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image_eq {Δ Δ' Δ'' : simplex_category } {φ : Δ ⟶ Δ''}
{e : Δ ⟶ Δ'} [epi e] {i : Δ' ⟶ Δ''} [mono i] (fac : e ≫ i = φ) :
image φ = Δ' | begin
haveI := strong_epi_of_epi e,
let e := image.iso_strong_epi_mono e i fac,
ext,
exact le_antisymm (len_le_of_epi (infer_instance : epi e.hom))
(len_le_of_mono (infer_instance : mono e.hom)),
end | lemma | simplex_category.image_eq | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_ι_eq {Δ Δ'' : simplex_category } {φ : Δ ⟶ Δ''}
{e : Δ ⟶ image φ} [epi e] {i : image φ ⟶ Δ''} [mono i] (fac : e ≫ i = φ) :
image.ι φ = i | begin
haveI := strong_epi_of_epi e,
rw [← image.iso_strong_epi_mono_hom_comp_ι e i fac,
simplex_category.eq_id_of_is_iso (image.iso_strong_epi_mono e i fac).hom, category.id_comp],
end | lemma | simplex_category.image_ι_eq | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category",
"simplex_category.eq_id_of_is_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
factor_thru_image_eq {Δ Δ'' : simplex_category } {φ : Δ ⟶ Δ''}
{e : Δ ⟶ image φ} [epi e] {i : image φ ⟶ Δ''} [mono i] (fac : e ≫ i = φ) :
factor_thru_image φ = e | by rw [← cancel_mono i, fac, ← image_ι_eq fac, image.fac] | lemma | simplex_category.factor_thru_image_eq | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"simplex_category"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Cat : simplex_category ⥤ Cat.{0} | simplex_category.skeletal_functor ⋙ forget₂ NonemptyFinLinOrd LinOrd ⋙
forget₂ LinOrd Lat ⋙ forget₂ Lat PartOrd ⋙ forget₂ PartOrd Preord ⋙ Preord_to_Cat | def | simplex_category.to_Cat | algebraic_topology | src/algebraic_topology/simplex_category.lean | [
"tactic.linarith",
"category_theory.skeletal",
"data.fintype.sort",
"order.category.NonemptyFinLinOrd",
"category_theory.functor.reflects_isomorphisms"
] | [
"Lat",
"LinOrd",
"NonemptyFinLinOrd",
"PartOrd",
"Preord",
"Preord_to_Cat",
"simplex_category",
"simplex_category.skeletal_functor"
] | This functor `simplex_category ⥤ Cat` sends `[n]` (for `n : ℕ`)
to the category attached to the ordered set `{0, 1, ..., n}` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simplicial_object | simplex_categoryᵒᵖ ⥤ C | def | category_theory.simplicial_object | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | The category of simplicial objects valued in a category `C`.
This is the category of contravariant functors from `simplex_category` to `C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
δ {n} (i : fin (n+2)) : X _[n+1] ⟶ X _[n] | X.map (simplex_category.δ i).op | def | category_theory.simplicial_object.δ | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.δ"
] | Face maps for a simplicial object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
σ {n} (i : fin (n+1)) : X _[n] ⟶ X _[n+1] | X.map (simplex_category.σ i).op | def | category_theory.simplicial_object.σ | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.σ"
] | Degeneracy maps for a simplicial object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_to_iso {n m : ℕ} (h : n = m) : X _[n] ≅ X _[m] | X.map_iso (eq_to_iso (by rw h)) | def | category_theory.simplicial_object.eq_to_iso | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | Isomorphisms from identities in ℕ. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_to_iso_refl {n : ℕ} (h : n = n) : X.eq_to_iso h = iso.refl _ | by { ext, simp [eq_to_iso], } | lemma | category_theory.simplicial_object.eq_to_iso_refl | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
δ_comp_δ {n} {i j : fin (n+2)} (H : i ≤ j) :
X.δ j.succ ≫ X.δ i = X.δ i.cast_succ ≫ X.δ j | by { dsimp [δ], simp only [←X.map_comp, ←op_comp, simplex_category.δ_comp_δ H] } | lemma | category_theory.simplicial_object.δ_comp_δ | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.δ_comp_δ"
] | The generic case of the first simplicial identity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
δ_comp_δ' {n} {i : fin (n+2)} {j : fin (n+3)} (H : i.cast_succ < j) :
X.δ j ≫ X.δ i = X.δ i.cast_succ ≫
X.δ (j.pred (λ hj, by simpa only [hj, fin.not_lt_zero] using H)) | by { dsimp [δ], simp only [←X.map_comp, ←op_comp, simplex_category.δ_comp_δ' H] } | lemma | category_theory.simplicial_object.δ_comp_δ' | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"fin.not_lt_zero",
"simplex_category.δ_comp_δ'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
δ_comp_δ'' {n} {i : fin (n+3)} {j : fin (n+2)} (H : i ≤ j.cast_succ) :
X.δ j.succ ≫ X.δ (i.cast_lt (nat.lt_of_le_of_lt (fin.le_iff_coe_le_coe.mp H) j.is_lt)) =
X.δ i ≫ X.δ j | by { dsimp [δ], simp only [←X.map_comp, ←op_comp, simplex_category.δ_comp_δ'' H] } | lemma | category_theory.simplicial_object.δ_comp_δ'' | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.δ_comp_δ''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
δ_comp_δ_self {n} {i : fin (n+2)} : X.δ i.cast_succ ≫ X.δ i = X.δ i.succ ≫ X.δ i | by { dsimp [δ], simp only [←X.map_comp, ←op_comp, simplex_category.δ_comp_δ_self] } | lemma | category_theory.simplicial_object.δ_comp_δ_self | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.δ_comp_δ_self"
