83 lines
1.9 KiB
Text
83 lines
1.9 KiB
Text
/-- Authors: Clive, Egbert --/
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import homotopy.connectedness homotopy.join
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open eq sigma pi function join is_conn is_trunc equiv is_equiv
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namespace retraction
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variables {A B C : Type} (r2 : B → C) (r1 : A → B)
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definition is_retraction_compose
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[Hr2 : is_retraction r2] [Hr1 : is_retraction r1] :
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is_retraction (r2 ∘ r1) :=
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begin
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cases Hr2 with s2 s2_is_right_inverse,
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cases Hr1 with s1 s1_is_right_inverse,
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fapply is_retraction.mk,
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{ exact s1 ∘ s2},
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{ intro b, esimp,
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calc
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r2 (r1 (s1 (s2 (b)))) = r2 (s2 (b)) : ap r2 (s1_is_right_inverse (s2 b))
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... = b : s2_is_right_inverse b
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}, /-- QED --/
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end
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definition is_retraction_compose_equiv_left [Hr2 : is_equiv r2] [Hr1 : is_retraction r1] : is_retraction (r2 ∘ r1) :=
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begin
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apply is_retraction_compose,
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end
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definition is_retraction_compose_equiv_right [Hr2 : is_retraction r2] [Hr1 : is_equiv r1] : is_retraction (r2 ∘ r1) :=
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begin
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apply is_retraction_compose,
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end
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end retraction
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namespace is_conn
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section
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open retraction
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universe variable u
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parameters (n : ℕ₋₂) {A : Type.{u}}
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parameter sec : ΠV : trunctype.{u} n,
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is_retraction (const A : V → (A → V))
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include sec
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protected definition intro : is_conn n A :=
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begin
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apply is_conn_of_map_to_unit,
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apply is_conn_fun.intro,
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intro P,
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refine is_retraction_compose_equiv_right (const A) (pi_unit_left P),
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end
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end
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end is_conn
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section Join_Theorem
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variables (X Y : Type)
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(m n : ℕ₋₂)
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[HXm : is_conn m X]
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[HYn : is_conn n Y]
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include HXm HYn
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theorem is_conn_join : is_conn (m +2+ n) (join X Y) :=
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begin
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apply is_conn.intro,
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intro V,
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apply is_retraction_of_is_equiv,
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apply is_equiv_of_is_contr_fun,
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intro f,
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refine is_contr_equiv_closed _ _,
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{exact unit},
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symmetry,
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exact sorry
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end
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end Join_Theorem
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