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fredrik-bakke committed Dec 9, 2024
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Expand Up @@ -33,7 +33,8 @@ open import synthetic-homotopy-theory.suspensions-of-types

The [axiom of choice](foundation.axiom-of-choice.md) implies the
[law of excluded middle](foundation.law-of-excluded-middle.md). This is often
referred to as {{#concept "Diaconescu's theorem" Agda=lem-AC-0}}.
referred to as
{{#concept "Diaconescu's theorem" WD="Diaconescu's theorem" WDID=Q3527059 Agda=lem-AC-0}}.

## Theorem

Expand All @@ -49,7 +50,7 @@ and `S` are the _poles_ of `ΣP`. There is a surjection from the
[inhabited](foundation.inhabited-types.md) family over `ΣP`. Applying the axiom
of choice to this family, we obtain a
[mere](foundation.propositional-truncations.md)
[section](fondation-core.sections.md) `s` of `f` which thus exhibits `P` as a
[section](foundation-core.sections.md) `s` of `f` which thus exhibits `P` as a
[logical equivalent](foundation.logical-equivalences.md) to `f⁻¹ N = f⁻¹ S`.
The latter is an [equation](foundation-core.identity-types.md) of booleans, and
the booleans have [decidable equality](foundation.decidable-equality.md) so `P`
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6 changes: 3 additions & 3 deletions src/synthetic-homotopy-theory/suspensions-of-types.lagda.md
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Expand Up @@ -55,12 +55,12 @@ open import synthetic-homotopy-theory.universal-property-suspensions

## Idea

The **suspension** of a type `X` is the
[pushout](synthetic-homotopy-theory.pushouts.md) of the
The {{#concept "suspension" WD="suspension" WDID=Q1307987 Agda=suspension}} of a
type `X` is the [pushout](synthetic-homotopy-theory.pushouts.md) of the
[span](foundation.spans.md)

```text
unit <-- X --> unit
1 <--- X ---> 1
```

Suspensions play an important role in synthetic homotopy theory. For example,
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