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\section*{polynomial functor}

\begin{quote}%
This entry is about a notion in [[category theory]]. For a different notion of the same name in ([[stable homotopy theory|stable]]) [[homotopy theory]] see at \emph{[[Goodwillie calculus]]}.


\end{quote}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{polynomial_functors}{}\section*{{Polynomial functors}}\label{polynomial_functors}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{example}{Example}\dotfill \pageref*{example} \linebreak
\noindent\hyperlink{ExamplesOnSets}{On sets}\dotfill \pageref*{ExamplesOnSets} \linebreak
\noindent\hyperlink{the_2category_of_polynomial_functors}{The 2-category of polynomial functors}\dotfill \pageref*{the_2category_of_polynomial_functors} \linebreak
\noindent\hyperlink{related_topics}{Related topics}\dotfill \pageref*{related_topics} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The concept of \emph{polynomial functor} is a [[categorification]] of that of \emph{[[polynomial]]}.

Polynomial endo-functors are used to encode a class of [[inductive types]] called \emph{[[W-types]]}, and also as the underlying data of [[polynomial monads]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $C$ be a [[locally cartesian closed category]]. A \textbf{polynomial functor} is specified by the data

\begin{displaymath}
W \overset{f}{\leftarrow} X \overset{g}{\to} Y \overset{h}{\to} Z
\end{displaymath}
in $C$. The resulting functor is the composite

\begin{displaymath}
C/W 
   \overset{f^*}{\to} 
  C/X 
   \overset{\Pi_g}{\to} 
  C/Y 
   \overset{\Sigma_h}{\to} 
  C/Z
  \,,
\end{displaymath}
where $\Pi_g$ and $\Sigma_h$ are the [[dependent product]] and [[dependent sum]] operations, right and left adjoint respectively to [[pullback\#PullbackFunctor|pullback functors]] $g^*$ and $h^*$.

When $W=Z$, this is a \textbf{polynomial endofunctor}.

Sometimes this general notion is called a \textbf{dependent polynomial functor}, with ``polynomial (endo)functor'' reserved for the ``one-variable'' case, when $W=Z=1$ is the [[terminal object]].

The data $f$, $g$, and $h$ that specify a polynomial functor is sometimes referred to as a \textbf{container} (or an \textbf{indexed container}, with \emph{container} reserved for the case $W=Z=1$). Other times \emph{container} is used as a synonym for ``polynomial functor''. Sometimes the data $f,g,h$ are instead referred to as a \textbf{polynomial} to distinguish them from the ``polynomial functor'' they determine.

If $g$ is an identity, the functor is sometimes called a \textbf{linear functor} or a \textbf{linear polynomial functor}. Note that this notion makes sense even if $C$ is not locally cartesian closed; all it needs are [[pullbacks]]. More generally, we can make sense of polynomial functors in any category with pullbacks if we restrict $g$ to be an [[exponentiable morphism]].

\hypertarget{example}{}\subsection*{{Example}}\label{example}

\hypertarget{ExamplesOnSets}{}\subsubsection*{{On sets}}\label{ExamplesOnSets}

For $C$ = [[Set]] the polynomial functor induced from a [[function]] $g$

\begin{displaymath}
\ast \stackrel{}{\leftarrow} X \stackrel{g}{\longrightarrow} Y
  \stackrel{}{\longrightarrow} \ast
\end{displaymath}
is given by

\begin{displaymath}
S \mapsto P_g(S) = \coprod_{y \in Y} S^{X_y}
  \,,
\end{displaymath}
where $X_y$ is the [[fiber]] of $g$ over $y \in Y$, and where the [[exponential object]] $S^{X_y}$ is the [[function set]] of [[functions]] from the fiber to $S$. The [[cardinality]] of the set on the right is

\begin{displaymath}
{\vert P_g(S)\vert}
   = 
  \sum_{y \in Y} {\vert S\vert}^{\vert X_y\vert}
\end{displaymath}
and it is in this sense that the concept of polynomial functor is a kind of [[categorification]] of that of [[polynomial]].