] | The special case of the first simplicial identity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
δ_comp_δ_self' {n} {j : fin (n+3)} {i : fin (n+2)} (H : j = i.cast_succ) :
X.δ j ≫ X.δ i = X.δ i.succ ≫ X.δ i | by { subst H, rw δ_comp_δ_self, } | lemma | category_theory.simplicial_object.δ_comp_δ_self' | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
δ_comp_σ_of_le {n} {i : fin (n+2)} {j : fin (n+1)} (H : i ≤ j.cast_succ) :
X.σ j.succ ≫ X.δ i.cast_succ = X.δ i ≫ X.σ j | by { dsimp [δ, σ], simp only [←X.map_comp, ←op_comp, simplex_category.δ_comp_σ_of_le H] } | lemma | category_theory.simplicial_object.δ_comp_σ_of_le | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.δ_comp_σ_of_le"
] | The second simplicial identity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
δ_comp_σ_self {n} {i : fin (n+1)} :
X.σ i ≫ X.δ i.cast_succ = 𝟙 _ | begin
dsimp [δ, σ],
simp only [←X.map_comp, ←op_comp, simplex_category.δ_comp_σ_self, op_id, X.map_id],
end | lemma | category_theory.simplicial_object.δ_comp_σ_self | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.δ_comp_σ_self"
] | The first part of the third simplicial identity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
δ_comp_σ_self' {n} {j : fin (n+2)} {i : fin (n+1)} (H : j = i.cast_succ):
X.σ i ≫ X.δ j = 𝟙 _ | by { subst H, rw δ_comp_σ_self, } | lemma | category_theory.simplicial_object.δ_comp_σ_self' | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
δ_comp_σ_succ {n} {i : fin (n+1)} :
X.σ i ≫ X.δ i.succ = 𝟙 _ | begin
dsimp [δ, σ],
simp only [←X.map_comp, ←op_comp, simplex_category.δ_comp_σ_succ, op_id, X.map_id],
end | lemma | category_theory.simplicial_object.δ_comp_σ_succ | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.δ_comp_σ_succ"
] | The second part of the third simplicial identity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
δ_comp_σ_succ' {n} {j : fin (n+2)} {i : fin (n+1)} (H : j = i.succ) :
X.σ i ≫ X.δ j = 𝟙 _ | by { subst H, rw δ_comp_σ_succ, } | lemma | category_theory.simplicial_object.δ_comp_σ_succ' | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
δ_comp_σ_of_gt {n} {i : fin (n+2)} {j : fin (n+1)} (H : j.cast_succ < i) :
X.σ j.cast_succ ≫ X.δ i.succ = X.δ i ≫ X.σ j | by { dsimp [δ, σ], simp only [←X.map_comp, ←op_comp, simplex_category.δ_comp_σ_of_gt H] } | lemma | category_theory.simplicial_object.δ_comp_σ_of_gt | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.δ_comp_σ_of_gt"
] | The fourth simplicial identity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
δ_comp_σ_of_gt' {n} {i : fin (n+3)} {j : fin (n+2)} (H : j.succ < i) :
X.σ j ≫ X.δ i = X.δ (i.pred (λ hi, by simpa only [fin.not_lt_zero, hi] using H)) ≫
X.σ (j.cast_lt ((add_lt_add_iff_right 1).mp (lt_of_lt_of_le
(by simpa only [fin.val_eq_coe, ← fin.coe_succ]
using fin.lt_iff_coe_lt_coe.mp H) i.is... | by { dsimp [δ, σ], simpa only [←X.map_comp, ←op_comp, simplex_category.δ_comp_σ_of_gt' H], } | lemma | category_theory.simplicial_object.δ_comp_σ_of_gt' | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"fin.coe_succ",
"fin.not_lt_zero",
"fin.val_eq_coe",
"simplex_category.δ_comp_σ_of_gt'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
σ_comp_σ {n} {i j : fin (n+1)} (H : i ≤ j) :
X.σ j ≫ X.σ i.cast_succ = X.σ i ≫ X.σ j.succ | by { dsimp [δ, σ], simp only [←X.map_comp, ←op_comp, simplex_category.σ_comp_σ H] } | lemma | category_theory.simplicial_object.σ_comp_σ | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.σ_comp_σ"
] | The fifth simplicial identity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
δ_naturality {X' X : simplicial_object C} (f : X ⟶ X') {n : ℕ} (i : fin (n+2)) :
X.δ i ≫ f.app (op [n]) = f.app (op [n+1]) ≫ X'.δ i | f.naturality _ | lemma | category_theory.simplicial_object.δ_naturality | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
σ_naturality {X' X : simplicial_object C} (f : X ⟶ X') {n : ℕ} (i : fin (n+1)) :
X.σ i ≫ f.app (op [n+1]) = f.app (op [n]) ≫ X'.σ i | f.naturality _ | lemma | category_theory.simplicial_object.σ_naturality | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whiskering (D : Type*) [category D] :
(C ⥤ D) ⥤ simplicial_object C ⥤ simplicial_object D | whiskering_right _ _ _ | def | category_theory.simplicial_object.whiskering | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | Functor composition induces a functor on simplicial objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
truncated (n : ℕ) | (simplex_category.truncated n)ᵒᵖ ⥤ C | def | category_theory.simplicial_object.truncated | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.truncated"
] | Truncated simplicial objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whiskering {n} (D : Type*) [category D] :
(C ⥤ D) ⥤ truncated C n ⥤ truncated D n | whiskering_right _ _ _ | def | category_theory.simplicial_object.truncated.whiskering | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | Functor composition induces a functor on truncated simplicial objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sk (n : ℕ) : simplicial_object C ⥤ simplicial_object.truncated C n | (whiskering_left _ _ _).obj simplex_category.truncated.inclusion.op | def | category_theory.simplicial_object.sk | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | The skeleton functor from simplicial objects to truncated simplicial objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const : C ⥤ simplicial_object C | category_theory.functor.const _ | abbreviation | category_theory.simplicial_object.const | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"category_theory.functor.const"
