Martin Escardo, 22nd and 24th January 2020, with further additions after that. The Cantor-Schröder-Bernstein for homotopy types, or ∞-groupoids, in Agda ------------------------------------------------------------------------- An unformalized version of parts of this file was published in the Journal of Homotopy and Related Structures, Springer, 28th June 2021. https://doi.org/10.1007/s40062-021-00284-6 There are two parts, which assume function extensionality but not univalence or the existence of propositional truncations (any assumption beyond MLTT is explicit in each claim). (1) A univalent-foundations version of Pierre Pradic and Chad E. Brown's argument that Cantor-Schröder-Bernstein implies excluded middle in constructive set theory. (Added 22nd January.) (https://arxiv.org/abs/1904.09193). Their proof, reproduced here, uses the compactness (also known as the searchability or omniscience) of ℕ∞. (See also Appendix II.) (2) A proof that excluded middle implies Cantor-Schröder-Bernstein for all homotopy types, or ∞-groupoids. (Added 24th January.) For any pair of types, if each one is embedded into the other, then they are equivalent. For this it is crucial that a map is an embedding if and only if its fibers are all propositions (rather than merely the map being left-cancellable). As far as we know, (2) is a new result. This part is the Agda version of https://arxiv.org/abs/2002.07079. Check our lecture notes to learn HoTT/UF with Agda: https://www.cs.bham.ac.uk/~mhe/HoTT-UF.in-Agda-Lecture-Notes/ \begin{code} {-# OPTIONS --safe --without-K #-} module CantorSchroederBernstein.CSB where open import CoNaturals.Type open import MLTT.Plus-Properties open import MLTT.Spartan open import Naturals.Properties open import TypeTopology.CompactTypes open import TypeTopology.GenericConvergentSequenceCompactness open import UF.Base open import UF.DiscreteAndSeparated open import UF.Embeddings open import UF.Equiv open import UF.ClassicalLogic open import UF.FunExt open import UF.Lower-FunExt open import UF.PropTrunc open import UF.Retracts open import UF.Sets open import UF.Subsingletons open import UF.Subsingletons-FunExt open import UF.Subsingletons-Properties \end{code} Our formulation of Cantor-Schröder-Bernstein: \begin{code} CSB : 𝓤 ̇ → 𝓥 ̇ → 𝓤 ⊔ 𝓥 ̇ CSB X Y = (X ↪ Y) × (Y ↪ X) → X ≃ Y Cantor-Schröder-Bernstein : (𝓤 𝓥 : Universe) → (𝓤 ⊔ 𝓥)⁺ ̇ Cantor-Schröder-Bernstein 𝓤 𝓥 = {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → CSB X Y \end{code} Part 1 ------ The following is Lemma 7 of the above reference, using retractions rather than surjections, for simplicity: \begin{code} Pradic-Brown-lemma : {X : 𝓤 ̇ } {A : 𝓥 ̇ } → retract (A + X) of X → is-Compact X → is-decidable A Pradic-Brown-lemma {𝓤} {𝓥} {X} {A} (r , s , η) c = γ e where P : X → 𝓤 ⊔ 𝓥 ̇ P x = Σ a ꞉ A , r x ＝ inl a d : (x : X) → is-decidable (P x) d x = equality-cases (r x) (λ (a : A) (u : r x ＝ inl a) → inl (a , u)) (λ (y : X) (v : r x ＝ inr y) → inr (λ (a , u) → +disjoint (inl a ＝⟨ u ⁻¹ ⟩ r x ＝⟨ v ⟩ inr y ∎))) e : is-decidable (Σ x ꞉ X , P x) e = c P d f : A → Σ x ꞉ X , P x f a = s (inl a) , a , η (inl a) γ : is-decidable (Σ x ꞉ X , P x) → is-decidable A γ (inl (x , a , u)) = inl a γ (inr φ) = inr (contrapositive f φ) \end{code} We formulate the following in more generality than we need here. Recall that a point x is h-isolated if the identity type x ＝ y is a subsingleton for every y. \begin{code} econstruction' : {X : 𝓤 ̇ } (P : 𝓥 ̇ ) (z : P → X) (s : X → X) → is-prop P → ((p : P) → is-h-isolated (z p)) → disjoint-images z s → is-embedding s → (X ↪ P + X) × (P + X ↪ X) econstruction' {𝓤} {𝓥} {X} P z s i h d e = ((f , j) , (g , k)) where f : X → P + X f = inr j : is-embedding f j = inr-is-embedding P X g : P + X → X g = cases z s l : is-embedding z l = maps-of-props-into-h-isolated-points-are-embeddings z i h k : is-embedding g k = disjoint-cases-embedding z s l e d \end{code} The level of generality we need here is the following. Recall that a point is x isolated if equality x ＝ y is decidable for every y. By Hedberg's Theorem, every isolated