On the other hand the dependent polynomial functor associated to

\begin{displaymath}
A \stackrel{p_1}{\leftarrow} X \stackrel{id}{\longrightarrow} X
  \stackrel{p_2}{\longrightarrow} B
\end{displaymath}
acts by

\begin{displaymath}
(S_a)_{a \in A}
  \mapsto
  \left(\coprod_{a \in A} S_a \times X_{a b} \right)_{b \in B}
  \,.
\end{displaymath}
Under [[cardinality]] this becomes [[matrix multiplication]] acting on [[vectors]] (with entries in the [[natural numbers]]). So in this case the dependent polynomial functor is a linear functor of several variables, an [[integral transform]].

\hypertarget{the_2category_of_polynomial_functors}{}\subsection*{{The 2-category of polynomial functors}}\label{the_2category_of_polynomial_functors}

Any polynomial functor, as defined above, is automatically equipped with a [[tensorial strength]], when the slice categories of $C$ are regarded as tensored over $C$ in the canonical way. The following theorem is proven in \hyperlink{GK}{Gambino--Kock}:

\begin{theorem}
\label{}\hypertarget{}{}
There is a [[bicategory]] whose objects are the objects of $C$, whose morphisms from $W$ to $Z$ are diagrams of the form

\begin{displaymath}
W \overset{f}{\leftarrow} X \overset{g}{\to} Y \overset{h}{\to} Z,
\end{displaymath}
and whose 2-morphisms are diagrams of the form

\begin{displaymath}
\itexarray{ & & X & \to & Y \\
  & \swarrow & \uparrow & & \mathllap{id}\uparrow & \searrow\\
  W && X' \times_{Y'} Y & \to & Y && Z\\
  &\nwarrow & \downarrow & & \downarrow & \nearrow \\
  && X' & \to & Y'. }
\end{displaymath}
This bicategory is equivalent to the 2-category whose objects are slice categories of $C$, whose morphisms are polynomial functors regarded as strong functors, and whose 2-morphisms are strength-respecting natural transformations.

\end{theorem}
In particular, ``being polynomial'' is a mere [[stuff, structure, property|property]] of a strong functor between slice categories. That is, the data $f,g,h$ are uniquely determined, up to isomorphism, by the strong functor they generate. (Note, though, that the property of ``being polynomial'' depends on a prior identification of the domain and codomain as being slice categories of some specified ambient category $C$. In particular, a functor $C/Z \to C/Z$ might be polynomial when its domain and codomain are regarded as slice categories of $C$, but not when they are regarded as slices of $C/Z$ over $1$ --- this happens when $W=Z$ but $h g \neq f$.)

Note that the above bicategory contains, as a locally full sub-bicategory, the usual bicategory of [[spans]]. Thus, as a special case, the bicategory of spans is equivalent to the 2-category of ``linear'' polynomial functors. Both of these are instances of [[Lack's coherence theorem]].

There is a particular subclass of the 2-morphism in this bicategory that is also interesting: a 2-morphism corresponds to a [[cartesian natural transformation]] if and only if the map $X' \times_{Y'} Y \to X$ is an isomorphism. Since anything isomorphic to a pullback is a pullback, in this case the diagram can be drawn more simply by omitting the upper square and merely asking that the lower square \emph{be} a pullback.

Furthermore, this bicategory is actually the horizontal bicategory of a [[double category]], indeed a [[framed bicategory]], in which the vertical arrows are the arrows of $C$, and the cells are diagrams as above but allowing also morphisms $W\to W'$ and $Z\to Z'$ on the left and right.

\hypertarget{related_topics}{}\subsection*{{Related topics}}\label{related_topics}

\begin{itemize}%
\item Polynomial endofunctors are important in the definition of [[W-types]] in categories.


\item Polynomial functors are a special case of [[parametric right adjoint|parametric right adjoints]].


\item Polynomial functors can be defined using [[exponentiable morphisms]] in a category that may not be locally cartesian closed. See also [[distributivity pullback]].


