# nLab coinductive type

foundations

## Foundational axioms

foundational axiom

# Contents

## Idea

The notion of coinductive types is dual to that of inductive types.

## Properties

### Categorical semantics

Where the categorical semantics of an inductive type is an initial algebra for an endofunctor, the semantics of a coinductive type is a terminal coalgebra of an endofunctor.

### Coinductive type formation in homotopy type theory

There is an obstacle to the complete dualization of the usual rules for inductive types in homotopy type theory, including dualizing the induction principle to a “coinduction principle”. For this some form of “codependent types” would be needed.

The universal property defining (internal) coinductive types in HoTT is dual to the one defining (internal) inductive types. One might hence assume that their existence is equivalent to a set of type-theoretic rules dual (in a suitable sense) to those given for external W-types… However, the rules for external W-types cannot be dualized in a naïve way, due to some asymmetry of HoTT related to dependent types as maps into a “type of types” (a universe) (ACS15)

However, it is possible instead to dualize the alternative characterization of inductive types as initial algebras for a notion of coinductive types as terminal coalgebras, and that can be formulated (and often constructed) in ordinary HoTT. In (ACS15) the authors proceed to construct coinductive types from indexed inductive types.

## References

• Benedikt Ahrens, Paolo Capriotti, Régis Spadotti, Non-wellfounded trees in Homotopy Type Theory, (arXiv:1504.02949)

• coinductives, discussion on Homotopy Type Theory group.

Revised on February 27, 2018 08:30:21 by David Corfield (147.147.173.215)