point is h-isolated. \begin{code} econstruction : {X : 𝓤 ̇ } (P : 𝓥 ̇ ) (x₀ : X) (s : X → X) → is-set X → is-prop P → is-isolated x₀ → ((x : X) → x₀ ≠ s x) → left-cancellable s → (X ↪ P + X) × (P + X ↪ X) econstruction {𝓤} {𝓥} {X} P x₀ s j i k d' lc = econstruction' P z s i h d e where z : P → X z p = x₀ h : (p : P) → is-h-isolated (z p) h p = isolated-is-h-isolated x₀ k d : disjoint-images z s d p = d' e : is-embedding s e = lc-maps-into-sets-are-embeddings s lc j \end{code} The Pradic-Brown argument uses the special case X = ℕ∞ with Zero and Succ, but, in Appendix II, we also consider X = ℕ with zero and succ. In the following, function extensionality is used to know that (1) ℕ∞ is a set, (2) its finite elements (in particular zero) are isolated, (3) ℕ∞ is compact. \begin{code} econstruction-ℕ∞ : funext 𝓤₀ 𝓤₀ → (P : 𝓤 ̇ ) → is-prop P → (ℕ∞ ↪ P + ℕ∞) × (P + ℕ∞ ↪ ℕ∞) econstruction-ℕ∞ fe P i = econstruction P Zero Succ (ℕ∞-is-set fe) i (finite-isolated fe zero) (x ↦ Zero-not-Succ) Succ-lc CSB-gives-EM : funext 𝓤₀ 𝓤₀ → (P : 𝓤 ̇ ) → is-prop P → CSB ℕ∞ (P + ℕ∞) → P + ¬ P CSB-gives-EM fe P i csb = γ where e : ℕ∞ ≃ P + ℕ∞ e = csb (econstruction-ℕ∞ fe P i) ρ : retract (P + ℕ∞) of ℕ∞ ρ = ≃-gives-▷ e γ : P + ¬ P γ = Pradic-Brown-lemma ρ (ℕ∞-Compact fe) \end{code} Hence if we assume Cantor-Schröder-Bernstein for the first universe 𝓤₀ and an arbitrary universe 𝓥, as formulated above, then we get excluded middle for propositions in the universe 𝓥: \begin{code} Cantor-Schröder-Bernstein-gives-EM : funext 𝓤₀ 𝓤₀ → Cantor-Schröder-Bernstein 𝓤₀ 𝓥 → EM 𝓥 Cantor-Schröder-Bernstein-gives-EM fe csb P i = CSB-gives-EM fe P i csb \end{code} We remark that if instead of requiring that we have a designated equivalence, we required that there is an unspecified equivalence in the formulation of Cantor-Schröder-Bernstein, we would still get excluded middle, because P + ¬ P is a proposition if P is: \begin{code} module wCSB-still-gives-EM (pt : propositional-truncations-exist) where open PropositionalTruncation pt public wCSB : 𝓤 ̇ → 𝓥 ̇ → 𝓤 ⊔ 𝓥 ̇ wCSB X Y = (X ↪ Y) × (Y ↪ X) → ∥ X ≃ Y ∥ wCantor-Schröder-Bernstein : (𝓤 𝓥 : Universe) → (𝓤 ⊔ 𝓥)⁺ ̇ wCantor-Schröder-Bernstein 𝓤 𝓥 = {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → wCSB X Y wCantor-Schröder-Bernstein-gives-EM : funext 𝓥 𝓤₀ → wCantor-Schröder-Bernstein 𝓤₀ 𝓥 → EM 𝓥 wCantor-Schröder-Bernstein-gives-EM {𝓥} fe w P i = γ where fe₀ : funext 𝓤₀ 𝓤₀ fe₀ = lower-funext 𝓥 𝓤₀ fe s : ∥ ℕ∞ ≃ P + ℕ∞ ∥ s = w (econstruction-ℕ∞ fe₀ P i) t : ℕ∞ ≃ P + ℕ∞ → P + ¬ P t e = Pradic-Brown-lemma (≃-gives-▷ e) (ℕ∞-Compact fe₀) γ : P + ¬ P γ = ∥∥-rec (decidability-of-prop-is-prop fe i) t s \end{code} Part 2 ------ The Cantor-Schröder-Bernstein Theorem holds for all homotopy types, or ∞-gropoids, in the presence of excluded middle. It is crucial here that embeddings have subsingleton fibers, so that e.g. the function is-g-point defined in the proof is property rather than data and hence we can apply univalent excluded middle to it. It is also worth remembering, for the sake of comparing the classical result for sets with its generalization to ∞-groupoids, that a map of types that are sets is an embedding if and only if it is left-cancellable. Our proof adapts Halmos' proof in his book Naive Set Theory to our more general situation. For foundational reasons, we make clear which instances of function extensionality and excluded middle are needed to conclude Cantor-Schröder-Bernstein for arbitrary universes 𝓤 and 𝓥. Added 28th January. To better understand this proof, the reader may consult the blog post https://homotopytypetheory.org/2020/01/26/the-cantor-schroder-bernstein-theorem-for-∞-groupoids/ first. However, we have tried to make the proof as understandable as we can here, and hopefully it should be possible to read it without reference to the blog post. \begin{code} EM-gives-Cantor-Schröder-Bernstein : Fun-Ext → EM (𝓤 ⊔ 𝓥) → Cantor-Schröder-Bernstein 𝓤 𝓥 EM-gives-Cantor-Schröder-Bernstein {𝓤} {𝓥} fe excluded-middle {X} {Y} ((f , f-is-emb) , (g , g-is-emb)) = need X ≃ Y which-is-given-by 𝒽 