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• Mathematical Proceedings of the CambridgePhilosophical Societyhttp://journals.cambridge.org/PSP

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NICOLA GAMBINO and JOACHIM KOCK

Mathematical Proceedings of the Cambridge Philosophical Society / Volume 154 / Issue 01 / January 2013, pp153 - 192DOI: 10.1017/S0305004112000394, Published online: 06 September 2012

How to cite this article:NICOLA GAMBINO and JOACHIM KOCK (2013). Polynomial functors and polynomial monads.Mathematical Proceedings of the Cambridge Philosophical Society, 154, pp 153-192 doi:10.1017/S0305004112000394

• Math. Proc. Camb. Phil. Soc. (2013), 154, 153192 c Cambridge Philosophical Society 2012doi:10.1017/S0305004112000394

First published online 6 September 2012

153

BY NICOLA GAMBINO

Dipartimento di Matematica e Informatica, Universita degli Studi di Palermo,via Archirafi 34, 90123 Palermo, Italy.

e-mail: ngambino@math.unipa.it

AND JOACHIM KOCK

Departament de Matematiques, Universitat Autonoma de Barcelona,08193 Bellaterra (Barcelona), Spain.

e-mail: kock@mat.uab.cat

(Received 8 February 2011; revised 28 May 2012)

Abstract

We study polynomial functors over locally cartesian closed categories. After setting up thebasic theory, we show how polynomial functors assemble into a double category, in fact aframed bicategory. We show that the free monad on a polynomial endofunctor is polynomial.The relationship with operads and other related notions is explored.

Introduction

Background. Notions of polynomial functor have proved useful in many areas of mathem-atics, ranging from algebra [37, 44] and topology [10, 53] to mathematical logic [18, 48]and theoretical computer science [2, 21, 26]. This paper deals with the notion of polynomialfunctor over locally cartesian closed categories. Before outlining our results, let us brieflymotivate this level of abstraction.

Among the devices used to organise and manipulate numbers, polynomials are ubiquitous.While formally a polynomial is a sequence of coefficients, it can be viewed also as a func-tion, and the fact that many operations on polynomial functions, including composition, canbe performed in terms of the coefficients alone is a crucial feature. The idea underpinningthe notion of a polynomial functor is to lift the machinery of polynomials and polynomialfunctions to the categorical level. An obvious notion results from letting the category of fi-nite sets take the place of the semiring of natural numbers, and defining polynomial functorsto be functors obtained by finite combinations of disjoint union and cartesian product. It isinteresting and fruitful to allow infinite sets. One reason is the interplay between induct-ively defined sets and polynomial functors. For example, the set of natural numbers can becharacterised as the least solution to the polynomial equation of sets

X 1 + X ,

• 154 N. GAMBINO AND J. KOCK

while the set of finite planar trees appears as least solution to the equation

X 1 +nN

Xn.

Hence, one arrives at considering as polynomial functors on the category of sets all thefunctors of the form

X aA

X Ba , (1)

where A is a set and (Ba | a A) is an A-indexed family of sets, which we represent as amap f : B A with Ba = f 1(a). It is natural to study also polynomial functors in manyvariables. A J -indexed family of polynomial functors in I -many variables has the form

(Xi | i I )

aA j

bBa

Xs(b) | j J , (2)

where the indexing refers to the diagram of sets

I Bs

f A

t J . (3)

This expression reduces to (1) when I and J are singleton sets. The functor specified in (2)is the composite of three functors: pullback along s, the right adjoint to pullback along f ,and the left adjoint to pullback along t . The categorical properties of these basic types offunctors allow us to manipulate polynomial functors like (2) in terms of their representingdiagrams (3); this is a key feature of the present approach to polynomial functors.

