Acyclic types are inhabited #1503
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Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
…acewhale/agda-unimath into Acyclic-types-are-inhabited
Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
…acewhale/agda-unimath into Acyclic-types-are-inhabited
Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
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| ### The space of sections of `f : A → B` is equivalent to the space of dependent maps `(b : B) → fiber f b` | ||
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| ```agda | ||
| module _ | ||
| {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) | ||
| where | ||
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| section-dependent-fiber : section f → (b : B) → fiber f b | ||
| pr1 (section-dependent-fiber (g , s) b) = g b | ||
| pr2 (section-dependent-fiber (g , s) b) = s b | ||
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| dependent-fiber-section : ((b : B) → fiber f b) → section f | ||
| pr1 (dependent-fiber-section g) b = pr1 (g b) | ||
| pr2 (dependent-fiber-section g) b = pr2 (g b) | ||
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| equiv-section-dependent-fiber : | ||
| section f ≃ ((b : B) → fiber f b) | ||
| pr1 equiv-section-dependent-fiber = section-dependent-fiber | ||
| pr1 (pr1 (pr2 equiv-section-dependent-fiber)) = dependent-fiber-section | ||
| pr2 (pr1 (pr2 equiv-section-dependent-fiber)) = refl-htpy | ||
| pr1 (pr2 (pr2 equiv-section-dependent-fiber)) = dependent-fiber-section | ||
| pr2 (pr2 (pr2 equiv-section-dependent-fiber)) = refl-htpy | ||
| ``` |
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Sorry, this is already formalized in foundation.split-surjective-maps:
Maybe you could add a link to that file in a ## See also section? See how this section is formatted elsewhere for inspiration.
src/synthetic-homotopy-theory/suspensions-of-propositions.lagda.md
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| is-inhabited-or-empty-is-finite-suspension : | ||
| is-finite (type-trunc-Set (suspension X)) → is-inhabited-or-empty X | ||
| is-inhabited-or-empty-is-finite-suspension p with number-of-elements-is-finite p in eq | ||
| ... | 0 = | ||
| ex-falso | ||
| ( is-nonempty-is-inhabited | ||
| ( is-inhabited-suspension X) | ||
| ( is-empty-set-truncation-is-empty | ||
| ( is-empty-is-zero-number-of-elements-is-finite p eq))) | ||
| ... | 1 = | ||
| inl | ||
| ( is-inhabited-suspension-is-0-connected | ||
| ( X) | ||
| ( is-contr-is-one-number-of-elements-is-finite p eq)) | ||
| ... | 2 = | ||
| inr | ||
| ( is-empty-trunc-suspension-has-two-elements | ||
| ( map-inv-equiv-has-cardinality-id-number-of-elements-is-finite | ||
| ( type-trunc-Set (suspension X)) | ||
| ( p) | ||
| ( 2) | ||
| ( eq))) | ||
| ... | succ-ℕ (succ-ℕ (succ-ℕ n)) = ex-falso {! is-upper-bound-finite-enumeration !} |
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Instead of using a with abstraction, you should instead define a lemma
is-decidable-has-count-trunc-set-suspension :
count (type-trunc-Set (suspension X)) → is-decidable X
and then instantiate the present theorem at that lemma to get your result.
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What's advertised on the tin. As bonus lemmas, we show that
Xis inhabited iffΣ Xis 0-connected, and is propositionally decidable ifftrunc-Set (Σ X)is finite.