The content discusses the use of parametricity, formulated in terms of cohesion, to derive induction principles for inductive and higher inductive types.
The key insights are:
Parametricity can be understood as an essentially modal aspect, connected to the concept of cohesion in category theory. This allows for a more axiomatic and synthetic treatment of parametricity, compared to previous analytic approaches.
By formalizing the notion of "graph types" which capture the idea of a predicate or relation as a type family, the author is able to prove a substitution lemma that serves as the basis for deriving induction principles.
The author demonstrates this approach by deriving induction principles for the natural numbers (N) and the circle (S1) higher inductive type, showing how the complexity of coherence conditions can be managed using parametricity.
The formalization is carried out in Agda, with careful attention paid to ensuring computational soundness and canonicity of the resulting definitions and theorems.
The overall contribution is a unifying framework for understanding and applying parametricity, with applications in solving problems around the complexity of coherence conditions in homotopy type theory and related fields.
Egy másik nyelvre
a forrásanyagból
arxiv.org
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