] | The constant simplicial object is the constant functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
augmented | comma (𝟭 (simplicial_object C)) (const C) | def | category_theory.simplicial_object.augmented | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | The category of augmented simplicial objects, defined as a comma category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
drop : augmented C ⥤ simplicial_object C | comma.fst _ _ | def | category_theory.simplicial_object.augmented.drop | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | Drop the augmentation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
point : augmented C ⥤ C | comma.snd _ _ | def | category_theory.simplicial_object.augmented.point | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | The point of the augmentation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_arrow : augmented C ⥤ arrow C | { obj := λ X,
{ left := (drop.obj X) _[0],
right := (point.obj X),
hom := X.hom.app _ },
map := λ X Y η,
{ left := (drop.map η).app _,
right := (point.map η),
w' := begin
dsimp,
rw ← nat_trans.comp_app,
erw η.w,
refl,
end } } | def | category_theory.simplicial_object.augmented.to_arrow | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | The functor from augmented objects to arrows. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
w₀ {X Y : augmented C} (f : X ⟶ Y) :
(augmented.drop.map f).app (op (simplex_category.mk 0)) ≫
Y.hom.app (op (simplex_category.mk 0)) =
X.hom.app (op (simplex_category.mk 0)) ≫ augmented.point.map f | by convert congr_app f.w (op (simplex_category.mk 0)) | lemma | category_theory.simplicial_object.augmented.w₀ | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.mk"
] | The compatibility of a morphism with the augmentation, on 0-simplices | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whiskering_obj (D : Type*) [category D] (F : C ⥤ D) :
augmented C ⥤ augmented D | { obj := λ X,
{ left := ((whiskering _ _).obj F).obj (drop.obj X),
right := F.obj (point.obj X),
hom := whisker_right X.hom F ≫ (functor.const_comp _ _ _).hom },
map := λ X Y η,
{ left := whisker_right η.left _,
right := F.map η.right,
w' := begin
ext,
dsimp,
rw [category.comp_id... | def | category_theory.simplicial_object.augmented.whiskering_obj | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | Functor composition induces a functor on augmented simplicial objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whiskering (D : Type u') [category.{v'} D] :
(C ⥤ D) ⥤ augmented C ⥤ augmented D | { obj := whiskering_obj _ _,
map := λ X Y η,
{ app := λ A,
{ left := whisker_left _ η,
right := η.app _,
w' := begin
ext n,
dsimp,
rw [category.comp_id, category.comp_id, η.naturality],
end }, }, } | def | category_theory.simplicial_object.augmented.whiskering | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | Functor composition induces a functor on augmented simplicial objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
augment (X : simplicial_object C) (X₀ : C) (f : X _[0] ⟶ X₀)
(w : ∀ (i : simplex_category) (g₁ g₂ : [0] ⟶ i),
X.map g₁.op ≫ f = X.map g₂.op ≫ f) : simplicial_object.augmented C | { left := X,
right := X₀,
hom :=
{ app := λ i, X.map (simplex_category.const i.unop 0).op ≫ f,
naturality' := begin
intros i j g,
dsimp,
rw ← g.op_unop,
simpa only [← X.map_comp, ← category.assoc, category.comp_id, ← op_comp] using w _ _ _,
end } } | def | category_theory.simplicial_object.augment | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category",
"simplex_category.const"
] | Augment a simplicial object with an object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
augment_hom_zero (X : simplicial_object C) (X₀ : C) (f : X _[0] ⟶ X₀) (w) :
(X.augment X₀ f w).hom.app (op [0]) = f | by { dsimp, rw [simplex_category.hom_zero_zero ([0].const 0), op_id, X.map_id, category.id_comp] } | lemma | category_theory.simplicial_object.augment_hom_zero | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.hom_zero_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cosimplicial_object | simplex_category ⥤ C | def | category_theory.cosimplicial_object | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category"
] | Cosimplicial objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
δ {n} (i : fin (n+2)) : X _[n] ⟶ X _[n+1] | X.map (simplex_category.δ i) | def | category_theory.cosimplicial_object.δ | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.δ"
] | Coface maps for a cosimplicial object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
σ {n} (i : fin (n+1)) : X _[n+1] ⟶ X _[n] | X.map (simplex_category.σ i) | def | category_theory.cosimplicial_object.σ | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.σ"
