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Ordinals.
finegeometer Aug 16, 2025
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finegeometer Aug 18, 2025
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12 changes: 12 additions & 0 deletions src/1Lab/Path/IdentitySystem.lagda.md
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Expand Up @@ -176,6 +176,18 @@ identity-system-gives-path {R = R} {r = r} ids =
∙ transport-refl _ )
```

In particular, swapping the arguments of the relation in an identity system
yields another identity system.

```agda
flip-identity-system :
∀ {ℓ ℓ'} {A : Type ℓ} {R : A → A → Type ℓ'} {r : ∀ a → R a a}
→ is-identity-system R r
→ is-identity-system (flip R) r
flip-identity-system ids =
equiv-path→identity-system $ identity-system-gives-path ids ∙e sym-equiv
```

## Based identity systems {defines="based-identity-system unary-identity-system"}

It is sometimes useful to characterise the *based* identity type at
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