where remark-f : type-of (f , f-is-emb) ＝ (X ↪ Y) remark-f = by-assumption remark-g : type-of (g , g-is-emb) ＝ (Y ↪ X) remark-g = by-assumption \end{code} In order to define 𝒽 : X ≃ Y, we use a notion of g-point. \begin{code} is-g-point : (x : X) → 𝓤 ⊔ 𝓥 ̇ is-g-point x = (x₀ : X) (n : ℕ) → ((g ∘ f) ^ n) x₀ ＝ x → fiber g x₀ \end{code} What is important for our purposes is that this is property rather than data, using the fact that g is an embedding, which means that its fibers are all propositions. \begin{code} recall : (x : X) → fiber g x ＝ (Σ y ꞉ Y , g y ＝ x) recall _ = by-definition also-recall : is-embedding g ＝ ((x : X) → is-prop (fiber g x)) also-recall = by-definition \end{code} We use the fact that propositions are closed under products, which requires function extensionality: \begin{code} being-g-point-is-prop : (x : X) → is-prop (is-g-point x) being-g-point-is-prop x = Π₃-is-prop fe (λ x₀ n (p : ((g ∘ f) ^ n) x₀ ＝ x) → need is-prop (fiber g x₀) which-is-given-by (g-is-emb x₀)) \end{code} By construction, considering x₀ = x and n = 0, we have that g is invertible at g-points, because, by definition, we have that ((g ∘ f) ^ 0) x ＝ x). \begin{code} g-is-invertible-at-g-points : (x : X) → is-g-point x → fiber g x g-is-invertible-at-g-points x γ = γ x 0 (by-definition ∶ ((g ∘ f) ^ 0) x ＝ x) \end{code} The fiber point is given by the first projection of the fiber: \begin{code} g⁻¹ : (x : X) → is-g-point x → Y g⁻¹ x γ = fiber-point (g-is-invertible-at-g-points x γ) \end{code} Because being a g-point is property, we can apply excluded middle to it: \begin{code} recall-the-notion-of-decidability : {𝓦 : Universe} {A : 𝓦 ̇ } → is-decidable A ＝ (A + ¬ A) recall-the-notion-of-decidability = by-definition δ : (x : X) → is-decidable (is-g-point x) δ x = excluded-middle (is-g-point x) (being-g-point-is-prop x) \end{code} The rest of the proof shows that the following function is an equivalence: \begin{code} h : X → Y h x = Cases (δ x) (γ ꞉ is-g-point x ↦ g⁻¹ x γ) (ν ꞉ ¬ is-g-point x ↦ f x) \end{code} For that purpose, it is enough to show that it is left-cancellable and split-surjective. To show that it is left-cancellable, we first show that g⁻¹ is a two-sided inverse in its domain of definition. That it is a right inverse follows from the definition of fiber, by taking the fiber identification, which is given by the second projection: \begin{code} g⁻¹-is-rinv : (x : X) (γ : is-g-point x) → g (g⁻¹ x γ) ＝ x g⁻¹-is-rinv x γ = fiber-identification (g-is-invertible-at-g-points x γ) \end{code} That it is a left inverse follows from the above and the fact that g, being an embedding, is left-cancellable: \begin{code} g⁻¹-is-linv : (y : Y) (γ : is-g-point (g y)) → g⁻¹ (g y) γ ＝ y g⁻¹-is-linv y γ = have (g (g⁻¹ (g y) γ) ＝⟨ g⁻¹-is-rinv (g y) γ ⟩ g y ∎) so-apply embeddings-are-lc g g-is-emb \end{code} We also need the following two facts to establish the left-cancellability of h: \begin{code} α : (x : X) → is-g-point (g (f x)) → is-g-point x α x γ = need is-g-point x which-is-given-by assume x₀ ∶ X and assume n ∶ ℕ and assume p ∶ ((g ∘ f) ^ n) x₀ ＝ x then (need fiber g x₀ which-is-given-by have ap (g ∘ f) p ∶ ((g ∘ f) ^ (succ n)) x₀ ＝ g (f x) so-apply γ x₀ (succ n)) f-g⁻¹-disjoint-images : (x x' : X) → ¬ is-g-point x → (γ : is-g-point x') → f x ≠ g⁻¹ x' γ f-g⁻¹-disjoint-images x x' ν γ p = have p ∶ f x ＝ g⁻¹ x' γ so need contradiction which-is-given-by have γ ∶ is-g-point x' which-is-impossible-by (v ∶ ¬ is-g-point x') where q : g (f x) ＝ x' q = have p ∶ f x ＝ g⁻¹ x' γ so-use (g (f x) ＝⟨ ap g p ⟩ g (g⁻¹ x' γ) ＝⟨ g⁻¹-is-rinv x' γ ⟩ x' ∎) u : ¬ is-g-point (g (f x)) u = have ν ∶ ¬ is-g-point x so-apply contrapositive (α x) v : ¬ is-g-point x' v = transport (- ↦ ¬ is-g-point -) q u \end{code} It is convenient to work with the following auxiliary function H and prove properties of H and then specialize them to h: \begin{code} H : (x : X) → is-decidable (is-g-point x) → Y H x d = Cases d (γ ꞉ is-g-point x ↦ g⁻¹ x γ) (ν ꞉ ¬ is-g-point x ↦ f x) notice-that : h ＝ x ↦ H x (δ x) notice-that = by-definition h-lc : left-cancellable h h-lc {x} {x'} = l (δ x) (δ x') where l : (d : is-decidable (is-g-point x )) (d' : is-decidable (is-g-point x')) → H x d ＝ H x' d' → x ＝ x' l (inl γ) (inl γ') p = have p ∶ g⁻¹ x γ ＝ g⁻¹ x' γ' so (x ＝⟨ (g⁻¹-is-rinv x γ)⁻¹ ⟩ g (g⁻¹ x γ) ＝⟨ ap g p ⟩ g (g⁻¹ x' γ') ＝⟨ g⁻¹-is-rinv x' γ' ⟩ x' ∎) l (inl γ) (inr ν') p = have p ∶ g⁻¹ x γ ＝ f x' which-is-impossible-by (- ↦ f-g⁻¹-disjoint-images x' x ν' γ (- ⁻¹)) l (inr ν) (inl γ') p = have p ∶ f x ＝ g⁻¹ x' γ' which-is-impossible-by f-g⁻¹-disjoint-images x x' ν γ' l (inr ν) (inr ν') p = have p ∶ f x ＝ f x' so-apply embeddings-are-lc f f-is-emb \end{code} Next we want to show that h is split surjective. For that purpose, we define the notion of f-point, which is data rather than property (as several x₀ and n are possible answers in general). (In particular, excluded middle can't be applied to the type f-point x, because excluded middle applies only to truth values.) \begin{code} f-point : (x : X) → 𝓤 ⊔ 𝓥 ̇ f-point x = Σ x₀ ꞉ X , (Σ n ꞉ ℕ , ((g ∘ f) ^ n) x₀ ＝ x) × ¬ fiber g x₀ \end{code} What is important for our argument is that non-f-points are g-points: \begin{code} non-f-point-is-g-point : (x : X) → ¬ f-point x → is-g-point x non-f-point-is-g-point x ν x₀ n p = need fiber g x₀ which-is-given-by (Cases (excluded-middle (fiber g x₀) (g-is-emb x₀)) (σ ꞉ fiber g x₀ ↦ σ) (u ꞉ ¬ fiber g x₀ ↦ have (x₀ , (n , p) , u) ∶ f-point x which-is-impossible-by (ν ∶ ¬ f-point x))) \end{code} We use the notion of f-point to prove the following, whose statement doesn't refer to the notion of f-point. \begin{code} claim : (y : Y) → ¬ is-g-point (g y) → Σ (x , p) ꞉ fiber f y , ¬ is-g-point x claim y ν = v where i : ¬¬ f-point (g y) i = have ν ∶ ¬ is-g-point (g y) so-apply contrapositive (non-f-point-is-g-point (g y)) ii : f-point (g y) → Σ (x , p) ꞉ fiber f y , ¬ is-g-point x ii (x₀ , (0 , p) , u) = have p ∶ x₀ ＝ g y so have (y , (p ⁻¹)) ∶ fiber g x₀ which-is-impossible-by (u ∶ ¬ fiber g x₀) ii (x₀ , (succ n , p) , u) = a , b where q : f (((g ∘ f) ^ n) x₀) ＝ y q = have p ∶ ((g ∘ f) ^ (succ n)) x₀ ＝ g y ∶ g (f (((g ∘ f) ^ n) x₀)) ＝ g y so-apply embeddings-are-lc g g-is-emb a : fiber f y a = ((g ∘ f) ^ n) x₀ , q b : ¬ is-g-point (((g ∘ f) ^ n) x₀) b = assume γ ∶ is-g-point (((g ∘ f) ^ n) x₀) then (have γ x₀ n refl ∶ fiber g x₀ which-is-impossible-by (u ∶ ¬ fiber g x₀)) iii : ¬¬ (Σ (x , p) ꞉ fiber f y , ¬ is-g-point x) iii = double-contrapositive ii i iv : is-prop (Σ (x , p) ꞉ fiber f y , ¬ is-g-point x) iv = have f-is-emb y ∶ is-prop (fiber f y) so-apply subtypes-of-props-are-props' pr₁ (pr₁-lc (λ {σ} → negations-are-props fe)) v : Σ (x , p) ꞉ fiber f y , ¬ is-g-point x v = double-negation-elim excluded-middle _ iv iii \end{code} With this we are ready to show that h is a split surjection. The idea is that, given y : Y, we check whether g y is a g-point or not, and if it is we map it to g y, and otherwise we map y to the point x : X given by the above claim. But then, of course, we also need to argue that this works. As above, we use the auxiliary function H for that purpose. \begin{code} h-split-surjection : (y : Y) → Σ x ꞉ X , h x ＝ y h-split-surjection y = x , p where a : is-decidable (is-g-point (g y)) → Σ x ꞉ X , ((d : is-decidable (is-g-point x)) → H x d ＝ y) a (inl γ) = g y , ψ where ψ : (d : is-decidable (is-g-point (g y))) → H (g y) d ＝ y ψ (inl γ') = H (g y) (inl γ') ＝⟨ by-definition ⟩ g⁻¹ (g y) γ' ＝⟨ g⁻¹-is-linv y γ' ⟩ y ∎ ψ (inr ν) = have ν ∶ ¬ is-g-point (g y) which-contradicts (γ ∶ is-g-point (g y)) a (inr ν) = x , ψ where w : Σ (x , p) ꞉ fiber f y , ¬ is-g-point x w = have ν ∶ ¬ is-g-point (g y) so-apply claim y x : X x = fiber-point (pr₁ w) p : f x ＝ y p = fiber-identification (pr₁ w) ψ : (d : is-decidable (is-g-point x)) → H x d ＝ y ψ (inl γ) = have γ ∶ is-g-point x which-is-impossible-by (pr₂ w ∶ ¬ is-g-point x) ψ (inr ν) = H x (inr ν) ＝⟨ by-definition ⟩ f x ＝⟨ p ⟩ y ∎ b : Σ x ꞉ X , ((d : is-decidable (is-g-point x)) → H x d ＝ y) b = a (δ (g y)) x : X x = pr₁ b p : h x ＝ y p = h x ＝⟨ by-construction ⟩ H x (δ x) ＝⟨ pr₂ b (δ x) ⟩ y ∎ \end{code} And because left-cancellable split