] | Codegeneracy maps for a cosimplicial object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
δ_comp_δ {n} {i j : fin (n+2)} (H : i ≤ j) :
X.δ i ≫ X.δ j.succ = X.δ j ≫ X.δ i.cast_succ | by { dsimp [δ], simp only [←X.map_comp, simplex_category.δ_comp_δ H], } | lemma | category_theory.cosimplicial_object.δ_comp_δ | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.δ_comp_δ"
] | The generic case of the first cosimplicial identity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
δ_comp_δ' {n} {i : fin (n+2)} {j : fin (n+3)} (H : i.cast_succ < j) :
X.δ i ≫ X.δ j = X.δ (j.pred (λ hj, by simpa only [hj, fin.not_lt_zero] using H)) ≫
X.δ i.cast_succ | by { dsimp [δ], simp only [←X.map_comp, ←op_comp, simplex_category.δ_comp_δ' H] } | lemma | category_theory.cosimplicial_object.δ_comp_δ' | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"fin.not_lt_zero",
"simplex_category.δ_comp_δ'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
δ_comp_δ'' {n} {i : fin (n+3)} {j : fin (n+2)} (H : i ≤ j.cast_succ) :
X.δ (i.cast_lt (nat.lt_of_le_of_lt (fin.le_iff_coe_le_coe.mp H) j.is_lt)) ≫ X.δ j.succ =
X.δ j ≫ X.δ i | by { dsimp [δ], simp only [←X.map_comp, ←op_comp, simplex_category.δ_comp_δ'' H] } | lemma | category_theory.cosimplicial_object.δ_comp_δ'' | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.δ_comp_δ''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
δ_comp_δ_self {n} {i : fin (n+2)} : X.δ i ≫ X.δ i.cast_succ = X.δ i ≫ X.δ i.succ | by { dsimp [δ], simp only [←X.map_comp, simplex_category.δ_comp_δ_self] } | lemma | category_theory.cosimplicial_object.δ_comp_δ_self | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.δ_comp_δ_self"
] | The special case of the first cosimplicial identity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
δ_comp_δ_self' {n} {i : fin (n+2)} {j : fin (n+3)} (H : j = i.cast_succ) :
X.δ i ≫ X.δ j = X.δ i ≫ X.δ i.succ | by { subst H, rw δ_comp_δ_self, } | lemma | category_theory.cosimplicial_object.δ_comp_δ_self' | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
δ_comp_σ_of_le {n} {i : fin (n+2)} {j : fin (n+1)} (H : i ≤ j.cast_succ) :
X.δ i.cast_succ ≫ X.σ j.succ = X.σ j ≫ X.δ i | by { dsimp [δ, σ], simp only [←X.map_comp, simplex_category.δ_comp_σ_of_le H] } | lemma | category_theory.cosimplicial_object.δ_comp_σ_of_le | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.δ_comp_σ_of_le"
] | The second cosimplicial identity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
δ_comp_σ_self {n} {i : fin (n+1)} :
X.δ i.cast_succ ≫ X.σ i = 𝟙 _ | begin
dsimp [δ, σ],
simp only [←X.map_comp, simplex_category.δ_comp_σ_self, X.map_id],
end | lemma | category_theory.cosimplicial_object.δ_comp_σ_self | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.δ_comp_σ_self"
] | The first part of the third cosimplicial identity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
δ_comp_σ_self' {n} {j : fin (n+2)} {i : fin (n+1)} (H : j = i.cast_succ) :
X.δ j ≫ X.σ i = 𝟙 _ | by { subst H, rw δ_comp_σ_self, } | lemma | category_theory.cosimplicial_object.δ_comp_σ_self' | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
δ_comp_σ_succ {n} {i : fin (n+1)} :
X.δ i.succ ≫ X.σ i = 𝟙 _ | begin
dsimp [δ, σ],
simp only [←X.map_comp, simplex_category.δ_comp_σ_succ, X.map_id],
end | lemma | category_theory.cosimplicial_object.δ_comp_σ_succ | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.δ_comp_σ_succ"
] | The second part of the third cosimplicial identity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
δ_comp_σ_succ' {n} {j : fin (n+2)} {i : fin (n+1)} (H : j = i.succ) :
X.δ j ≫ X.σ i = 𝟙 _ | by { subst H, rw δ_comp_σ_succ, } | lemma | category_theory.cosimplicial_object.δ_comp_σ_succ' | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
δ_comp_σ_of_gt {n} {i : fin (n+2)} {j : fin (n+1)} (H : j.cast_succ < i) :
X.δ i.succ ≫ X.σ j.cast_succ = X.σ j ≫ X.δ i | by { dsimp [δ, σ], simp only [←X.map_comp, simplex_category.δ_comp_σ_of_gt H] } | lemma | category_theory.cosimplicial_object.δ_comp_σ_of_gt | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.δ_comp_σ_of_gt"
] | The fourth cosimplicial identity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
δ_comp_σ_of_gt' {n} {i : fin (n+3)} {j : fin (n+2)} (H : j.succ < i) :
X.δ i ≫ X.σ j = X.σ (j.cast_lt ((add_lt_add_iff_right 1).mp (lt_of_lt_of_le
(by simpa only [fin.val_eq_coe, ← fin.coe_succ]
using fin.lt_iff_coe_lt_coe.mp H) i.is_le))) ≫
X.δ (i.pred (λ hi, by simpa only [fin.not_lt_zero, hi] usi... | by { dsimp [δ, σ], simpa only [←X.map_comp, ←op_comp, simplex_category.δ_comp_σ_of_gt' H], } | lemma | category_theory.cosimplicial_object.δ_comp_σ_of_gt' | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"fin.coe_succ",
"fin.not_lt_zero",
"fin.val_eq_coe",
"simplex_category.δ_comp_σ_of_gt'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
σ_comp_σ {n} {i j : fin (n+1)} (H : i ≤ j) :
X.σ i.cast_succ ≫ X.σ j = X.σ j.succ ≫ X.σ i | by { dsimp [δ, σ], simp only [←X.map_comp, simplex_category.σ_comp_σ H] } | lemma | category_theory.cosimplicial_object.σ_comp_σ | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.σ_comp_σ"
] | The fifth cosimplicial identity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
