W-types

Idea

In type theory, by a $\mathcal{W}$-type [Martin-Löf (1982), pp. 171, (1984), pp. 43] one means a type which is defined inductively in a $\mathcal{W}$ell-founded way based on a type $C$ of “constructors” and a type of $A$ of “arities”. As such, $\mathcal{W}$-types are special kinds of inductive types (see below).

The same terminology “$\mathcal{W}$-type” is used [Moerdijk & Palmgren (2000)] for objects in (suitable pre-)toposes which provide categorical semantics for the type-theoretic notion (see further below).

Concretely, the terms/elements of a $\mathcal{W}$-type may be thought of as “rooted well-founded trees” with a certain branching type; different $\mathcal{W}$-types are distinguished by their branching signatures: In categorical semantics in Sets, a branching signature is represented by a family of sets $\{A_c\}_{c \in C}$ such that

• each node of the tree is labeled with an element $c \colon C$ – referring to a constructor;

• if a node is labeled by $c$ then it has exactly ${|A_c|}$ outgoing edges, each labeled by some $a \colon A_c$ – the arity of the constructor $c$.

If one discards the requirement that the trees be well-founded, then the notion of $\mathcal{W}$-type becomes that of a coinductive type called an M-type (presumably since “M” is like a “W” upside down).

In practice, $\mathcal{W}$-types are used to:

1. provide a constructive counterpart of the classical notion of a well-ordering

2. uniformly define a variety of well-behaved inductive types.

More complex inductive types, with multiple constructors that are assumed only to be strictly positive, can be reduced to $\mathcal{W}$-types, at least in the presence of other structure such as sum types and function extensionality; see for instance Abbott, Altenkirch & Ghani (2004). This can even be extended to inductive families.

In most set theoretic mathematical foundations, $\mathcal{W}$-types can be proven to exist, but in predicative mathematics and in type theory, where this is not the case, they are often introduced axiomatically (as usual for inductive types more generally).

Definition

There are two slightly different formulations of W-types:

1. $\mathcal{W}$-types in type theory

2. $\mathcal{W}$-types in categories

$\mathcal{W}$-types in type theory

(due to Martin-Löf (1984), pp. 43; streamlined review e.g. in Awodey, Gambino & Sojakova (2012, §2))

$\mathcal{W}$-types in categories

Plain version

We discuss the categorical semantics of $\mathcal{W}$-types, due to Moerdijk & Palmgren (2000).

Here one describes a $\mathcal{W}$-type as an initial algebra for an endofunctor on an ambient category $\mathcal{C}$. The family $\{A_c\}_{c\in C}$ can be thought of as a morphism

in some category $\mathcal{C}$ (such that the fiber over $c\in C$ is $A_c$), and the endofunctor in question is the composite

where

• $A^*$ denotes context extension, hence the pullback functor (a.k.a. $A\times -$);

• $\Pi_f$ denotes the interpretation of the dependent product, i.e. the right base change along $f$;

• $\Sigma_C$ the denotes the interpretation of the dependent sum, i.e. the left base change given by the forgetful functor from $\mathcal{C}_{/C}$ to $\mathcal{C}$.

Equivalently, it is the composite

where $\bigcirc_f$ is the reader monad $\Pi_f \circ f^*$. In other words, the dependent product is not actually dependent.

Such a composite is called a polynomial endofunctor. Explicitly, it is the functor $X \mapsto \sum_{c : C} X^{A_c}$.

This definition makes sense in any locally cartesian closed category, although the W-type (the initial algebra) may or may not exist in any given such category. (A non-elementary construction of them is given by the transfinite construction of free algebras.)

The above definition is most useful when the category $\mathcal{C}$ is not just locally cartesian closed but is a Π-pretopos, since often we want to use at least coproducts in constructing $A$ and $C$. For example, a natural numbers object (Exp. ) is a $\mathcal{W}$-type specified by one of the coproduct inclusions $1\to 1+1$, and the list object $L X$ is a $\mathcal{W}$-type specified by $X\to X+1$. More generally, endofunctors that look like polynomials in the traditional sense:

can be constructed as polynomial endofunctors in the above sense for any lextensive category, and hence in any $\Pi$-pretopos. A $\Pi$-pretopos in which all W-types exist is called a ΠW-pretopos.

In a topos with a natural numbers object (NNO), W-types for any polynomial endofunctor can be constructed as certain sets of well-founded trees; thus every topos with a NNO is a ΠW-pretopos. This applies in particular in the topos Set (unless one is a predicativist, in which case $Set$ is not a topos and W-types in it must be postulated explicitly).

Dependent $\mathcal{W}$-types

The above has a natural generalization to dependent or indexed W-types (Gambino & Hyland (2004)) with a type $C$ of indices:

given a diagram of the form

there is the dependent polynomial functor

This reduces to the above for $C = \ast$ the terminal object.

Notice that we do not necessarily have $g f = h$, so this is not just a polynomial endofunctor of $\mathcal{C}/_{C}$ considered as a lccc in its own right. If we do have $g f = h$, then $C$ is called a type of parameters instead of indices.

Examples

[Martin-Löf (1984), pp. 45, Dybjer (1997, p. 330, 333)]

Properties

Related concepts

• M-type

• predicative topos

References

General

The original definition in type theory is due to

• Per Martin-Löf, pp. 171 of: Constructive Mathematics and Computer Programming, in: Proceedings of the Sixth International Congress of Logic, Methodology and Philosophy of Science (1979), Studies in Logic and the Foundations of Mathematics 104 (1982) 153-175 $[$doi:10.1016/S0049-237X(09)70189-2, ISBN:978-0-444-85423-0$]$

• Per Martin-Löf (notes by Giovanni Sambin), pp. 43 of: Intuitionistic type theory, Lecture notes Padua 1984, Bibliopolis, Napoli (1984) [pdf, pdf]

In homotopy type theory:

• Jasper Hugunin, Why Not W?, Leibniz International Proceedings in Informatics (LIPIcs) 188 (2021) [doi:10.4230/LIPIcs.TYPES.2020.8, pdf]

On the h-level of W-types:

• Nils Anders Danielsson, Positive h-levels are closed under W (2012) [web]

• Jasper Hugunin, IWTypes, https://github.com/jashug/IWTypes

which also computes the identity types of W-types (and more generally indexed W-types).

W-types in Coq: wiki