surjections are equivalences, we are done: \begin{code} 𝒽 : X ≃ Y 𝒽 = h , lc-split-surjections-are-equivs h h-lc h-split-surjection \end{code} We record the following special case: \begin{code} EM-gives-Cantor-Schröder-Bernstein₀ : Fun-Ext → EM 𝓤₀ → Cantor-Schröder-Bernstein 𝓤₀ 𝓤₀ EM-gives-Cantor-Schröder-Bernstein₀ = EM-gives-Cantor-Schröder-Bernstein \end{code} If the type X in the proof is connected, then every map of X into a set is constant. In particular, the property of being a g-point is constant, because the type of truth values is a set (assuming univalence for subsingletons). Hence, by excluded middle, it is constantly true or constantly false, and so h = g⁻¹ or h = f, which means that one of the embeddings f and g is already an equivalence. Mike Shulman observed that this is true even without excluded middle: If X is connected and we have an embedding g : Y → X and any function at all f : X → Y, then g is an equivalence. In fact, for any x : X, we have ∥ g (f (x)) = x ∥ since X is connected; thus g is (non-split) surjective. But a surjective embedding is an equivalence. \begin{code} module CSB-for-connected-types-without-EM (pt : propositional-truncations-exist) where open PropositionalTruncation pt public open import UF.Connected pt open import UF.ImageAndSurjection pt \end{code} We say that X is weakly connected if ∥ x ＝ x' ∥ for all x and x' in X, and that it is connected if additionally ∥ X ∥ is pointed. \begin{code} lemma : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) (g : Y → X) → is-wconnected X → is-embedding g → is-equiv g lemma f g w e = surjective-embeddings-are-equivs g e s where a : ∀ x → ∥ g (f (x)) ＝ x ∥ a x = w (g (f x)) x s : is-surjection g s x = ∥∥-functor (λ p → (f x , p)) (a x) cCSB : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → is-wconnected Y → CSB X Y cCSB {𝓤} {𝓥} {X} {Y} w ((f , i) , (g , _)) = γ where γ : X ≃ Y γ = f , lemma g f w i \end{code} Of course, we can instead assume that X is wconnected: \begin{code} cCSB' : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → is-wconnected X → CSB X Y cCSB' w e = ≃-sym (cCSB w (pr₂ e , pr₁ e)) \end{code} Another direct corollary is that weakly connected types are Dedekind finite: \begin{code} wconnected-types-are-Dedekind-finite : {X : 𝓤 ̇ } → is-wconnected X → (f : X → X) → is-embedding f → is-equiv f wconnected-types-are-Dedekind-finite w f = lemma f f w \end{code} APPENDIX I ---------- The above is an attempt to make the proof more readable and match the blog post. Here is a more concise version of the above in a more direct Agda style which some will prefer (and which could be made even more concise by avoiding auxiliary definitions used for the purpose of indicating types explicitly). \begin{code} EM-gives-Cantor-Schröder-Bernstein' : Fun-Ext → EM (𝓤 ⊔ 𝓥) → Cantor-Schröder-Bernstein 𝓤 𝓥 EM-gives-Cantor-Schröder-Bernstein' {𝓤} {𝓥} fe excluded-middle {X} {Y} ((f , f-is-emb) , (g , g-is-emb)) = 𝒽 where is-g-point : (x : X) → 𝓤 ⊔ 𝓥 ̇ is-g-point x = (x₀ : X) (n : ℕ) → ((g ∘ f) ^ n) x₀ ＝ x → fiber g x₀ g-is-invertible-at-g-points : (x : X) (γ : is-g-point x) → fiber g x g-is-invertible-at-g-points x γ = γ x 0 refl g⁻¹ : (x : X) → is-g-point x → Y g⁻¹ x γ = fiber-point (g-is-invertible-at-g-points x γ) g⁻¹-is-rinv : (x : X) (γ : is-g-point x) → g (g⁻¹ x γ) ＝ x g⁻¹-is-rinv x γ = fiber-identification (g-is-invertible-at-g-points x γ) g⁻¹-is-linv : (y : Y) (γ : is-g-point (g y)) → g⁻¹ (g y) γ ＝ y g⁻¹-is-linv y γ = embeddings-are-lc g g-is-emb (g⁻¹-is-rinv (g y) γ) α : (x : X) → is-g-point (g (f x)) → is-g-point x α x γ x₀ n p = γ x₀ (succ n) (ap (g ∘ f) p) f-g⁻¹-disjoint-images : (x x' : X) → ¬ is-g-point x → (γ : is-g-point x') → f x ≠ g⁻¹ x' γ f-g⁻¹-disjoint-images x x' ν γ p = 𝟘-elim (v γ) where q = g (f x) ＝⟨ ap g p ⟩ g (g⁻¹ x' γ) ＝⟨ g⁻¹-is-rinv x' γ ⟩ x' ∎ u : ¬ is-g-point (g (f x)) u = contrapositive (α x) ν v : ¬ is-g-point x' v = transport (- ↦ ¬ is-g-point -) q u being-g-point-is-prop : (x : X) → is-prop (is-g-point x) being-g-point-is-prop x = Π₃-is-prop fe (λ x₀ _ _ → g-is-emb x₀) δ : (x : X) → is-decidable (is-g-point x) δ x = excluded-middle (is-g-point x) (being-g-point-is-prop x) H : (x : X) → is-decidable (is-g-point x) → Y H x (inl γ) = g⁻¹ x γ H x (inr _) = f x h : X → Y h x = H x (δ x) h-lc : left-cancellable h h-lc {x} {x'} = l (δ x) (δ x') where l : (d : is-decidable (is-g-point x)) (d' : is-decidable (is-g-point x')) → H x d ＝ H x' d' → x ＝ x' l (inl γ) (inl γ') p = x ＝⟨ (g⁻¹-is-rinv x γ)⁻¹ ⟩ g (g⁻¹ x γ) ＝⟨ ap g p ⟩ g (g⁻¹ x' γ') ＝⟨ g⁻¹-is-rinv x' γ' ⟩ x' ∎ l (inl γ) (inr ν') p = 𝟘-elim (f-g⁻¹-disjoint-images x' x ν' γ (p ⁻¹)) l (inr ν) (inl γ') p = 𝟘-elim (f-g⁻¹-disjoint-images x x' ν γ' p) l (inr ν) (inr ν') p = embeddings-are-lc f f-is-emb p f-point : (x : X) → 𝓤 ⊔ 𝓥 ̇ f-point x = Σ x₀ ꞉ X , (Σ n ꞉ ℕ , ((g ∘ f) ^ n) x₀ ＝ x) × ¬ fiber g x₀ non-f-point-is-g-point : (x : X) → ¬ f-point x → is-g-point x non-f-point-is-g-point x ν x₀ n p = Cases (excluded-middle (fiber g x₀) (g-is-emb x₀)) (λ (σ : fiber g x₀) → σ) (λ (u : ¬ fiber g x₀) → 𝟘-elim (ν (x₀ , (n , p) , u))) claim : (y : Y) → ¬ is-g-point (g y) → Σ (x , p) ꞉ fiber f y , ¬ is-g-point x claim y ν = v where i : ¬¬ f-point (g y) i = contrapositive (non-f-point-is-g-point (g y)) ν ii : f-point (g y) → Σ (x , p) ꞉ fiber f y , ¬ is-g-point x ii (x₀ , (0 , p) , u) = 𝟘-elim (u (y , (p ⁻¹))) ii (x₀ , (succ n , p) , u) = a , b where q : f (((g ∘ f) ^ n) x₀) ＝ y q = embeddings-are-lc g g-is-emb p a : fiber f y a = ((g ∘ f) ^ n) x₀ , q b : ¬ is-g-point (((g ∘ f) ^ n) x₀) b γ = 𝟘-elim (u (γ x₀ n refl)) iii : ¬¬ (Σ (x , p) ꞉ fiber f y , ¬ is-g-point x) iii = double-contrapositive ii i iv : is-prop (Σ (x , p) ꞉ fiber f y , ¬ is-g-point x) iv = subtypes-of-props-are-props' pr₁ (pr₁-lc (λ {σ} → negations-are-props fe)) (f-is-emb y) v : Σ (x , p) ꞉ fiber f y , ¬ is-g-point x v = double-negation-elim excluded-middle _ iv iii h-split-surjection : (y : Y) → Σ x ꞉ X , h x ＝ y h-split-surjection y = x , p where a : is-decidable (is-g-point (g y)) → Σ x ꞉ X , ((d : is-decidable (is-g-point x)) → H x d ＝ y) a (inl γ) = g y , ψ where ψ : (d : is-decidable (is-g-point (g y))) → H (g y) d ＝ y ψ (inl γ') = g⁻¹-is-linv y γ' ψ (inr ν) = 𝟘-elim (ν γ) a (inr ν) = x , ψ where w : Σ (x , p) ꞉ fiber f y , ¬ is-g-point x w = claim y ν x : X x = fiber-point (pr₁ w) ψ : (d : is-decidable (is-g-point x)) → H x d ＝ y ψ (inl γ) = 𝟘-elim (pr₂ w γ) ψ (inr ν) = fiber-identification (pr₁ w) b : Σ x ꞉ X , ((d : is-decidable (is-g-point x)) → H x d ＝ y) b = a (δ (g y)) x : X x = pr₁ b p : h x ＝ y p = h x ＝⟨ by-construction ⟩ H x (δ x) ＝⟨ pr₂ b (δ x) ⟩ y ∎ 𝒽 : X ≃ Y 𝒽 = h , lc-split-surjections-are-equivs h h-lc h-split-surjection \end{code} APPENDIX II ----------- Added 17th Feb 2020. A stronger result is added below, 18th Feb 2020, with a technically and conceptually simpler proof. (But 19th Feb 2020 below shows that this 17th Feb 2020 argument has some merits.) Coming back to part 1, we consider what follows if we assume CSB for types with decidable equality (which are necessarily sets) only. Such types are called discrete. We adapt an argument in Johnstone's Sketches of an Elephant Volume 2 (Lemma D.4.1.12). See also van Atten https://www.sciencedirect.com/science/article/pii/S0019357718303276 for BKS⁺ defined below (strong Brouwer-Kripke Schema) and the fact that together with Markov Principle it implies excluded middle (attributed to Moschovakis). The terminology "Rosolini-data" is in connection with the Rosolini dominance from synthetic domain theory and topology. \begin{code} Rosolini-data : 𝓤 ̇ → 𝓤 ⁺ ̇ Rosolini-data {𝓤} P = Σ A ꞉ (ℕ → 𝓤 ̇ ) , ((n : ℕ) → is-decidable (A n)) × is-prop (Σ A) × (P ↔ Σ A) \end{code} Notice this is data on P rather than property of P because multiple A's apply to the same P, when P holds. Notice also that we don't need to require that each A n is a proposition, as this is automatic because ℕ is a set: \begin{code} is-prop-total-gives-is-prop-each : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) → is-set X → is-prop (Σ A) → (x : X) → is-prop (A x) is-prop-total-gives-is-prop-each A j i x a a' = t where q : (x , a) ＝ (x , a') q = i (x , a) (x , a') t = a ＝⟨ by-definition ⟩ transport A refl a ＝⟨ ap (- ↦ transport A - a) (j refl (ap pr₁ q)) ⟩ transport A (ap pr₁ q) a ＝⟨ from-Σ-＝' q ⟩ a' ∎ \end{code} Here a typal, rather than propositional, version of BKS⁺, which is data-valued rather than propositionally valued, arises. \begin{code} dBKS⁺ : (𝓤 : Universe) → 𝓤 ⁺ ̇ dBKS⁺ 𝓤 = (P : 𝓤 ̇ ) → is-prop P → Rosolini-data P \end{code} It is convenient to work with the following formulation of Markov Principle that avoids ∃ (and hence propositional truncations), which is easily seen to be equivalent to the traditional formulation using ∃ (using the fact that unique choice just holds (trivially) in HoTT/UF). \begin{code} MP : (𝓤 : Universe) → 𝓤 ⁺ ̇ MP 𝓤 = (A : ℕ → 𝓤 ̇ ) → ((n : ℕ) → is-decidable (A n)) → is-prop (Σ A) → ¬¬ Σ A → Σ A \end{code} The following, which derives double negation elimination from dBKS⁺ and MP, is formulated and proved in pure (spartan) MLTT: \begin{code} dBKS⁺-and-MP-give-DNE : dBKS⁺ 𝓤 → MP 𝓤 → DNE 𝓤 dBKS⁺-and-MP-give-DNE {𝓤} bks mp P i = γ (bks P i) where γ : (Σ A ꞉ (ℕ → 𝓤 ̇ ) , ((n : ℕ) → is-decidable (A n)) × is-prop (Σ A) × (P ↔ Σ A)) → ¬¬ P → P γ (A , d , j , f , g) = δ where f' : ¬¬ P → ¬¬ Σ A f' = double-contrapositive f h : ¬¬ Σ A → Σ A h = mp A d j δ : ¬¬ P → P δ = g ∘ h ∘ f' \end{code} But the following, which derives excluded middle, needs function extensionality: \begin{code} dBKS⁺-and-MP-give-EM : funext 𝓤 𝓤₀ → dBKS⁺ 𝓤 → MP 𝓤 → EM 𝓤 dBKS⁺-and-MP-give-EM fe bks MP = DNE-gives-EM fe (dBKS⁺-and-MP-give-DNE bks MP) \end{code} So dBKS⁺ "almost" gives excluded middle in some sense. We now show that CSB for discrete types gives dBKS⁺: \begin{code} blemma : (P : 𝓤 ̇ ) {X : 𝓥 ̇ } → is-set X → is-prop P → X ≃ P + X → Σ A ꞉ (X → 𝓤 ⊔ 𝓥 ̇ ) , ((x : X) → is-decidable (A x)) × is-prop (Σ A) × (P ↔ Σ A) blemma {𝓤} {𝓥 } P {X} j i (f , (s , η) , (r , ε)) = A , d , l , (φ , γ) where A : X → 𝓤 ⊔ 𝓥 ̇ A x = Σ p ꞉ P , f x ＝ inl p d : (x : X) → is-decidable (A x) d x = equality-cases (f x) (λ (p : P) (u : f x ＝ inl p) → inl (p , u)) (λ (y : X) (v : f x ＝ inr y) → inr (λ (a , u) → +disjoint (inl a ＝⟨ u ⁻¹ ⟩ f x ＝⟨ v ⟩ inr y ∎))) k : (x : X) → is-prop (A x) k x = Σ-is-prop i (λ p → +-is-set P X (props-are-sets i) j) l : is-prop (Σ A) l (x , p , u) (x' , p' , u') = t where q : x ＝ x' q = equivs-are-lc f ((s , η) , (r , ε)) (f x ＝⟨ u ⟩ inl p ＝⟨ ap inl (i p p') ⟩ inl p' ＝⟨ u' ⁻¹ ⟩ f x' ∎) t : x , p , u ＝ x' , p' , u' t = to-subtype-＝ k q φ : P → Σ A φ p = s (inl p) , p , η (inl p) γ : Σ A → P γ (x , p , u) = p rlemma : (P : 𝓤 ̇ ) → is-prop P → ℕ ≃ P + ℕ → Rosolini-data P rlemma P = blemma P ℕ-is-set discrete-Cantor-Schröder-Bernstein : (𝓤 𝓥 : Universe) → (𝓤 ⊔ 𝓥)⁺ ̇ discrete-Cantor-Schröder-Bernstein 𝓤 𝓥 = {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → is-discrete X → is-discrete Y → CSB X Y econstruction-ℕ : (P : 𝓤 ̇ ) → is-prop P → (ℕ ↪ P + ℕ) × (P + ℕ ↪ ℕ) econstruction-ℕ P i = econstruction P zero succ ℕ-is-set i (ℕ-is-discrete zero) (λ (x : ℕ) (p : zero ＝ succ x) → positive-not-zero x (p ⁻¹)) succ-lc dlemma : (P : 𝓥 ̇ ) → discrete-Cantor-Schröder-Bernstein 𝓤₀ 𝓥 → is-prop P → ℕ ≃ P + ℕ dlemma P csb i = csb ℕ-is-discrete (+-is-discrete (props-are-discrete i) ℕ-is-discrete) (econstruction-ℕ P i) discrete-CSB-gives-dBKS⁺ : discrete-Cantor-Schröder-Bernstein 𝓤₀ 𝓥 → dBKS⁺ 𝓥 discrete-CSB-gives-dBKS⁺ csb P i = γ where e : ℕ ≃ P + ℕ e = dlemma P csb i γ : Rosolini-data P γ = rlemma P i e \end{code} Added 18th Feb 2020. We make the 17 Feb 2020 delopment sharper, at the expense of assuming propositional extensionality (univalence for propositions). If we have a uniform way to get an equivalence ℕ ≃ P + ℕ for any proposition P, then given by a function φ : (P : 𝓤 ̇ ) → is-prop P → ℕ ≃ P + ℕ, we can use φ to decide P for any proposition P. To see this, first consider P = 𝟙, and let x be the natural number that is mapped to inl ⋆ by the equivalence given by φ. Then, for an arbitrary proposition P, if the equivalence maps x to inl p for some p, we have that P holds. Otherwise, if it maps x to inr y for some y : ℕ, then P can't hold, for if it did we