δ_naturality {X' X : cosimplicial_object C} (f : X ⟶ X') {n : ℕ} (i : fin (n+2)) :
X.δ i ≫ f.app (simplex_category.mk (n+1)) =
f.app (simplex_category.mk n) ≫ X'.δ i | f.naturality _ | lemma | category_theory.cosimplicial_object.δ_naturality | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
σ_naturality {X' X : cosimplicial_object C} (f : X ⟶ X') {n : ℕ} (i : fin (n+1)) :
X.σ i ≫ f.app (simplex_category.mk n) =
f.app (simplex_category.mk (n+1)) ≫ X'.σ i | f.naturality _ | lemma | category_theory.cosimplicial_object.σ_naturality | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whiskering (D : Type*) [category D] :
(C ⥤ D) ⥤ cosimplicial_object C ⥤ cosimplicial_object D | whiskering_right _ _ _ | def | category_theory.cosimplicial_object.whiskering | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | Functor composition induces a functor on cosimplicial objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
truncated (n : ℕ) | simplex_category.truncated n ⥤ C | def | category_theory.cosimplicial_object.truncated | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.truncated"
] | Truncated cosimplicial objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sk (n : ℕ) : cosimplicial_object C ⥤ cosimplicial_object.truncated C n | (whiskering_left _ _ _).obj simplex_category.truncated.inclusion | def | category_theory.cosimplicial_object.sk | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.truncated.inclusion"
] | The skeleton functor from cosimplicial objects to truncated cosimplicial objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const : C ⥤ cosimplicial_object C | category_theory.functor.const _ | abbreviation | category_theory.cosimplicial_object.const | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"category_theory.functor.const"
] | The constant cosimplicial object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
augmented | comma (const C) (𝟭 (cosimplicial_object C)) | def | category_theory.cosimplicial_object.augmented | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | Augmented cosimplicial objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
drop : augmented C ⥤ cosimplicial_object C | comma.snd _ _ | def | category_theory.cosimplicial_object.augmented.drop | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | Drop the augmentation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
point : augmented C ⥤ C | comma.fst _ _ | def | category_theory.cosimplicial_object.augmented.point | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | The point of the augmentation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_arrow : augmented C ⥤ arrow C | { obj := λ X,
{ left := (point.obj X),
right := (drop.obj X) _[0],
hom := X.hom.app _ },
map := λ X Y η,
{ left := (point.map η),
right := (drop.map η).app _,
w' := begin
dsimp,
rw ← nat_trans.comp_app,
erw ← η.w,
refl,
end } } | def | category_theory.cosimplicial_object.augmented.to_arrow | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | The functor from augmented objects to arrows. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whiskering_obj (D : Type*) [category D] (F : C ⥤ D) :
augmented C ⥤ augmented D | { obj := λ X,
{ left := F.obj (point.obj X),
right := ((whiskering _ _).obj F).obj (drop.obj X),
hom := (functor.const_comp _ _ _).inv ≫ whisker_right X.hom F },
map := λ X Y η,
{ left := F.map η.left,
right := whisker_right η.right _,
w' := begin
ext,
dsimp,
rw [category.id_comp... | def | category_theory.cosimplicial_object.augmented.whiskering_obj | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | Functor composition induces a functor on augmented cosimplicial objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whiskering (D : Type u') [category.{v'} D] :
(C ⥤ D) ⥤ augmented C ⥤ augmented D | { obj := whiskering_obj _ _,
map := λ X Y η,
{ app := λ A,
{ left := η.app _,
right := whisker_left _ η,
w' := begin
ext n,
dsimp,
rw [category.id_comp, category.id_comp, η.naturality],
end }, }, } | def | category_theory.cosimplicial_object.augmented.whiskering | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | Functor composition induces a functor on augmented cosimplicial objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
augment (X : cosimplicial_object C) (X₀ : C) (f : X₀ ⟶ X.obj [0])
(w : ∀ (i : simplex_category) (g₁ g₂ : [0] ⟶ i),
f ≫ X.map g₁ = f ≫ X.map g₂) : cosimplicial_object.augmented C | { left := X₀,
right := X,
hom :=
{ app := λ i, f ≫ X.map (simplex_category.const i 0),
naturality' := begin
intros i j g,
dsimp,
simpa [← X.map_comp] using w _ _ _,
end } } | def | category_theory.cosimplicial_object.augment | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category",
"simplex_category.const"
] | Augment a cosimplicial object with an object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
augment_hom_zero (X : cosimplicial_object C) (X₀ : C) (f : X₀ ⟶ X.obj [0]) (w) :
(X.augment X₀ f w).hom.app [0] = f | by { dsimp, rw [simplex_category.hom_zero_zero ([0].const 0), X.map_id, category.comp_id] } | lemma | category_theory.cosimplicial_object.augment_hom_zero | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"simplex_category.hom_zero_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
