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Polynomial functor (type theory)

Polynomial functor (type theory) is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Polynomial functor (type theory) rather than just read about it. In short: 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.

Key takeaways

Polynomial functor (type theory) belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Polynomial functor (type theory) to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Polynomial functor (type theory) from memory before moving on to harder problems.

Reference excerpt

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; 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 or a container. The polynomial functor associated to the container (A, B) is defined as follows:

P : U ⟶ U X ⟼ ∑ a : A ( B ( a ) → X ) {\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. The action of P on functions is defined by

P : ( X → Y ) ⟶ ( P X → P Y ) f ⟼ ( ( a , g ) ↦ ( a , g ∘ f ) ) {\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).

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:

P ( f ∘ g ) = P f ∘ P g P ( i d X ) = i d P X {\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 i d X {\displaystyle {\mathsf {id}}_{X}} is the identity function on the type X.

Inline citations

References 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. {{cite journal}}: Cite uses deprecated parameter |citeseerx= (help) 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].

External links An extensive collection of Notes on Polynomial Functors

Worked examples

Example 1 — a first encounter with Polynomial functor (type theory)

Start with the simplest possible case. Write down what Polynomial functor (type theory) claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Polynomial functor (type theory) before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Polynomial functor (type theory) ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Polynomial functor (type theory)

In research: Polynomial functor (type theory) appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Polynomial functor (type theory) in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Polynomial functor (type theory) is common in secondary-school and first-year university syllabi. It links to neighbouring topics Type theory, so understanding it makes those chapters shorter.
In everyday life: Look for Polynomial functor (type theory) outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Polynomial functor (type theory) in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Polynomial functor (type theory) means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Polynomial functor (type theory) out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Polynomial functor (type theory) in simple terms?

Why does Polynomial functor (type theory) matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Polynomial functor (type theory)?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Polynomial functor (type theory).