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Category theory

Polynomial Functors

• Nelson Niu, David I. Spivak. Polynomial Functors: A Mathematical Theory of Interaction.
  • David Spivak. Polynomial functors: Lecture 2/9
• Topos Institute GitHub: Poly
• nLab: Polynomial functor
• David Spivak. It’s Proly like Poly but better
• Sophie Libkind. David Spivak. A polynomial account of Bayesian update

Representable functor

For a set {$S$}, we denote by {$y^S:\mathbf{Set}\rightarrow\mathbf{Set}$} the functor that sends each set {$X$} to the set {$S=\mathbf{Set}(S,X)$} and each function {$h:X\rightarrow Y$} to the function {$h^S:X^S\rightarrow Y^S$} that sends {$g:S\rightarrow X$} to {$h\circ g:S\rightarrow Y$}.

• We say that it is a representable functor, we denote it {$y^S$}, we call it the functor represented by {$S$}, and we say that it is a pure power.
• We call {$S$} the representing set of {$y^S$}.

Polynomial functor

A polynomial functor (or simply polynomial) is a functor {$p:\mathbf{Set}\rightarrow\mathbf{Set}$} such that there exists a set {$I$} of sets, an {$I$}-indexed family of sets {$(p[i])_{i\in I}$}, and an isomorphism

{$$p\cong\sum_{i\in I}y^{p[i]}$$}

to the corresponding {$I$}-indexed sum of representables. Up to isomorphism, a polynomial functor is just a sum of representables.