simplicial_cosimplicial_equiv : (simplicial_object C)ᵒᵖ ≌ (cosimplicial_object Cᵒᵖ) | functor.left_op_right_op_equiv _ _ | def | category_theory.simplicial_cosimplicial_equiv | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | The anti-equivalence between simplicial objects and cosimplicial objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cosimplicial_simplicial_equiv : (cosimplicial_object C)ᵒᵖ ≌ (simplicial_object Cᵒᵖ) | functor.op_unop_equiv _ _ | def | category_theory.cosimplicial_simplicial_equiv | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | The anti-equivalence between cosimplicial objects and simplicial objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simplicial_object.augmented.right_op (X : simplicial_object.augmented C) :
cosimplicial_object.augmented Cᵒᵖ | { left := opposite.op X.right,
right := X.left.right_op,
hom := X.hom.right_op } | def | category_theory.simplicial_object.augmented.right_op | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"opposite.op"
] | Construct an augmented cosimplicial object in the opposite
category from an augmented simplicial object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cosimplicial_object.augmented.left_op (X : cosimplicial_object.augmented Cᵒᵖ) :
simplicial_object.augmented C | { left := X.right.left_op,
right := X.left.unop,
hom := X.hom.left_op } | def | category_theory.cosimplicial_object.augmented.left_op | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | Construct an augmented simplicial object from an augmented cosimplicial
object in the opposite category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simplicial_object.augmented.right_op_left_op_iso (X : simplicial_object.augmented C) :
X.right_op.left_op ≅ X | comma.iso_mk X.left.right_op_left_op_iso (eq_to_iso $ by simp) (by tidy) | def | category_theory.simplicial_object.augmented.right_op_left_op_iso | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | Converting an augmented simplicial object to an augmented cosimplicial
object and back is isomorphic to the given object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cosimplicial_object.augmented.left_op_right_op_iso (X : cosimplicial_object.augmented Cᵒᵖ) :
X.left_op.right_op ≅ X | comma.iso_mk (eq_to_iso $ by simp) X.right.left_op_right_op_iso (by tidy) | def | category_theory.cosimplicial_object.augmented.left_op_right_op_iso | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | Converting an augmented cosimplicial object to an augmented simplicial
object and back is isomorphic to the given object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simplicial_to_cosimplicial_augmented :
(simplicial_object.augmented C)ᵒᵖ ⥤ cosimplicial_object.augmented Cᵒᵖ | { obj := λ X, X.unop.right_op,
map := λ X Y f,
{ left := f.unop.right.op,
right := f.unop.left.right_op,
w' := begin
ext x,
dsimp,
simp_rw ← op_comp,
congr' 1,
exact (congr_app f.unop.w (op x)).symm,
end } } | def | category_theory.simplicial_to_cosimplicial_augmented | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | A functorial version of `simplicial_object.augmented.right_op`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cosimplicial_to_simplicial_augmented :
cosimplicial_object.augmented Cᵒᵖ ⥤ (simplicial_object.augmented C)ᵒᵖ | { obj := λ X, opposite.op X.left_op,
map := λ X Y f, quiver.hom.op $
{ left := f.right.left_op,
right := f.left.unop,
w' := begin
ext x,
dsimp,
simp_rw ← unop_comp,
congr' 1,
exact (congr_app f.w x.unop).symm,
end} } | def | category_theory.cosimplicial_to_simplicial_augmented | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [
"opposite.op",
"quiver.hom.op"
] | A functorial version of `cosimplicial_object.augmented.left_op`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simplicial_cosimplicial_augmented_equiv :
(simplicial_object.augmented C)ᵒᵖ ≌ cosimplicial_object.augmented Cᵒᵖ | equivalence.mk
(simplicial_to_cosimplicial_augmented _)
(cosimplicial_to_simplicial_augmented _)
(nat_iso.of_components (λ X, X.unop.right_op_left_op_iso.op) $ λ X Y f,
by { dsimp, rw ←f.op_unop, simp_rw ← op_comp, congr' 1, tidy })
(nat_iso.of_components (λ X, X.left_op_right_op_iso) $ by tidy) | def | category_theory.simplicial_cosimplicial_augmented_equiv | algebraic_topology | src/algebraic_topology/simplicial_object.lean | [
"algebraic_topology.simplex_category",
"category_theory.arrow",
"category_theory.limits.functor_category",
"category_theory.opposites"
] | [] | The contravariant categorical equivalence between augmented simplicial
objects and augmented cosimplicial objects in the opposite category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sSet : Type (u+1) | simplicial_object (Type u) | def | sSet | algebraic_topology | src/algebraic_topology/simplicial_set.lean | [
"algebraic_topology.simplicial_object",
"algebraic_topology.topological_simplex",
"category_theory.limits.presheaf",
"category_theory.limits.types",
"category_theory.yoneda",
"topology.category.Top.limits.basic"
] | [] | The category of simplicial sets.
