fix error
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3 changed files with 28 additions and 3 deletions
24
TODO.txt
24
TODO.txt
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@ -8,3 +8,27 @@ talk with Favonia about:
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- higher cube filling strategies
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- higher cube filling strategies
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- HIT equivalences
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- HIT equivalences
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- algebra
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- algebra
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/-
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Adjointness:
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Σ X ⟶ Y X ∧ Y → Z
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======= ===========
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X → Ω Y X → (Y → Z)
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Spectrum: Y : ℕ → Type* with e : Ω Yₙ₊₁ ≃* Yₙ.
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HOMOLOGY:
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Hₙ(X, Y) :≡? ∥ X ∧ Ω² (Y (n+2)) ∥₀ ≃ ∥ X ∧ Y n ∥₀
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Eilenberg Steenrod-axioms:
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H : ℤ → Type* → AbGroup
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- functorial in second argument
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- (optional): respects pointed equivalences and pointed homotopies
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axioms:
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- the canonical map
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-
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- Given (Xᵢ)ᵢ : I → Type* (satisfying AC?) the canonical functor
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⊕ hₙ
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-/
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@ -1,6 +1,6 @@
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-- Authors: Floris van Doorn
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-- Authors: Floris van Doorn
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import homotopy.EM algebra.category.functor.equivalence ..pointed ..pointed_pi
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import homotopy.EM algebra.category.functor.equivalence types.pointed2 ..pointed_pi ..pointed
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open eq equiv is_equiv algebra group nat pointed EM.ops is_trunc trunc susp function is_conn
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open eq equiv is_equiv algebra group nat pointed EM.ops is_trunc trunc susp function is_conn
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@ -617,7 +617,8 @@ namespace EM
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(trivial_homotopy_group_of_is_trunc (ptrunc 0 A) !zero_lt_succ), exact sorry
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(trivial_homotopy_group_of_is_trunc (ptrunc 0 A) !zero_lt_succ), exact sorry
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-- rexact isomorphism_of_equiv (equiv_of_isomorphism z) sorry
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-- rexact isomorphism_of_equiv (equiv_of_isomorphism z) sorry
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},
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},
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{ apply @is_conn_fun_trunc_elim, apply is_conn_fun_tr }
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{ apply @is_conn_fun_trunc_elim, apply is_conn_fun_tr },
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{ apply is_trunc_pfiber }
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end
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end
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definition pfiber_postnikov_map_succ (A : Type*) (n : ℕ) :
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definition pfiber_postnikov_map_succ (A : Type*) (n : ℕ) :
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@ -15,7 +15,7 @@ namespace cohomology
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/- The cohomology of X with coefficients in Y is
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/- The cohomology of X with coefficients in Y is
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trunc 0 (A →* Ω[2] (Y (n+2)))
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trunc 0 (A →* Ω[2] (Y (n+2)))
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In the file arrow_group (in algebra) we construct the group structor on this type.
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In the file arrow_group (in algebra) we construct the group structure on this type.
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-/
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-/
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definition cohomology (X : Type*) (Y : spectrum) (n : ℤ) : AbGroup :=
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definition cohomology (X : Type*) (Y : spectrum) (n : ℤ) : AbGroup :=
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AbGroup_trunc_pmap X (Y (n+2))
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AbGroup_trunc_pmap X (Y (n+2))
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