[Merged by Bors] - refactor: move morphisms in StructuredArrow to a lower universe #6397
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I also believe this is a nice improvement. Thanks! bors merge |
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This PR changes the definition
```lean
abbrev fromPUnit (X : C) : Discrete PUnit.{v + 1} ⥤ C :=
(Functor.const _).obj X
```
to
```lean
abbrev fromPUnit (X : C) : Discrete PUnit.{w + 1} ⥤ C :=
(Functor.const _).obj X
```
and then redefines
```lean
def StructuredArrow (S : D) (T : C ⥤ D) :=
Comma (Functor.fromPUnit S) T
```
to
```lean
def StructuredArrow (S : D) (T : C ⥤ D) :=
Comma (Functor.fromPUnit.{0} S) T
```
The effect of this is that given `{C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₂} D] (S : D) (T : C ⥤ D)`, the morphisms of `StructuredArrow S T` no longer live in `max v₁ v₂`, but in `v₁`, as they should. Thus, this PR is basically a better version of #6388.
My guess for why we used to have the larger universe is that back in the day, the universes for limits were much more restricted, so stating the results of `Limits/Comma.lean` was not possible if the morphisms of `StructuredArrow` live in `v₁`. Luckily, by now it is possible to state everything correctly, and this PR adjusts `Limits/Comma.lean` and `Limits/Over.lean` accordingly.
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This PR changes the definition
abbrev fromPUnit (X : C) : Discrete PUnit.{v + 1} ⥤ C := (Functor.const _).obj Xto
abbrev fromPUnit (X : C) : Discrete PUnit.{w + 1} ⥤ C := (Functor.const _).obj Xand then redefines
to
The effect of this is that given
{C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₂} D] (S : D) (T : C ⥤ D), the morphisms ofStructuredArrow S Tno longer live inmax v₁ v₂, but inv₁, as they should. Thus, this PR is basically a better version of #6388.My guess for why we used to have the larger universe is that back in the day, the universes for limits were much more restricted, so stating the results of
Limits/Comma.leanwas not possible if the morphisms ofStructuredArrowlive inv₁. Luckily, by now it is possible to state everything correctly, and this PR adjustsLimits/Comma.leanandLimits/Over.leanaccordingly.