polynomial monad (changes)

Showing changes from revision #2 to #3:
Added | ~~Removed~~ | ~~Chan~~ged

A *polynomial monad* is a monad whose underlying endofunctor is a polynomial functor polynomial functor. This is of course equivalent to being a monad in the category of polynomial functors.

A basic example is the free-monoid monad, Example 1.9. It is exhibited by the polynomial $1\leftarrow \mathbb{N}^\prime\rightarrow \mathbb{N}\rightarrow 1$ where the middle arrow is such that for all $n\in \mathbb{N}$ its fiber over $n$ has cardinality $n$.

One can construct the free monad on a polynomial endofunctor.

An extensive category $E$ (which in particular has finite sums) has *W-types* iff every polynomial functor in a single variable on $E$ has an initial algebra. The “W” in the name of this notion refers to the fact Martin-Löf’s types of wellfounded trees (translated into category theory) are initial algebras for polynomial endofunctors in a single variable. Initial algebras for (general) polynomial functors correspond to *Petersson-Synek tree types*.

- Nicola Gambino and Joachim Kock (2009); Polynomial functors and polynomial monads; arXiv.

Last revised on February 27, 2013 at 23:31:57. See the history of this page for a list of all contributions to it.