This is the category of contravariant functors from
`simplex_category` to `Type u`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
standard_simplex : simplex_category ⥤ sSet | yoneda | def | sSet.standard_simplex | algebraic_topology | src/algebraic_topology/simplicial_set.lean | [
"algebraic_topology.simplicial_object",
"algebraic_topology.topological_simplex",
"category_theory.limits.presheaf",
"category_theory.limits.types",
"category_theory.yoneda",
"topology.category.Top.limits.basic"
] | [
"sSet",
"simplex_category"
] | The `n`-th standard simplex `Δ[n]` associated with a nonempty finite linear order `n`
is the Yoneda embedding of `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
as_order_hom {n} {m} (α : Δ[n].obj m) :
order_hom (fin (m.unop.len+1)) (fin (n+1)) | α.to_order_hom | def | sSet.as_order_hom | algebraic_topology | src/algebraic_topology/simplicial_set.lean | [
"algebraic_topology.simplicial_object",
"algebraic_topology.topological_simplex",
"category_theory.limits.presheaf",
"category_theory.limits.types",
"category_theory.yoneda",
"topology.category.Top.limits.basic"
] | [
"order_hom"
] | The `m`-simplices of the `n`-th standard simplex are
the monotone maps from `fin (m+1)` to `fin (n+1)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
boundary (n : ℕ) : sSet | { obj := λ m, {α : Δ[n].obj m // ¬ function.surjective (as_order_hom α)},
map := λ m₁ m₂ f α, ⟨f.unop ≫ (α : Δ[n].obj m₁),
by { intro h, apply α.property, exact function.surjective.of_comp h }⟩ } | def | sSet.boundary | algebraic_topology | src/algebraic_topology/simplicial_set.lean | [
"algebraic_topology.simplicial_object",
"algebraic_topology.topological_simplex",
"category_theory.limits.presheaf",
"category_theory.limits.types",
"category_theory.yoneda",
"topology.category.Top.limits.basic"
] | [
"function.surjective.of_comp",
"sSet"
] | The boundary `∂Δ[n]` of the `n`-th standard simplex consists of
all `m`-simplices of `standard_simplex n` that are not surjective
(when viewed as monotone function `m → n`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
boundary_inclusion (n : ℕ) :
∂Δ[n] ⟶ Δ[n] | { app := λ m (α : {α : Δ[n].obj m // _}), α } | def | sSet.boundary_inclusion | algebraic_topology | src/algebraic_topology/simplicial_set.lean | [
"algebraic_topology.simplicial_object",
"algebraic_topology.topological_simplex",
"category_theory.limits.presheaf",
"category_theory.limits.types",
"category_theory.yoneda",
"topology.category.Top.limits.basic"
] | [] | The inclusion of the boundary of the `n`-th standard simplex into that standard simplex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
horn (n : ℕ) (i : fin (n+1)) : sSet | { obj := λ m,
{ α : Δ[n].obj m // set.range (as_order_hom α) ∪ {i} ≠ set.univ },
map := λ m₁ m₂ f α, ⟨f.unop ≫ (α : Δ[n].obj m₁),
begin
intro h, apply α.property,
rw set.eq_univ_iff_forall at h ⊢, intro j,
apply or.imp _ id (h j),
intro hj,
exact set.range_comp_subset_range _ _ hj,
end⟩ } | def | sSet.horn | algebraic_topology | src/algebraic_topology/simplicial_set.lean | [
"algebraic_topology.simplicial_object",
"algebraic_topology.topological_simplex",
"category_theory.limits.presheaf",
"category_theory.limits.types",
"category_theory.yoneda",
"topology.category.Top.limits.basic"
] | [
"sSet",
"set.eq_univ_iff_forall",
"set.range",
"set.range_comp_subset_range"
] | `horn n i` (or `Λ[n, i]`) is the `i`-th horn of the `n`-th standard simplex, where `i : n`.
It consists of all `m`-simplices `α` of `Δ[n]`
for which the union of `{i}` and the range of `α` is not all of `n`
(when viewing `α` as monotone function `m → n`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
horn_inclusion (n : ℕ) (i : fin (n+1)) :
Λ[n, i] ⟶ Δ[n] | { app := λ m (α : {α : Δ[n].obj m // _}), α } | def | sSet.horn_inclusion | algebraic_topology | src/algebraic_topology/simplicial_set.lean | [
"algebraic_topology.simplicial_object",
"algebraic_topology.topological_simplex",
"category_theory.limits.presheaf",
"category_theory.limits.types",
"category_theory.yoneda",
"topology.category.Top.limits.basic"
] | [] | The inclusion of the `i`-th horn of the `n`-th standard simplex into that standard simplex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
S1 : sSet | limits.colimit $ limits.parallel_pair
((standard_simplex.map $ simplex_category.δ 0) : Δ[0] ⟶ Δ[1])
(standard_simplex.map $ simplex_category.δ 1) | def | sSet.S1 | algebraic_topology | src/algebraic_topology/simplicial_set.lean | [
"algebraic_topology.simplicial_object",
"algebraic_topology.topological_simplex",
"category_theory.limits.presheaf",
"category_theory.limits.types",
"category_theory.yoneda",
"topology.category.Top.limits.basic"
] | [
"sSet",
"simplex_category.δ"
] | The simplicial circle. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
truncated (n : ℕ) | simplicial_object.truncated (Type u) n | def | sSet.truncated | algebraic_topology | src/algebraic_topology/simplicial_set.lean | [
"algebraic_topology.simplicial_object",
"algebraic_topology.topological_simplex",
"category_theory.limits.presheaf",
"category_theory.limits.types",
"category_theory.yoneda",
"topology.category.Top.limits.basic"
] | [] | Truncated simplicial sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sk (n : ℕ) : sSet ⥤ sSet.truncated n | simplicial_object.sk n | def | sSet.sk | algebraic_topology | src/algebraic_topology/simplicial_set.lean | [
"algebraic_topology.simplicial_object",
"algebraic_topology.topological_simplex",
"category_theory.limits.presheaf",
"category_theory.limits.types",
"category_theory.yoneda",
"topology.category.Top.limits.basic"
] | [
"sSet",
"sSet.truncated"
] | The skeleton functor on simplicial sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
