nLab
polynomial functor

Definition

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

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

in $C$. The resulting functor is the composite

$$C/W\stackrel{f^{*}}{\to }C/X\stackrel{{\Pi }_g}{\to }C/Y\stackrel{{\Sigma }_h}{\to }C/Z.$$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 $h$ which 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”.

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.

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

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

and whose 2-morphisms are diagrams of the form

$$\begin{array}{ccccc}& & X& \to & Y\\ & \swarrow & \uparrow & & \mathrm{id}\uparrow & \searrow \\ W& & X\prime {\times }_{Y\prime }Y& \to & Y& & Z\\ & \nwarrow & \downarrow & & \downarrow & \nearrow \\ & & X\prime & \to & Y\prime .\end{array}$$\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.

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.

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 $X\mapsto {\prod }_{d\in t}P(S^{\mathrm{arity}(d)})$. Kripke models are coalgebras of functor $K:X\mapsto P(\mathrm{Prop})\times P(X)$ where $\mathrm{Prop}$ 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

References

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

• Joachim Kock (2009); Polynomial functors and trees; arXiv.

• Joachim Kock, André Joyal, Michael Batanin, Jean-François Mascari (2010); Polynomial functors and opetopes; Advances in Mathematics, vol 224, pages 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.