nLab
coinductive type

Skip the Navigation Links | Home Page | All Pages | Recently Revised | Authors | Feeds | Export |

Context

Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-level 1-type/h-prop
proofgeneralized elementprogram
conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator monadmodal type theory, monad (in computer science)

homotopy levels

semantics

Deduction and Induction

deductive reasoning, deduction

inductive reasoning

Foundations

foundations

The basis of it all

Foundational axioms

foundational axiom

Removing axioms

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.

Created on October 4, 2012 23:54:26 by Urs Schreiber (82.169.65.155)
Edit | Views: Print | TeX | Source | Linked from: predicative mathematics, natural numbers object, terminal coalgebra of an endofunctor, initial algebra of an endofunctor, corecursion, coinduction, free monoid, induction, W-type, internal logic of an (infinity,1)-topos, identity type, calculus of constructions, product type, inductive type, higher inductive type, proof, negative type, dependent sum type, induction - contents, M-type, coinductive definition, inductive definition