[Merged by Bors] - feat(CategoryTheory): functors that are dense at an object

[joelriou]

Given a functor F : C ⥤ D and Y : D, we say that F is dense at Y if Y identifies to the colimit of all F.obj X for X : C and f : F.obj X ⟶ Y, i.e. the obvious natural transformation makes the identity functor of D a pointwise left Kan extension of F along F at Y.

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[github-actions]

PR summary 74170ab45e

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.CategoryTheory.Functor.KanExtension.DenseAt (new file)	466

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Declarations diff

+ DenseAt
+ DenseAt.ofIso
+ DenseAt.ofNatIso
+ DenseAt.postcompEquivalence
+ DenseAt.precompEquivalence
+ congr_isDenseAt
+ instance : E.isPointwiseLeftKanExtensionAt.IsClosedUnderIsomorphisms
+ instance : E.isPointwiseRightKanExtensionAt.IsClosedUnderIsomorphisms
+ instance : F.isDenseAt.IsClosedUnderIsomorphisms := by
+ isDenseAt
+ isDenseAt_eq_isPointwiseLeftKanExtensionAt
+ isPointwiseLeftKanExtensionAt
+ isPointwiseRightKanExtensionAt
++ hom_ext'

You can run this locally as follows

## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for script/declarations_diff.sh contains some details about this script.

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No changes to technical debt.

You can run this locally as

./scripts/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[robin-carlier]

Will review more thoroughly later, but I really think that having "canonical" in the naming is a recipe for disaster in terms of understanding and should be avoided at all cost. I don’t believe this is standard terminology at all.

Given that this is used to define dense functors, maybe F.DenseAt Y would be a more reasonable way to name this?

I am also rather worried by the fact that the design here goes with ad hoc definitions and does not even mention (not even in docstrings!) the link with Kan extensions.

The definition here is that the (left extension defined by the) inverse right unitor F.rightUnitor.inv exhibits 𝟭 D as a pointwise left Kan extension of F along itself at Y (Kerodon phrases the definition of dense functors in that language as well).

The ad hoc definitions here means that we’ll be having more than one way of saying the same thing, and means that depending on how many properties for dense functors we want, we may have to duplicate API etc.

In fact, replacing your definition by

abbrev CanonicalColimit (Y : D) : Type _ := 
  (Functor.LeftExtension.mk (𝟭 D) F.rightUnitor.inv).IsPointwiseLeftKanExtensionAt Y

makes the proof of hom_ext go through.

Given that

def canonicalCoconeIso (Y : D) :
    canonicalCocone F Y ≅ (Functor.LeftExtension.mk (𝟭 D) F.rightUnitor.inv).coconeAt Y :=
  Cocones.ext (.refl _) (by cat_disch)

also goes through, I think it’s safe to say that we don’t lose a lot by taking Kan extensions as a definition here (and we gain the link with a well-established API).

Hence I also think the file here belongs in the Functor/KanExtension subfolder.

[joelriou]

Thanks! Indeed, I should have made the connection with Kan extensions! The name "canonical colimit" is used throughout the book by Adámek and Rosický, but indeed, Functor.DenseAt looks better.

[robin-carlier]

Thanks!

maintainer delegate

[github-actions]

🚀 Pull request has been placed on the maintainer queue by robin-carlier.

[joelriou]

Thanks @robin-carlier for the review!

[jcommelin]

Thanks 🎉

bors merge

[mathlib-bors]

Pull request successfully merged into master.

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