] _)
(functor_id F₂ x)
(id_disp _).
|
Proposition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/SetGroupoidComprehension.v
|
transportf_functor_disp_mor_id
| null |
transportf_functor_disp_mor_comp
{C₁ C₂ : category}
{D : disp_cat C₂}
{F₁ F₂ : C₁ ⟶ C₂}
(p : F₁ = F₂)
{x y z : C₁}
{f : x --> y}
{g : y --> z}
{xx : D (F₁ x)}
{yy : D (F₁ y)}
{zz : D (F₁ z)}
(ff : xx -->[ #F₁ f ] yy)
(gg : yy -->[ #F₁ g ] zz)
: (transportf_functor_disp_mor
p
(transportb (λ z, _ -->[ z ] _) (functor_comp F₁ _ _) (ff ;; gg))
=
transportb
(λ z, _ -->[ z ] _)
(functor_comp F₂ _ _)
(transportf_functor_disp_mor p ff ;; transportf_functor_disp_mor p gg))%mor_disp.
|
Proposition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/SetGroupoidComprehension.v
|
transportf_functor_disp_mor_comp
| null |
isofib_comprehension_pb_mor_data
: functor_data G₀ (total_category (reindex_disp_cat F D)).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/SetGroupoidComprehension.v
|
isofib_comprehension_pb_mor_data
| null |
isofib_comprehension_pb_mor_laws
: is_functor isofib_comprehension_pb_mor_data.
|
Proposition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/SetGroupoidComprehension.v
|
isofib_comprehension_pb_mor_laws
| null |
isofib_comprehension_pb_mor
: G₀ ⟶ total_category (reindex_disp_cat F D).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/SetGroupoidComprehension.v
|
isofib_comprehension_pb_mor
| null |
isofib_comprehension_pb_mor_pr1
: isofib_comprehension_pb_mor
∙ total_functor (reindex_disp_cat_disp_functor F D)
=
H₁.
|
Proposition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/SetGroupoidComprehension.v
|
isofib_comprehension_pb_mor_pr1
| null |
isofib_comprehension_pb_mor_pr2
: isofib_comprehension_pb_mor
∙ pr1_category (reindex_disp_cat F D)
=
H₂.
|
Proposition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/SetGroupoidComprehension.v
|
isofib_comprehension_pb_mor_pr2
| null |
isofib_comprehension_pb_mor_unique_ob
: H' ~ isofib_comprehension_pb_mor.
|
Lemma
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/SetGroupoidComprehension.v
|
isofib_comprehension_pb_mor_unique_ob
| null |
isofib_comprehension_pb_mor_unique
: H' = isofib_comprehension_pb_mor.
|
Proposition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/SetGroupoidComprehension.v
|
isofib_comprehension_pb_mor_unique
| null |
disp_cat_isofib_comprehension_cartesian
: is_cartesian_disp_functor disp_cat_isofib_comprehension.
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/SetGroupoidComprehension.v
|
disp_cat_isofib_comprehension_cartesian
| null |
disp_cat_split_isofib_comprehension_data
: disp_functor_data
(functor_identity _)
disp_cat_split_isofib
(disp_codomain _).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/SetGroupoidComprehension.v
|
disp_cat_split_isofib_comprehension_data
|
* 5. Comprehension for split isofibrations
|
disp_cat_split_isofib_comprehension_axioms
: disp_functor_axioms disp_cat_split_isofib_comprehension_data.
|
Proposition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/SetGroupoidComprehension.v
|
disp_cat_split_isofib_comprehension_axioms
| null |
disp_cat_split_isofib_comprehension
: disp_functor
(functor_identity _)
disp_cat_split_isofib
(disp_codomain _).
|
Definition
|
CategoryTheory
|
[
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] |
UniMath/CategoryTheory/DisplayedCats/Examples/SetGroupoidComprehension.v
|
disp_cat_split_isofib_comprehension
| null |
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