nLab
polynomial functor

This entry is about a notion in category theory. For a different notion of the same name in (stable) homotopy theory see at Goodwillie calculus.

Idea

The concept of polynomial functor is a categorification of that of polynomial.

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

Definition

Let $C$ be a locally cartesian closed category. A polynomial functor is specified by the data

$$W \overset{f}{\leftarrow} X \overset{g}{\to} Y \overset{h}{\to} Z$$

in $C$. The resulting functor is the composite

$$C/W \overset{f^*}{\to} C/X \overset{\Pi_g}{\to} C/Y \overset{\Sigma_h}{\to} C/Z \,,$$

where $\Pi_g$ and $\Sigma_h$ are the dependent product and dependent sum operations, right and left adjoint respectively to pullback functors $g^*$ and $h^*$.

When $W=Z$, this is a polynomial endofunctor.

Sometimes this general notion is called a 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 container (or an indexed container, with container reserved for the case $W=Z=1$). Other times container is used as a synonym for “polynomial functor”. Sometimes the data $f,g,h$ are instead referred to as a polynomial to distinguish them from the “polynomial functor” they determine.

If $g$ is an identity, the functor is sometimes called a linear functor or a 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.

Example

On sets

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

$$\ast \stackrel{}{\leftarrow} X \stackrel{g}{\longrightarrow} Y \stackrel{}{\longrightarrow} \ast$$

is given by

$$S \mapsto P_g(S) = \coprod_{y \in Y} S^{X_y} \,,$$

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

$${\vert P_g(S)\vert} = \sum_{y \in Y} {\vert S\vert}^{\vert X_y\vert}$$

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

$$A \stackrel{p_1}{\leftarrow} X \stackrel{id}{\longrightarrow} X \stackrel{p_2}{\longrightarrow} B$$

acts by

$$(S_a)_{a \in A} \mapsto \left(\coprod_{a \in A} S_a \times X_{a b} \right)_{b \in B} \,.$$

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.

The 2-category 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 Gambino–Kock:

Theorem

There is a bicategory whose objects are the objects of $C$, whose morphisms from $W$ to $Z$ are diagrams of the form

$$W \overset{f}{\leftarrow} X \overset{g}{\to} Y \overset{h}{\to} Z,$$

and whose 2-morphisms are diagrams of the form

$$\array{ & & X & \to & Y \\ & \swarrow & \uparrow & & \mathllap{id}\uparrow & \searrow\\ W && X' \times_{Y'} Y & \to & Y && Z\\ &\nwarrow & \downarrow & & \downarrow & \nearrow \\ && X' & \to & Y'. }$$

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.

In particular, “being polynomial” is a mere 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 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.

Related topics

• Polynomial endofunctors are important in the definition of W-types in categories.

• Polynomial functors are a special case of parametric right adjoints.

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

• Kripke frames $(R,S)$ (with a transition relation $R$ of arity $2$) as studied in modal logic are coalgebras for the power-set functor $P$. Kripke frames for a more general 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.

Examples

• symmetric polynomial, elementary symmetric polynomial

Related entries

• polynomial monad

• integral transform

• Tambara functor

• polynomial (infinity,1)-functor

References

The relation of plain polynomial functors to trees is discussed in

• Joachim Kock, Polynomial functors and trees, (2009) (arXiv:0807.2874)

The category, $Poly$, of one-variable polynomial functors on $Set$ and its application to dynamical systems is considered in

• David Spivak, Poly: An abundant categorical setting for mode-dependent dynamics, (arXiv:2005.01894)

Dependent (multivariate) polynomial functors are considered in

• Nicola Gambino and Joachim Kock, Polynomial functors and polynomial monads, (2009) (arXiv:0906.4931).

Talks on polynomial functors are available at

• Workshop on Polynomial Functors, Topos Institute, 15–19 March 2021. (website)

A book in progress and a course on polynomial functors are available at

• David Spivak, Nelson Niu, Polynomial Functors: A General Theory of Interaction, Topos Institute, 15 July - 26 August 2021. (website, pdf)

Generalization to homotopy theory and higher category theory is discussed in

• Joachim Kock, 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. (arXiv:1210.0828)

• David Gepner, Rune Haugseng, Joachim Kock, ∞-Operads as Analytic Monads, (arXiv:1712.06469)

• Mark Weber, Operads as polynomial 2-monads (arXiv:1412.7599)

• Benno van den Berg, Ieke Moerdijk, W-types in Homotopy Type Theory (arXiv:1307.2765)

See also

• Joachim Kock, André Joyal, Michael Batanin, Jean-François Mascari, Polynomial functors and opetopes, Advances in Mathematics, vol 224, 2010, pp 2690–2737. (arXiv:0706.1033)

• Yde Venema, Algebras and Coalgebras, §6 (p.332-426) in Blackburn, van Benthem, Wolter, Handbook of modal logic, Elsevier, 2007.

• Mark Weber, Polynomials in categories with pullbacks, Theory and Applications of Categories, Vol. 30, 2015, No. 16, pp 533-598. (journal)

• Charles Walker, Universal properties of polynomials, (arXiv:1806.10477)

• Ross Street, Polynomials as spans, Cahiers de topologie et géométrie différentielle catégoriques, Vol. LXI-2 (2020), pp 113-153 (pdf)