Understanding Subdistinctiveness in Type Theory and Homotopy Type Theory
In the context of type theory and homotopy type theory, a notion that has been introduced by Vladimir Voevodsky and his collaborators is the concept of "subdistinctiveness".
Roughly speaking, the distinctiveness of a type is a measure of how much the type stands out from other types in the sense that it has a lot of structure that is not shared with other types. For example, the type `Nat` (natural numbers) is highly distinctive because it has a lot of structure that is not shared with other types, such as the fact that it is a linear order and that it has a successor function.
On the other hand, the type `Set` (sets) is less distinctive because it does not have as much structure that is not shared with other types. In fact, `Set` is often considered to be a "universal" type in the sense that it can be used to encode any other type, which means that it does not have as much structure that is unique to itself.
The subdistinctiveness of a type is a measure of how much the type is like other types in the sense that it has less structure that is not shared with other types. For example, the type `Fin Nat` (finite natural numbers) is less distinctive than `Nat` because it has fewer structures that are not shared with other types. In fact, `Fin Nat` can be considered to be a "special case" of `Nat` in the sense that it is a subset of `Nat` and it has fewer elements.
The subdistinctiveness of a type can be measured using a variety of methods, such as the size of the type, the number of structure that the type has, etc. For example, the type `Fin Nat` is less distinctive than `Nat` because it has a smaller size (it only contains the finite natural numbers) and it has fewer structures (it does not have a successor function).
In general, the concept of subdistinctiveness is useful for understanding the relationships between different types in a type theory, and it can be used to reason about the properties of types and their relationships to other types. For example, one can use the concept of subdistinctiveness to prove that certain types are "essentially" the same as other types, or to show that certain types are "essentially" distinct from other types.
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