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.

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

${\displaystyle {\begin{aligned}P:(X\to Y)&\longrightarrow (PX\to PY)\\f\qquad &\longmapsto \left((a,g)\mapsto (a,g\circ f)\right)\end{aligned}}}$

Note that this assignment is only truly functorial in extensional type theories (see #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:

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

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

• Abbott, Michael; Altenkirch, Thorsten; Ghani, Neil (2005). "Containers: Constructing strictly positive types". Theoretical Computer Science. 342 (1): 4. CiteSeerX 10.1.1.166.34. doi:10.1016/j.tcs.2005.06.002.
• Ahrens, Benedikt; Capriotti, Paolo; Spadotti, Régis (2015-04-12). Non-wellfounded trees in Homotopy Type Theory. Leibniz International Proceedings in Informatics (LIPIcs). Vol. 38. pp. 17–30. arXiv:1504.02949. doi:10.4230/LIPIcs.TLCA.2015.17. ISBN 9783939897873. S2CID 15020752.
• 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].

• An extensive collection of Notes on Polynomial Functors