|logic||category theory||type theory|
|true||terminal object/(-2)-truncated object||h-level 0-type/unit type|
|false||initial object||empty type|
|proposition||(-1)-truncated object||h-proposition, mere proposition|
|cut rule||composition of classifying morphisms / pullback of display maps||substitution|
|cut elimination? for implication||counit for hom-tensor adjunction||beta reduction|
|introduction rule for implication||unit for hom-tensor adjunction||eta conversion|
|disjunction||coproduct ((-1)-truncation of)||sum type (bracket type of)|
|implication||internal hom||function type|
|negation||internal hom into initial object||function type into empty type|
|universal quantification||dependent product||dependent product type|
|existential quantification||dependent sum ((-1)-truncation of)||dependent sum type (bracket type of)|
|equivalence||path space object||identity type|
|equivalence class||quotient||quotient type|
|induction||colimit||inductive type, W-type, M-type|
|higher induction||higher colimit||higher inductive type|
|completely presented set||discrete object/0-truncated object||h-level 2-type/preset/h-set|
|set||internal 0-groupoid||Bishop set/setoid|
|universe||object classifier||type of types|
|modality||closure operator, (idemponent) monad||modal type theory, monad (in computer science)|
|linear logic||(symmetric, closed) monoidal category||linear type theory/quantum computation|
|proof net||string diagram||quantum circuit|
|(absence of) contraction rule||(absence of) diagonal||no-cloning theorem|
|synthetic mathematics||domain specific embedded programming language|
|type theory||category theory|
|natural deduction||universal construction|
|function type||internal hom|
Like any type constructor in type theory, a function type is specified by rules saying when we can introduce it as a type, how to construct terms of that type, how to use or “eliminate” terms of that type, and how to compute when we combine the constructors with the eliminators.
The type formation rule to build a function type is easy:
Function types are almost always defined as a negative type. In this presentation, primacy is given to the eliminators. The natural eliminator of a function type says that we can apply it to any input:
The constructor is then determined as usual for a negative type: to construct a term of , we have to specify how it behaves when applied to any input. In other words, we should have a term of type containing a free variable of type . This yields the usual “-abstraction” constructor:
The β-reduction rule is the obvious one (the ur-example of all -reductions), saying that when we evaluate a -abstraction, we do it by substituting for the bound variable.
If we want an η-conversion rule, the natural one says that every function is a -abstraction:
It is also possible to present function types as a positive type. However, this requires a stronger metatheory, such as a logical framework. We use the same constructor (-abstraction), but now the eliminator says that to define an operation using a function, it suffices to say what to do in the case that that function is a lambda abstraction.
This rule cannot be formulated in the usual presentation of type theory, since it involves a “higher-order judgment”: the context of the term involves a “term of type containing a free variable of type ”. However, it is possible to make sense of it. In dependent type theory, we need additionally to allow to depend on .
The natural -reduction rule for this eliminator is
and its -conversion rule looks something like
Here is a term containing a free variable of type . By substituting for , we obtain a term of type which depends on “a term containing a free variable ”. We then apply the positive eliminator at , and the -rule says that this can be computed by just substituting for in .
As usual, the positive and negative formulations are equivalent in a suitable sense. They have the same constructor, while we can formulate the eliminators in terms of each other:
The conversion rules also correspond.
In dependent type theory, this definition of only gives us a properly typed dependent eliminator if the negative function type satisfies -conversion. As usual, if it satisfies propositional eta-conversion then we can transport along that instead—and conversely, the dependent eliminator allows us to prove propositional -conversion. This is the content of Propositions 3.5, 3.6, and 3.7 in (Garner).
In logic, functions types express implication. More precisely, for two propositions, under propositions as types the implication is the function type (or rather the bracket type of that if one wishes to force this to be of type again ).
A textbook account in the context of programming languages is in section III of