nLab function type

Idea

In type theory a function type $X \to Y$ for two types $X,Y$ is the type of functions from $X$ to $Y$.

In a model of the type theory in categorical semantics, this is an exponential object. In set theory, it is a function set. In dependent type theory, it is a special case of a dependent product type.

Overview

	type theory	category theory
	syntax	semantics
	natural deduction	universal construction
	function type	internal hom
type formation	$\frac{\vdash\: X \colon Type \;\;\;\;\; \vdash\; A\colon Type}{\vdash \; \left(X \to A\right) \colon Type}$
term introduction	$\frac{x \colon X \;\vdash\; a(x) \colon A}{\vdash (x \mapsto a\left(x\right)) \colon \left(X \to A\right) }$
term elimination	$\frac{\vdash\; f \colon \left(X \to A\right)\;\;\;\; \vdash \; x \colon X}{\;\;\;\vdash\; f(x) \colon A}$
computation rule	$(y \mapsto a(y))(x) = a(x)$

Definition

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:

$$\frac{A\colon Type \qquad B \colon Type}{A\to B\colon Type}$$

As a negative type

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:

$$\frac{f\colon A\to B \qquad a\colon A}{f(a) \colon B}$$

The constructor is then determined as usual for a negative type: to construct a term of $A\to B$, we have to specify how it behaves when applied to any input. In other words, we should have a term of type $B$ containing a free variable of type $A$. This yields the usual “$\lambda$-abstraction” constructor:

$$\frac{x\colon A\vdash b\colon B}{\lambda x.b\colon A\to B}$$

The ∞-reduction rule is the obvious one (the ur-example of all $\beta$-reductions), saying that when we evaluate a $\lambda$-abstraction, we do it by substituting for the bound variable.

$$(\lambda x.b)(a) \;\to_\beta\; b[a/x]$$

If we want an ∞-conversion rule, the natural one says that every function is a $\lambda$-abstraction:

$$\lambda x.f(x) \;\to_\eta\; f$$

As a positive type

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 ($\lambda$-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.

$$\frac{(x\colon A \vdash b\colon B) \vdash c\colon C \qquad f\colon A\to B}{funsplit(c,f)\colon C}$$

This rule cannot be formulated in the usual presentation of type theory, since it involves a “higher-order judgment”: the context of the term $c\colon C$ involves a “term of type $B$ containing a free variable of type $A$”. However, it is possible to make sense of it. In dependent type theory, we need additionally to allow $C$ to depend on $A\to B$.

The natural $\beta$-reduction rule for this eliminator is

$$funsplit(c, \lambda x.g) \;\to_\beta c[g/b]$$

and its $\eta$-conversion rule looks something like

$$funsplit(c[\lambda x.b / g], f) \;\to_\eta\; c[f/g].$$

Here $g\colon A\to B \vdash c\colon C$ is a term containing a free variable of type $A\to B$. By substituting $\lambda x.b$ for $g$, we obtain a term of type $C$ which depends on “a term $b\colon B$ containing a free variable $x\colon A$”. We then apply the positive eliminator at $f\colon A\to B$, and the $\eta$-rule says that this can be computed by just substituting $f$ for $g$ in $c$.

Positive versus negative

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:

$$\begin{aligned} f(a) &\coloneqq funsplit(b[a/x], f)\\ funsplit(c, f) &\coloneqq c[f(x)/b] \end{aligned}$$

The conversion rules also correspond.

In dependent type theory, this definition of $funsplit$ only gives us a properly typed dependent eliminator if the negative function type satisfies $\eta$-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 $\eta$-conversion. This is the content of Propositions 3.5, 3.6, and 3.7 in (Garner).

As a special case of the dependent product

In dependent type theory a function type $A \to B$ is the special case the dependent product over $a : A$ for the special case that $B$ is regarded as an $A$-dependent type that actually happens to be $A$-independent. The rules are given as follows:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B \; \mathrm{type} \quad \Gamma, a:A, b:A \vdash B[a/x] \equiv B[b/x] \; \mathrm{type}}{\Gamma \vdash A \to B \; \mathrm{type}}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B \; \mathrm{type} \quad \Gamma, a:A, b:A \vdash B[a/x] \equiv B[b/x] \; \mathrm{type}}{\Gamma \vdash A \to B \equiv \prod_{x:A} B\; \mathrm{type}}$$

In categorical semantics this is the statement that a section of a product projection $A \times B \to A$ is equivalently just a morphism $A \to B$.

As types of anafunctions

In dependent type theory, in the same way that one could define equivalence types as types of one-to-one correspondences, one could also define function types as types of anafunctions. This requires both identity types and heterogeneous identity types being defined first, which we shall write as $a =_A b$ and $x =_{B}^{p} y$ respectively for $a:A$, $b:A$, $p:a =_A b$, $x:B(a)$, and $y:B(b)$.

Rules for function types

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash A \to B \; \mathrm{type}} \qquad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, f:A \to B, x:A, y:B \vdash \mathcal{F}_{A, B}(f, x, y) \; \mathrm{type}}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash f(x):B}{\Gamma \vdash x \mapsto f(x):A \to B}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash f(x):B}{\Gamma, x:A \vdash \alpha(x):\mathcal{F}_{A, B}(x \mapsto f(x), x, f(x))}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, f:A \to B, x:A \vdash \mathrm{ev}(f, x):B}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, f:A \to B, x:A \vdash \beta(f, x):\mathcal{F}_{A, B}(f, x, \mathrm{ev}(f, x))}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, f:A \to B, x:A, y:B, u:\mathcal{F}_{A, B}(f, x, y) \vdash \kappa(f, x, y, u):\mathrm{ev}(f, x) =_B y}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, f:A \to B, x:A, y:B, u:\mathcal{F}_{A, B}(f, x, y) \vdash \eta(f, x, y, u):\beta(f, x) =_{\mathcal{F}_{A, B}(f, x)}^{\kappa(f, x, y, u)} u}$$

By the rules for dependent sum types and dependent product types, one could derive that

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, f:A \to B \vdash \eta(f):\prod_{x:A} \mathrm{isContr}\left(\sum_{y:B} \mathcal{F}_{A, B}(f, x, y) \right)}$$

which is precisely the statement that $\mathcal{F}_{A, B}(f)$ is an anafunction for all functions $f:A \to B$.

Properties

Relation to dependent product types

A function type is the special case of a dependent product type for the case where the dependent type does not actually depend.

$$(X \to A) = \prod_{x \colon X} A \,.$$

See also at function monad.

Application in logic

In logic, functions types express implication. More precisely, for $\phi, \psi$ two propositions, under propositions as types the implication $\phi \Rightarrow \psi$ is the function type $\phi \to \psi$ (or rather the bracket type of that if one wishes to force this to be of type $Prop$ again ).

Related concepts

• function application

• function monad

• dependent product type

• equivalence type, embedding type

• partial function type

• lambda calculus

• implication, linear implication

• extension type

References

A textbook account in the context of programming languages is in section III of

• Robert Harper, Practical Foundations for Programming Languages

See also

• Richard Garner, On the strength of dependent products in the type theory of Martin-Löf.