polynomial monad



Higher algebra

2-Category theory



A polynomial monad is a monad on a category CC whose underlying endofunctor is a polynomial functor and whose unit and multiplication are cartesian natural transformations. This is of course equivalent to being a monad in the bicategory of polynomial functors and cartesian transformations.


A basic example is the free-monoid monad (Gambino-Kock 09, Example 1.9). This is exhibited by the polynomial 1 11\leftarrow \mathbb{N}^\prime\rightarrow \mathbb{N}\rightarrow 1 where the middle arrow is such that for all nn\in \mathbb{N} its fiber over nn has cardinality nn.

One can construct the free monad on a polynomial endofunctor, and it is a polynomial monad.


Parametric right adjointness

Every polynomial monad is a p.r.a. monad.


Since all polynomial functors preserve pullbacks, a polynomial monad is necessarily a cartesian monad. Note that cartesian transformations between polynomial functors also have an explicit description: they are given by diagrams

X Y W Z X Y \array{ & & X & \to & Y\\ & \swarrow & & & & \searrow \\ W && \downarrow && \downarrow && Z \\ & \nwarrow & & & & \nearrow \\ & & X' & \to & Y' }

in which the middle square is a pullback. Thus, all the data of a polynomial monad can be described very concretely. This gives a convenient way to produce cartesian monads, and indeed very many cartesian monads arising in practice are in fact polynomial. Polynomial monads have close relations with a number of other notions.

Relation to clubs

Polynomial monads (on CC rather than a slice of it) are, in some sense, the most natural base monads for the theory of clubs. Although the club construction can be performed over any cartesian monad, if this monad is polynomial, then all clubs over it are also polynomial, since polynomial functors are a sieve with respect to cartesian transformations. Moreover, the club monoidal structure can be seen as just an instance of the explicit construction of composition for polynomials (see 1.11 of (Gambino-Kock)).

Relation to generalized multicategories

The bicategory of polynomial functors and cartesian transformations is in fact the horizontal bicategory of a double category Poly̲\underline{Poly}, whose vertical arrows are the morphisms of CC and whose cells are diagrams like

W X Y Z W X Y Z \array{ W & \leftarrow & X & \to & Y & \to & Z \\ \downarrow && \downarrow && \downarrow &&\downarrow\\ W' & \leftarrow & X' & \to & Y' & \to & Z' }

in which the middle square (only) is a pullback. Suppose W=Z=1W'=Z'=1, so the bottom polynomial is determined by a single morphism g:XYg':X'\to Y', and that W=ZW=Z. Then to give the rest of the diagram is to give an object YY, a morphism YZY\to Z, and a morphism X× YYZX' \times_{Y'} Y \to Z. By adjointness, the latter is equivalent to giving a morphism YΠ g(Z×X)Y\to \Pi_{g'}(Z\times X'); but this codomain is just TZT Z, where TT is the polynomial functor determined by gg'.

Thus, a polynomial equipped with a cartesian transformation to 1XY11 \leftarrow X' \to Y' \to 1 is exactly a “TT-span” such as arises in the theory of generalized multicategories. A similar argument shows that if TT is a polynomial monad, then the slice double category of Poly̲\underline{Poly} over gg' is equivalent to the double category of TT-spans (a.k.a. the “horizontal Kleisli” double category). Therefore, polynomial monads cartesianly-over TT, being the horizontal monads in this double category, are precisely TT-multicategories. (Such polynomial monads on CC itself, rather than any slice, are the TT-clubs mentioned above; in general a TT-multicategory TX 0X 1X 0T X_0 \leftarrow X_1 \to X_0 is identified with the monad it induces on C/X 0C/X_0.)

Relation to object classifiers

Polynomial monads have a natural interpretation in terms of object classifiers. Specifically, given any polynomial 1BgA11 \leftarrow B \xrightarrow{g} A \to 1, generating a polynomial functor P 1,g,1P_{1,g,1}, we can consider the class S gS_g of all pullbacks of gg. A cartesian unit IdP 1,g,1Id \to P_{1,g,1} says that id 1:11id_1 : 1\to 1 is a pullback of gg, and therefore so are all identities. Similarly, the composite P 1,g,1P 1,g,1P_{1,g,1} \circ P_{1,g,1} involves the classifying object for a pair of composable S gS_g-morphisms, and so a cartesian multiplication P 1,g,1P 1,g,1P 1,g,1P_{1,g,1} \circ P_{1,g,1} \to P_{1,g,1} tells us that S gS_g is closed under composition.

Thus, a polynomial monad (on CC/1C \cong C/1) can be regarded as a morphism gg together with a coherent way to make S gS_g into a category. More precisely, consider the slice category Cart(C)/gCart(C)/g, where Cart(C)Cart(C) is the category whose objects are morphisms of CC and whose morphisms are pullback squares. This comes with source and target functors Cart(C)/gCCart(C)/g \rightrightarrows C. To make P 1,g,1P_{1,g,1} into a polynomial monad is then equivalent to giving unit and composition functors enhancing Cart(C)/gCCart(C)/g \rightrightarrows C to a double category such that the forgetful map Cart(C)/gCart(C)Cart(C)/g \to Cart(C) is a double functor.

This claim could stand some independent verification.

For example, consider the free monoid monad above determined by g:g:\mathbb{N}\prime \to \mathbb{N}. To exhibit a function as a pullback of this gg is to say that its fibers are finite and have been equipped with bijections to canonical finite sets, which we may as well think of as giving them linear orders. To give the monad structure on this functor is equivalent to noting that every identity map has ordered finite fibers, and the composite of functions with ordered finite fibers again has ordered finite fibers, in a way that is associative, unital, and stable under pullback.

Other interesting examples include:

  • The monad for symmetric strict monoidal categories on CatCat: its S gS_g consists of discrete (op)fibrations with ordered finite fibers in which all reindexing functions are bijections.
  • The monad for cartesian strict monoidal categories on CatCat: its S gS_g consists of discrete fibrations with ordered finite fibers.
  • The monad for cocartesian strict monoidal categories on CatCat: its S gS_g consists of discrete opfibrations with ordered finite fibers.

The class of discrete Conduche functors with ordered finite fibers is almost an example; it wants to be classified by the bicategory FinSpanFinSpan of spans between finite sets, but that is not an object of CatCat. Thus, a polynomial monad in CatCat determined by a discrete Conduche functor with ordered finite fibers is a reasonable substitute for the nonexistent notion of “FinSpanFinSpan-club”.

Identifying the classes of morphisms corresponding to standard polynomial monads like these also tells us how to identify when a polynomial monad is induced by a club (or more generally a generalized multicategory) over them: when the gg for that monad belongs to the appropriate class, compatibly. For instance, if SS is the monad for cartesian strict monoidal categories, then a polynomial monad on CatCat is an SS-club just when the morphism gg in its defining polynomial is a discrete fibration with ordered finite fibers, and its composition and unit respect that structure.


A fairly comprehensive discussion of the notion is due to

The homotopy theory of algebras over polynomial monads is in

An application to the theory of opetopes in discussed in

Polynomial monads as a “generalization of small categories” for homotopy theory are studied in

  • Michael Batanin, Florian De Leger; Grothendieck’s homotopy theory, polynomial monads and delooping of spaces of long knots, 2017: arxiv

Last revised on December 30, 2018 at 01:08:48. See the history of this page for a list of all contributions to it.