natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory
Extensional type theory denotes the flavor of type theory in which identity types are demanded to be propositions / of h-level 1. In other words, they are determined by their extensions — the collection of pairs of points which are equal. Type theory which is not extensional is called intensional type theory.
The word “extensional” in type theory (even when applied to identity types) sometimes refers instead to the axiom of function extensionality. In general this property is orthogonal to the one considered here: function extensionality can hold or fail in both extensional and intensional type theory. In particular, homotopy type theory is intensional in that identity types are crucially not demanded to be propositions, but function extensionality is often assumed (in terms of these intensional identity types, of course) — in particular, it follows from the univalence axiom.
The basic definition of identity types as an inductive type family makes them intensional. There are two sorts of ways to make them extensional: definitionally or propositionally.
In a definitionally extensional type theory, any inhabitant of an identity type $p:Id_A(x,y)$ induces a definitional equality between $x$ and $y$. In other words, we have an “equality reflection rule” of the form
At first, this may appear to be only a “skeletality” assumption, since it does not assert explicitly that $p$ is reflexivity rather than a nontrivial loop. However, we can derive this with the induction rule for identity types. Consider the dependent type
This is well-typed because the reflection rule applied to $p$ yields a definitional equality $x\equiv y$, so that we have $refl(x):Id_A(x,y)$. Moreover, substituting $x$ for $y$ and $refl(x)$ for $p$ yields the type $Id_{Id_A(x,x)}(refl(x),refl(x))$, which is inhabited by $refl(refl(x))$.
Thus, by induction on identity, we have a term in the above type, witnessing a propositional equality between $p$ and $refl(x)$. Finally, applying the equality reflection rule again, we get a definitional equality $p\equiv refl(x)$.
On the other hand, if in addition to the equality reflection rule we postulate that all equality proofs are definitionally equal to reflexivity, then we can derive the induction rule for identity types. Definitionally extensional type theory is often presented in this form.
A different, also equivalent, way of presenting definitionally extensional type theory is with a definitional eta-conversion rule for the identity types; see here.
In a propositionally extensional type theory, we still distinguish definitional and propositional equality, but no two terms can be propositionally equal in more than one way (up to propositional equality). In the language of homotopy type theory, this means that all types are h-sets. There are a number of equivalent ways to force this to be true by adding axioms? to type theory.
We can add as an axiom the “uniqueness of identity proofs” (axiom UIP) property that any two inhabitants of the same identity type $Id_A(x,y)$ are themselves equal (in the corresponding higher identity type).
We can add Streicher’s axiom K which says that any inhabitant of a self-equality type $Id_A(x,x)$ is (propositionally) equal to the identity/reflexivity equality $1_x$. (Axiom K is so named because $K$ comes after $J$, and $J$ usually denotes the eliminator for identity types.)
In the presence of dependent sum types, we can add an axiom saying that if $(a,b_1)$ and $(a,b_2)$ are pairs in a dependent sum $\sum_{x\colon A} B(x)$ with the same first component, and the identity type $Id_{\sum_{x\colon A} B(x)}((a,b_1), (a,b_2))$ is inhabited, then so is $Id_a(b_1,b_2)$.
We can allow definition of functions by a strong form of pattern matching, as in unmodified Agda. The relevant “extra strength” is to allow output types of a pattern match which are only well-defined after the pattern has been matched.
Propositionally extensional type theory has some of the attributes of intensional type theory, and many type theorists use “extensional type theory” to refer only to the definitional version.
Only the intensional, but not the extensional, Martin-Löf type theory is decidable. See intensional type theory for more on this.
Homotopy Type System is dependent type theory with a mix of extensional and intensional identity types.
Among the most thorough recent treatments of extensional type theory are
Ieke Moerdijk, E. Palmgren, Wellfounded trees in categories. Ann. Pure Appl. Logic, 104:189-218 (2000)
Ieke Moerdijk, E. Palmgren, Type theories, toposes and constructive set theory: predicative aspects of AST, Ann. Pure Appl. Logic, 114:155-201, (2002)