Tradeoff of defining types as subobjects

posts/tradeoff-of-defining-types-as-subobjects.md

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+---
+author: 'Yaël Dillies'
+category: 'Design patterns'
+date: 2025-12-10 14:00:00 UTC+02:00
+description: 'Comparison of the different ways to define types from existing supertypes'
+has_math: true
+link: ''
+slug: tradeoff-of-defining-types-as-subobjects
+tags: ''
+title: Tradeoffs of defining types as subobjects
+type: text
+---
+
+It often happens in formalisation that a type of interest is a subobject of another type of interest.
+For example, the unit circle in the complex plane is naturally a submonoid[^1].
+
+What is the best way of defining this unit circle on top of `Complex`?
+This blog post examines the pros and cons of the available designs.
+
+<!-- TEASER_END -->
+
+## The available designs
+
+We will illustrate the various designs using the "circle in the complex plane" example.
+
+The first option, most loyal to the description of the unit circle
+as a submonoid of the complex plane, is to simply define the unit circle as a `Submonoid Complex`.
+We will call the "Subobject design".
+
+```lean
+-- Subobject design
+def circle : Submonoid Complex where
+  carrier := {x : Complex | ‖x‖ = 1}
+  one_mem' := sorry
+  mul_mem' := sorry
+```
+
+The second obvious choice would be to define the unit circle as a custom structure.
+We call this the "Custom Structure" design.
+
+```lean
+-- Custom Structure design
+structure Circle : Type where
+  val : Complex
+  norm_val : ‖val‖ = 1
+```
+
+Finally, an intermediate option would be to define it as the coercion to `Type` of a subobject.
+We call this the "Coerced Subobject" design.
+
+```lean
+-- Coerced Subobject design
+def Circle : Type := circle -- possibly replacing `circle` by its definition
+```
+
+## Dot notation
+
+In the Subobject design, `x : circle` really means `x : ↥circle`,
+namely that `x` has type `Subtype _`.
+This means that dot notation `x.foo` resolves to `Subtype.foo x`
+rather than the expected `circle.foo x`.
+
+The Coerced Subobject and Custom Structure designs do not suffer from this (unexpected) behavior.
+
+See [#15021](https://github.com/leanprover-community/mathlib4/pull/15021) for an example
+where switching from the Subobject design (`ProbabilityTheory.kernel`)
+to the Custom Structure design (`ProbabilityTheory.Kernel`) was motivated by access to dot notation.
+
+## Custom named projections
+
+In the Subobject and Coerced Subobject designs, the fields of `↥circle`/`Circle`
+are `val` and `property`, as coming from `Subtype`.
+This leads to potentially uninformative code of the form
+```lean
+-- Subobject design
+def foo : circle where
+  val := 1
+  property := by simp
+
+-- Coerced subobject design
+def bar : Circle where
+  val := 1
+  property := by simp
+```
+
+In the Custom Structure design, projections can be custom-named, leading to more informative code:
+```lean
+def baz : Circle where
+  val := 1
+  norm_val := by simp
+```
+
+## Generic instances
+
+In the Coerced Subobject and Custom Structure designs, generic instances about subobjects
+do not apply and must be copied over manually.
+In the Coerced Subobject design, they can easily be transferred/derived:
+```lean
+def Circle : Type := circle
+deriving TopologicalSpace
+
+instance : MetricSpace Circle := Subtype.metricSpace
+```
+
+In the Custom Structure design, these instances must either
+be copied over manually or transferred using some kind of isomorphism.
+
+The Subobject design, by definition, lets all generic instances apply.
+
+## Conclusion
+
+Here is a table compiling the above discussion.
+
+| Aspect | Subobject | Coerced Subobject | Custom Structure |
+|--|--|--|--|
+| Dot notation | ✗ | ✓ | ✓ |
+| Custom named projections | ✗ | ✗ | ✓ |
+| Generic instances | automatic | easy to transfer | hard to transfer but could be improved |
+
+In conclusion, the Custom Structure design is superior in performance and usability
+at the expense of a one-time overhead cost when transferring generic instances.
+The Subobject design is still viable for types that see little use,
+so as to avoid the instance transferring cost.
+Finally, the Coerced Subobject is a good middle ground
+when having custom named projections is not particularly necessary.
+It could also reasonably be used as an intermediate step
+when refactoring from the Subobject design to the Custom Structure design.
+
+## Further concerns
+
+Subobjects are not the only ones having a coercion to `Type`:
+Concrete categories are categories whose objects can be interpreted as types,
+and there usually is a coercion from such a category to `Type`.
+
+For example,
+[`AlgebraicGeometry.Scheme`](https://leanprover-community.github.io/mathlib4_docs/find/?pattern=AlgebraicGeometry.Scheme#doc)
+sees widespread use in algebraic geometry and
+[`AlgebraicGeometry.Proj`](https://leanprover-community.github.io/mathlib4_docs/find/?pattern=AlgebraicGeometry.Proj#doc)
+is a term of type `Scheme` that is also used as a type.
+
+This raises issues of its own, as `Proj 𝒜` is (mathematically)
+also a smooth manifold for some graded algebras `𝒜`.
+Since terms only have one type, we can't hope to have
+both `Proj 𝒜 : Scheme` and `Proj 𝒜 : SmoothMnfld`[^2].
+One option would be to have a separate `DifferentialGeometry.Proj 𝒜` of type `SmoothMnfld`.
+Another one would be to provide the instances
+saying that `AlgebraicGeometry.Proj 𝒜` is a smooth manifold.
+
+[^1]: Nevermind that it is actually a subgroup. Mathlib currently can't talk about subgroups of fields.
+[^2]: The `SmoothMnfld` category does not exist yet in Mathlib