Polynomial functor (type theory)

In type theory, a polynomial functor (or container functor) is a kind of endofunctor of a category of types that is intimately related to the concept of inductive and coinductive types. Specifically, all W-types (resp. M-types) are (isomorphic to) initial algebras (resp. final coalgebras) of such functors.

Polynomial functors have been studied in the more general setting of a pretopos with Σ-types,[1] this article deals only with the applications of this concept inside the category of types of a Martin-Löf style type theory.

Definition

Let U be a universe of types, let A : U, and let B : A → U be a family of types indexed by A. The pair (A, B) is sometimes called a signature[2] or a container.[3] The polynomial functor associated to the container (A, B) is defined as follows:[4][5][6]

$${\displaystyle {\begin{aligned}P:U&\longrightarrow U\\X&\longmapsto \sum _{a:A}(B(a)\to X)\end{aligned}}.}$$

Any functor naturally isomorphic to P is called a container functor.[7] The action of P on functions is defined by

$${\displaystyle {\begin{aligned}P^{*}:(X\to Y)&\longrightarrow (P^{*}X\to P^{*}Y)\\(f,(a,g))&\longmapsto (a,g\circ f)\end{aligned}}.}$$

Note that this assignment is not only truly functorial in extensional type theories (see #Properties).

Properties

In intensional type theories, such functions are not truly functors, because the universe type is not strictly a category (the field of homotopy type theory is dedicated to exploring how the universe type behaves more like a higher category). However, it is functorial up to propositional equalities, that is, the following identity types are inhabited:

$${\displaystyle {\begin{aligned}P^{*}(f\circ g)&=P^{*}f\circ P^{*}g\\P^{*}({\mathsf {id}}_{X})&={\mathsf {id}}_{PX}\end{aligned}}.}$$

for any functions f and g and any type X, where ${\displaystyle {\mathsf {id}}_{X}}$ is the identity function on the type X.[8]

Inline citations

1. Moerdijk, Ieke; Palmgren, Erik (2000). "Wellfounded trees in categories". Annals of Pure and Applied Logic. 104 (1–3): 189–218. doi:10.1016/s0168-0072(00)00012-9. hdl:2066/129036.
2. Ahrens, Definition 1. sfn error: no target: CITEREFAhrens (help)
3. Abbot, p. 4. sfn error: no target: CITEREFAbbot (help)
4. Univalent Foundations Program 2013, Equation 5.4.6.
5. Ahrens, Definition 2. sfn error: no target: CITEREFAhrens (help)
6. Awodey 2012, p. 8. sfn error: no target: CITEREFAwodey2012 (help)
7. Abbot, p. 10. sfn error: no target: CITEREFAbbot (help)
8. Awodey 2015. sfn error: no target: CITEREFAwodey2015 (help)

References

• Univalent Foundations Program (2013). Homotopy Type Theory: Univalent Foundations of Mathematics. Institute for Advanced Study. p. 159.
• Awodey, Steve; Gambino, Nicola; Sojakova, Kristina (2012-01-18). "Inductive types in homotopy type theory". arXiv:1201.3898 [math.LO].
• Awodey, Steve; Gambino, Nicola; Sojakova, Kristina (2015-04-21). "Homotopy-initial algebras in type theory". arXiv:1504.05531 [math.LO].
• Ahrens, Benedikt; Capriotti, Paolo; Spadotti, Régis (2015-04-12). "Non-wellfounded trees in Homotopy Type Theory". arXiv:1504.02949. doi:10.4230/LIPIcs.TLCA.2015.17. Cite journal requires |journal= (help)
• Abbott, Michael; Altenkirch, Thorsten; Ghani, Neil (2005). "Containers: Constructing strictly positive types". Theoretical Computer Science. 342 (1): 4. doi:10.1016/j.tcs.2005.06.002.

External links

• An extensive collection of Notes on Polynomial Functors