would have p : P, and hence P ＝ 𝟙 by propositional extensionality, and the equivalence would have to map x to inl p, which is different from the value inr y of the equivalence at x. There is nothing that depends on the nature of the specific type ℕ in the above argument, and hence we formulate this uniformity lemma with arbitrary types X and Y, although we will apply it to X = Y = ℕ. In order to simplify the calculational details of this proof, we work with the type T of true propositions, which is contractible with center of contraction 𝟙. \begin{code} ulemma : funext 𝓤 𝓤 → propext 𝓤 → {X : 𝓥 ̇ } {Y : 𝓦 ̇ } → ((P : 𝓤 ̇ ) → is-prop P → X ≃ P + Y) → EM 𝓤 ulemma {𝓤} fe pe {X} {Y} φ = em where T : 𝓤 ⁺ ̇ T = Σ P ꞉ 𝓤 ̇ , is-prop P × P c : (t : T) → (𝟙 , 𝟙-is-prop , ⋆) ＝ t c = 𝟙-is-true-props-center fe pe f : T → X f (P , i , p) = ⌜ φ P i ⌝⁻¹ (inl p) x : X x = f (𝟙 , 𝟙-is-prop , ⋆) ν : (P : 𝓤 ̇ ) (i : is-prop P) (y : Y) → ⌜ φ P i ⌝ x ＝ inr y → ¬ P ν P i y r p = γ where a : x ＝ f (P , i , p) a = ap f (c (P , i , p)) b = inr y ＝⟨ r ⁻¹ ⟩ ⌜ φ P i ⌝ x ＝⟨ ap ⌜ φ P i ⌝ a ⟩ ⌜ φ P i ⌝ (f (P , i , p)) ＝⟨ by-definition ⟩ ⌜ φ P i ⌝ (⌜ φ P i ⌝⁻¹ (inl p)) ＝⟨ ≃-sym-is-rinv (φ P i) (inl p) ⟩ inl p ∎ γ : 𝟘 γ = +disjoint' b em : (P : 𝓤 ̇ ) → is-prop P → P + ¬ P em P i = equality-cases (⌜ φ P i ⌝ x) (λ (p : P) (l : ⌜ φ P i ⌝ x ＝ inl p) → inl p) (λ (y : Y) (r : ⌜ φ P i ⌝ x ＝ inr y) → inr (ν P i y r)) discrete-CSB-gives-EM : funext 𝓥 𝓥 → propext 𝓥 → discrete-Cantor-Schröder-Bernstein 𝓤₀ 𝓥 → EM 𝓥 discrete-CSB-gives-EM {𝓥} fe pe csb = ulemma fe pe φ where φ : (P : 𝓥 ̇ ) → is-prop P → ℕ ≃ P + ℕ φ P = dlemma P csb \end{code} Thus, in particular, decidable equality is not enough to get a constructive version of CSB. Even with decidable equality of the given types, one still needs full excluded middle. Discussion ---------- The Pradic-Brown argument, although it requires a non-discrete type to work, has the advantage that if we weaken the statement of CSB to say that an unspecified (rather than designated) equivalence exists, for any two given embeddings in opposite directions, (X ↪ Y) × (Y ↪ X) → ∥ X ≃ Y ∥. one still gets excluded middle, as already remarked above. And it is also nice and clear and short. Our argument doesn't work with this weakening of the hypothesis, as in this case it is no longer possible to define the function φ in the proof (without choice, which is stronger than what we want to prove, namely excluded middle) to apply the uniformity lemma. The reason is that Pradic and Brown use only one instance of CSB, for a given proposition, whereas we use a family of instances. In any case, in the other direction, excluded middle does give CSB with a designated equivalence in the conclusion, as previously shown above. Added 19th Feb 2020: In light of the above discussion, notice that the 17th Feb 2020 development has its merits, after all, compared to the 18th Feb 2020 development. We don't get excluded middle if we weaken CSB, but we do get BKS⁺. \begin{code} module discrete-wCSB-gives-BKS⁺ (pt : propositional-truncations-exist) where \end{code} We open the following module only to have access to the definition of wCSB: \begin{code} open wCSB-still-gives-EM pt discrete-wCantor-Schröder-Bernstein : (𝓤 𝓥 : Universe) → (𝓤 ⊔ 𝓥)⁺ ̇ discrete-wCantor-Schröder-Bernstein 𝓤 𝓥 = {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → is-discrete X → is-discrete Y → wCSB X Y \end{code} We now consider the propositional version of BKS⁺: \begin{code} is-Rosolini : 𝓤 ̇ → 𝓤 ⁺ ̇ is-Rosolini P = ∥ Rosolini-data P ∥ BKS⁺ : (𝓤 : Universe) → 𝓤 ⁺ ̇ BKS⁺ 𝓤 = (P : 𝓤 ̇ ) → is-prop P → is-Rosolini P discrete-wCSB-gives-BKS⁺ : discrete-wCantor-Schröder-Bernstein 𝓤₀ 𝓥 → BKS⁺ 𝓥 discrete-wCSB-gives-BKS⁺ w P i = γ where s : ∥ ℕ ≃ P + ℕ ∥ s = w ℕ-is-discrete (+-is-discrete (props-are-discrete i) ℕ-is-discrete) (econstruction-ℕ P i) γ : is-Rosolini P γ = ∥∥-functor (rlemma P i) s \end{code} Notice that BKS⁺ also implies excluded middle in the presence of MP, because EM is a proposition (in any case, this was already observed by Moschovakis, as discussed above).