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[Merged by Bors] - feat: CostructuredArrow is locally small #6388

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@TwoFX TwoFX commented Aug 5, 2023


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@TwoFX TwoFX added awaiting-review The author would like community review of the PR easy < 20s of review time. See the lifecycle page for guidelines. t-category-theory Category theory labels Aug 5, 2023
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bors merge

@github-actions github-actions bot added ready-to-merge This PR has been sent to bors. and removed awaiting-review The author would like community review of the PR labels Aug 6, 2023
bors bot pushed a commit that referenced this pull request Aug 6, 2023
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@bors bors bot changed the title feat: CostructuredArrow is locally small [Merged by Bors] - feat: CostructuredArrow is locally small Aug 6, 2023
@bors bors bot closed this Aug 6, 2023
@bors bors bot deleted the locallysmall-costructuredarrow branch August 6, 2023 03:13
bors bot pushed a commit that referenced this pull request Aug 28, 2023
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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