Although the theory of polynomial functors over Set is already rich and interesting, onefinal abstraction is called for: we may as well work in any category with finite limits wherepullback functors have both adjoints. These are the locally cartesian closed categories, andwe develop the theory in this setting, applicable not only to some current developments inoperad theory and higher-dimensional algebra [35, 36], but also in mathematical logic ,and in theoretical computer science [2, 21]. We hasten to point out that since the categoryof vector spaces is not locally cartesian closed, our theory does not immediately apply tovarious notions of polynomial functor that have been studied in that context [44, 53].

Main results. Our general goal is to present a mathematically efficient account of the funda-mental properties of polynomial functors over locally cartesian closed categories, which canserve as a reference for further developments. With this general aim, we begin our expositionby including some known results that either belong to folklore or were only available in thecomputer science literature (but not in their natural generality), giving them a unified treat-ment and streamlined proofs. These results mainly concern the diagram representation ofstrong natural transformations between polynomial functors, and versions of some of theseresults can be found in Abbotts thesis .

Having laid the groundwork, our first main result is to assemble polynomial functors into adouble category, in fact a framed bicategory in the sense of Shulman , hence providing aconvenient and precise way of handling the base change operation for polynomial functors.There are two biequivalent versions of this framed bicategory: one is the strict framed 2-category of polynomial functors, the other is the (nonstrict) bicategory of their representingdiagrams.

Our second main result states that the free monad on a polynomial functor is a polyno-mial monad. This result extends to general polynomial functors the corresponding result

• Polynomial functors and polynomial monads 155

for polynomial functors in a single variable  and for finitary polynomial functors onthe category of sets [35, 36]. We also observe that free monads enjoy a double-categoricaluniversal property which is stronger than the bicategorical universal property that a prioricharacterises them.

The final section gives some illustration of the usefulness of the double-category view-point in applications. We give a purely diagrammatic comparison between Burroni P-spans , and polynomials over P (for P a polynomial monad). This yields in turn aconcise equivalence between polynomial monads over P and P-multicategories [12, 43],with base change (multifunctors) conveniently built into the theory. Operads are a specialcase of this.

Related work. Polynomial functors and closely related notions have been reinvented severaltimes by workers in different contexts, unaware of the fact that such notions had already beenconsidered elsewhere. To help unifying the disparate developments, we provide many point-ers to the literature, although surveying the different developments in any detail is outsidethe scope of this paper.

We should say first of all that our notions of polynomial and polynomial functor are almostexactly the same as the notions of container and container functor introduced in theoreticalcomputer science by Abbott, Altenkirch and Ghani [1, 2, 3, 4] to provide semantics for re-cursive data types, and studied further in . The differences, mostly stylistic, are explainedin Section 218. A predecessor to containers were the shapely types of Jay and Cockett which we revisit in Sections 316317. The importance of polynomial functors for depend-ent type theory was first observed by Moerdijk and Palmgren , cf. Section 43. Theirpolynomial functors are what we call polynomial functors in one variable.

The use of polynomial functors in program semantics goes back at least to Manes andArbib , and was recently explored from a different viewpoint under the name inter-action systems in the setting of dependent type theory by Hancock and Setzer  andby Hyvernat , where polynomials are also given a game-theoretic interpretation. Themorphisms there are certain bisimulations, more general than the strong natural transforma-tions used in the present work.

Within category theory, many related notions have been studied. In Section 118 we listsix equivalent characterisations of polynomial functors over Set, and briefly comment on thecontexts of the related notions: familially representable functors of Diers  and CarboniJohnstone  (see also [43, appendix C]), and local right adjoints of Lamarche ,Taylor , and Weber [61, 62], a notion that in the present setting is equivalent to thenotion of parametric right adjoint of Street . We also comment on the relationship withspecies and analytic functors [9, 29], and with Girards normal functors .

Tambara  studied a notion of polynomial motivated by representation theory andgroup cohomology, where the three operations are, respectively, restriction, trace (ad-ditive transfer), and norm (multiplicative transfer). In Section 123, we give an algebraic-theory interpretation of o