\item Kripke frames $(R,S)$ (with a transition relation $R$ of arity $2$) as studied in [[modal logic]] are [[coalgebra]]s for the power-set functor $P$. Kripke frames for a more general \emph{modal similarity type} $t$ are a coalgebras of a functor of the form $X\mapsto \product_{d\in t} P(S^{arity(d)})$. Kripke models are coalgebras of functor $K:X\mapsto P(Prop)\times P(X)$ where $Prop$ is the set of propositional variables of the logic in consideration. In particular all the functors appearing here are polynomial functors. So, at least in some aspects, the study of modal logics reduces to the study of (certain) polynomial functors



\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item [[symmetric polynomial]], [[elementary symmetric polynomial]]

\end{itemize}
\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[polynomial monad]]


\item [[integral transform]]


\item [[Tambara functor]]


\item [[polynomial (infinity,1)-functor]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The relation of plain polynomial functors to [[trees]] is discussed in

\begin{itemize}%
\item [[Joachim Kock]], \emph{Polynomial functors and trees}, (2009) (\href{http://arxiv.org/abs/0807.2874}{arXiv:0807.2874})

\end{itemize}
Dependent (multivariate) polynomial functors are considered in

\begin{itemize}%
\item [[Nicola Gambino]] and [[Joachim Kock]], \emph{Polynomial functors and polynomial monads}, (2009) (\href{http://arxiv.org/abs/0906.4931}{arXiv:0906.4931}).

\end{itemize}
Generalization to [[homotopy theory]] and [[higher category theory]] is discussed in

\begin{itemize}%
\item [[Joachim Kock]], \emph{Data types with symmetries and polynomial functors over groupoids}, 28th Conference on the Mathematical Foundations of Programming Semantics (Bath, June 2012); in Electronic Notes in Theoretical Computer Science. (\href{http://arxiv.org/abs/1210.0828}{arXiv:1210.0828})


\item [[David Gepner]], [[Rune Haugseng]], [[Joachim Kock]], \emph{∞-Operads as Analytic Monads}, (\href{https://arxiv.org/abs/1712.06469}{arXiv:1712.06469})


\item [[Mark Weber]], \emph{Operads as polynomial 2-monads} (\href{http://arxiv.org/abs/1412.7599}{arXiv:1412.7599})


\item [[Benno van den Berg]], [[Ieke Moerdijk]], \emph{W-types in Homotopy Type Theory} (\href{http://arxiv.org/abs/1307.2765}{arXiv:1307.2765})



\end{itemize}
See also

\begin{itemize}%
\item [[Joachim Kock]], [[André Joyal]], [[Michael Batanin]], [[Jean-François Mascari]], \emph{Polynomial functors and opetopes}, Advances in Mathematics, vol 224, pages 2690--2737, (2010) (\href{http://arxiv.org/abs/0706.1033}{arXiv:0706.1033})

\end{itemize}
\begin{itemize}%
\item [[Yde Venema]], \emph{Algebras and Coalgebras}, \S{}6 (p.332-426) in Blackburn, van Benthem, Wolter, \emph{Handbook of modal logic}, Elsevier, 2007.


\item [[Mark Weber]], \emph{Polynomials in categories with pullbacks}, \href{http://tac.mta.ca/tac/volumes/30/16/30-16abs.html}{TAC}


\item Charles Walker, \emph{Universal properties of polynomials}, \href{https://arxiv.org/abs/1806.10477}{arxiv}



\end{itemize}
[[!redirects polynomial functor]] [[!redirects polynomial functors]] [[!redirects polynomial endofunctor]] [[!redirects polynomial endofunctors]]

[[!redirects dependent polynomial functor]] [[!redirects dependent polynomial functors]] [[!redirects dependent polynomial endofunctor]] [[!redirects dependent polynomial endofunctors]]

[[!redirects linear polynomial functor]] [[!redirects linear polynomial functors]] [[!redirects linear polynomial endofunctor]] [[!redirects linear polynomial endofunctors]]

[[!redirects linear dependent polynomial functor]] [[!redirects linear dependent polynomial functors]] [[!redirects linear dependent polynomial endofunctor]] [[!redirects linear dependent polynomial endofunctors]]

[[!redirects dependent linear polynomial functor]] [[!redirects dependent linear polynomial functors]] [[!redirects dependent linear polynomial endofunctor]] [[!redirects dependent linear polynomial endofunctors]]

[[!redirects container]] [[!redirects containers]]



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