augmented | simplicial_object.augmented (Type u) | abbreviation | sSet.augmented | algebraic_topology | src/algebraic_topology/simplicial_set.lean | [
"algebraic_topology.simplicial_object",
"algebraic_topology.topological_simplex",
"category_theory.limits.presheaf",
"category_theory.limits.types",
"category_theory.yoneda",
"topology.category.Top.limits.basic"
] | [] | The category of augmented simplicial sets, as a particular case of
augmented simplicial objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
standard_simplex : simplex_category ⥤ sSet.augmented | { obj := λ Δ,
{ left := sSet.standard_simplex.obj Δ,
right := terminal _,
hom := { app := λ Δ', terminal.from _, }, },
map := λ Δ₁ Δ₂ θ,
{ left := sSet.standard_simplex.map θ,
right := terminal.from _, }, } | def | sSet.augmented.standard_simplex | algebraic_topology | src/algebraic_topology/simplicial_set.lean | [
"algebraic_topology.simplicial_object",
"algebraic_topology.topological_simplex",
"category_theory.limits.presheaf",
"category_theory.limits.types",
"category_theory.yoneda",
"topology.category.Top.limits.basic"
] | [
"sSet.augmented",
"simplex_category"
] | The functor which sends `[n]` to the simplicial set `Δ[n]` equipped by
the obvious augmentation towards the terminal object of the category of sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Top.to_sSet : Top ⥤ sSet | colimit_adj.restricted_yoneda simplex_category.to_Top | def | Top.to_sSet | algebraic_topology | src/algebraic_topology/simplicial_set.lean | [
"algebraic_topology.simplicial_object",
"algebraic_topology.topological_simplex",
"category_theory.limits.presheaf",
"category_theory.limits.types",
"category_theory.yoneda",
"topology.category.Top.limits.basic"
] | [
"Top",
"sSet",
"simplex_category.to_Top"
] | The functor associating the singular simplicial set to a topological space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sSet.to_Top : sSet ⥤ Top | colimit_adj.extend_along_yoneda simplex_category.to_Top | def | sSet.to_Top | algebraic_topology | src/algebraic_topology/simplicial_set.lean | [
"algebraic_topology.simplicial_object",
"algebraic_topology.topological_simplex",
"category_theory.limits.presheaf",
"category_theory.limits.types",
"category_theory.yoneda",
"topology.category.Top.limits.basic"
] | [
"Top",
"sSet",
"simplex_category.to_Top"
] | The geometric realization functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sSet_Top_adj : sSet.to_Top ⊣ Top.to_sSet | colimit_adj.yoneda_adjunction _ | def | sSet_Top_adj | algebraic_topology | src/algebraic_topology/simplicial_set.lean | [
"algebraic_topology.simplicial_object",
"algebraic_topology.topological_simplex",
"category_theory.limits.presheaf",
"category_theory.limits.types",
"category_theory.yoneda",
"topology.category.Top.limits.basic"
] | [
"Top.to_sSet",
"sSet.to_Top"
] | Geometric realization is left adjoint to the singular simplicial set construction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sSet.to_Top_simplex :
(yoneda : simplex_category ⥤ _) ⋙ sSet.to_Top ≅ simplex_category.to_Top | colimit_adj.is_extension_along_yoneda _ | def | sSet.to_Top_simplex | algebraic_topology | src/algebraic_topology/simplicial_set.lean | [
"algebraic_topology.simplicial_object",
"algebraic_topology.topological_simplex",
"category_theory.limits.presheaf",
"category_theory.limits.types",
"category_theory.yoneda",
"topology.category.Top.limits.basic"
] | [
"sSet.to_Top",
"simplex_category",
"simplex_category.to_Top"
] | The geometric realization of the representable simplicial sets agree
with the usual topological simplices. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
index_set (Δ : simplex_categoryᵒᵖ) | Σ (Δ' : simplex_categoryᵒᵖ), { α : Δ.unop ⟶ Δ'.unop // epi α } | def | simplicial_object.splitting.index_set | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | The index set which appears in the definition of split simplicial objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk {Δ Δ' : simplex_category} (f : Δ ⟶ Δ') [epi f] : index_set (op Δ) | ⟨op Δ', f, infer_instance⟩ | def | simplicial_object.splitting.index_set.mk | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [
"simplex_category"
] | The element in `splitting.index_set Δ` attached to an epimorphism `f : Δ ⟶ Δ'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
e | A.2.1 | def | simplicial_object.splitting.index_set.e | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | The epimorphism in `simplex_category` associated to `A : splitting.index_set Δ` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext' : A = ⟨A.1, ⟨A.e, A.2.2⟩⟩ | by tidy | lemma | simplicial_object.splitting.index_set.ext' | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext (A₁ A₂ : index_set Δ) (h₁ : A₁.1 = A₂.1)
(h₂ : A₁.e ≫ eq_to_hom (by rw h₁) = A₂.e) : A₁ = A₂ | begin
rcases A₁ with ⟨Δ₁, ⟨α₁, hα₁⟩⟩,
rcases A₂ with ⟨Δ₂, ⟨α₂, hα₂⟩⟩,
simp only at h₁,
subst h₁,
simp only [eq_to_hom_refl, comp_id, index_set.e] at h₂,
simp only [h₂],
end | lemma | simplicial_object.splitting.index_set.ext | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : index_set Δ | ⟨Δ, ⟨𝟙 _, by apply_instance,⟩⟩ | def | simplicial_object.splitting.index_set.id | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | The distinguished element in `splitting.index_set Δ` which corresponds to the
identity of `Δ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_id : Prop | A = id _ | def | simplicial_object.splitting.index_set.eq_id | algebraic_topology | src/algebraic_topology/split_simplicial_object.lean | [
"algebraic_topology.simplicial_object",
"category_theory.limits.shapes.finite_products"
] | [] | The condition that an element `splitting.index_set Δ` is the distinguished
element `splitting.index_set.id Δ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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