Skip to content

Commit

Permalink
A historical note on the extended fundamental theorem of identity typ…
Browse files Browse the repository at this point in the history
…es (#1313)

Just adding a small historical note since it is explained this theorem
was stated and proved at the Regensburg summer school, but the unbased
version was not.
  • Loading branch information
fredrik-bakke authored Feb 10, 2025
1 parent 8ba9c35 commit b4cc266
Showing 1 changed file with 9 additions and 5 deletions.
Original file line number Diff line number Diff line change
Expand Up @@ -41,8 +41,9 @@ open import foundation.universe-levels

## Idea

The {{#concept "Regensburg extension" Agda=extended-fundamental-theorem-id}} of
the
The
{{#concept "Regensburg extension" Disambiguation="of the fundamental theorem of identity types" Agda=extended-fundamental-theorem-id}}
of the
[fundamental theorem of identity types](foundation.fundamental-theorem-of-identity-types.md)
asserts that for any [subuniverse](foundation.subuniverses.md) `P`, and any
[pointed](structured-types.pointed-types.md)
Expand Down Expand Up @@ -129,11 +130,14 @@ over a not necessarily pointed or inhabited type `A` whose elements are all
merely equal. In other words, `A` is any π₀-trivial type. The characterization
asserts that the following are equivalent:

1. For every `x : A` and every family of maps out of the identity types
`f : (y : A) (x = y) B x y`, then for every `y : A` the `f y` is a
`𝒫`-map.
1. For every `x : A`, every family of maps out of the identity types
`f : (y : A) (x = y) B x y`, is a family of `𝒫`-maps.
2. For every `x : A` the type `Σ A (B x)` is `𝒫`-separated.

This unbased extension of the fundamental theorem was not stated or proved at
the Regensburg summer school, but was later contributed by
[Fredrik Bakke](https://www.ntnu.edu/employees/fredrik.bakke) in January 2025.

```agda
module _
{l1 l2 l3 : Level} (𝒫 : subuniverse (l1 ⊔ l2) l3)
Expand Down

0 comments on commit b4cc266

Please sign in to comment.