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Of Polynomial Functors Joachim Kock Notes on Polynomial Functors Very preliminary version: 2009-08-05 23:56 PLEASE CHECK IF A NEWER VERSION IS AVAILABLE! http://mat.uab.cat/~kock/cat/polynomial.html PLEASE DO NOT WASTE PAPER WITH PRINTING! Joachim Kock Departament de Matemàtiques Universitat Autònoma de Barcelona 08193 Bellaterra (Barcelona) SPAIN e-mail: [email protected] VERSION 2009-08-05 23:56 Available from http://mat.uab.cat/~kock This text was written in alpha.ItwastypesetinLATEXinstandardbook style, with mathpazo and fancyheadings.Thefigureswerecodedwiththetexdraw package, written by Peter Kabal. The diagrams were set using the diagrams package of Paul Taylor, and with XY-pic (Kristoffer Rose and Ross Moore). Preface Warning. Despite the fancy book layout, these notes are in VERY PRELIMINARY FORM In fact it is just a big heap of annotations. Many sections are very sketchy, and on the other hand many proofs and explanations are full of trivial and irrelevant details.Thereisalot of redundancy and bad organisation. There are whole sectionsthathave not been written yet, and things I want to explain that I don’t understand yet... There may also be ERRORS here and there! Feedback is most welcome. There will be a real preface one day IamespeciallyindebtedtoAndréJoyal. These notes started life as a transcript of long discussions between An- dré Joyal and myself over the summer of 2004 in connection with[64]. With his usual generosity he guided me through the basic theory of poly- nomial functors in a way that made me feel I was discovering it all by myself. I was fascinated by the theory, and I felt very privileged for this opportunity, and by writing down everything I learned (including many details I filled in myself), I hoped to benefit others as well. Soon the subject became one of my main research lines and these notes began to grow out of proportions. The next phase of the manuscript was as a placetodump basic stuff I needed to work out in relation to my research. At present it is just a heap of annotations, and the reader is recommended to read rather the three research papers that have seen the light in the interim: Polynomial functors and opetopes [64]withJoyal,Batanin,andMascari,Polynomial func- tors and trees [63](single-authored),andPolynomial functors and polynomial [Rough draft, version 2009-08-05 23:56.] [Input file: polynomial.tex, 2009-08-05 23:56] iv monads [39]withGambino.(Iwillgraduallyincorporatethematerialfrom these papers into the notes in some form.) Finally, a third aspect, which has been present all along, is the ambi- tion that these notes can grow into something more unified — a book! I would like to survey connections with and applications to combinatorics, computer science, topology, and whatever else turns up. Thiswillrequire of course that I learn more about these connections and applications, and Iam Ihavebenefitedfromconversationsandemailcorrespondencewith Anders Kock, Clemens Berger, Ieke Moerdijk, Juan Climent Vidal, Mark Weber, Martin Hyland, Michael Batanin, Nicola Gambino, Tom Fiore, Krzysztof Kapulkin. IamverythankfultoJesúsHernandoPérezAlcázarfororganising my mini-course on polynomial functors in Santa Marta, Colombia, in August 2009 — a perfect opportunity for me to synthesise some of this material. Barcelona, August 2009 JOACHIM KOCK [email protected] [Rough draft, version 2009-08-05 23:56.] [Input file: polynomial.tex, 2009-08-05 23:56] Contents Preface iii Introduction 1 Historical remarks ........................ 4 IPolynomialfunctorsinonevariable 9 0.0 Prologue: natural numbers and finite sets ........... 11 1Basictheoryofpolynomialsinonevariable 17 1.1 Definition ............................. 19 1.2 Examples .............................. 24 1.3 Other viewpoints ......................... 28 1.4 Basic operations on polynomials ................ 30 1.5 Composition of polynomials (substitution) .......... 34 1.6 Differential calculus ........................ 40 1.7 Properties of polynomial functors ................ 43 2Categoriesofpolynomialfunctorsinonevariable 45 2.1 The category Set[X] of polynomial functors .......... 46 Cartesian morphisms ....................... 48 2.2 Sums, products .......................... 54 2.3 Algebra of polynomial functors: categorification and Burnside semirings 56 2.4 Composition ............................ 60 2.5 The subcategory Poly:onlycartesiannaturaltransformations 60 Products in Poly .......................... 63 Differentiation in Poly ...................... 65 [Rough draft, version 2009-08-05 23:56.] [Input file: polynomial.tex, 2009-08-05 23:56] vi CONTENTS 3Aside:Polynomialfunctorsandnegativesets 67 3.1 Negative sets ............................ 67 3.2 The geometric series revisited .................. 74 3.3 Moduli of punctured Riemann spheres ............. 77 4Algebras 81 4.1 Initial algebras, least fixpoints .................. 81 Functoriality of least fixpoints .................. 84 4.2 Natural numbers, free monoids ................. 84 4.3 Tree structures as least fixpoints ................. 89 4.4 Induction, well-founded trees .................. 93 4.5 Transfinite induction ....................... 95 4.6 Free-forgetful ........................... 99 5Polynomialmonadsandoperads 105 5.1 Polynomial monads ........................105 Cartesian monads .........................105 The free monad on a polynomial endofunctor (one variable) 110 Examples ..............................112 5.2 Classical definition of operads ..................114 5.3 The monoidal category of collections ..............115 5.4 Finitary polynomial functors and collections .........118 Equivalence of monoidal categories ..............120 5.5 The free operad on a collection .................121 5.6 P-operads .............................122 6[Polynomialfunctorsincomputerscience] 125 6.1 Data types .............................125 Shapely types ...........................133 6.2 Program semantics ........................135 7[Species...] 139 7.1 Introduction to species and analytical functors ........139 7.2 Polynomial functors and species ................140 [Rough draft, version 2009-08-05 23:56.] [Input file: polynomial.tex, 2009-08-05 23:56] CONTENTS vii II Polynomial functors in many variables 141 8Polynomialsinmanyvariables 143 8.1 Introductory discussion .....................143 8.2 The pullback functor and its adjoints ..............146 Coherence .............................155 8.3 Beck-Chevalley and distributivity ................156 8.4 Further Beck-Chevalley gymnastics ...............165 The twelve ways of a square ...................165 The six ways of a pair of squares ................167 One more lemma .........................168 8.5 Composition ............................170 Rewrite systems and coherence .................175 8.6 Basic properties ..........................178 8.7 Examples ..............................180 The free-category functor on graphs with fixed object set ..182 9Examples 185 9.1 Linear functors (matrices) ....................185 9.2 Finite polynomials: the Lawvere theory of comm. semirings 191 Lawvere theories .........................192 Proof of Tambara’s theorem ...................197 9.3 Differential calculus of polynomial functors ..........200 Introduction ............................200 Partial derivatives .........................200 Homogeneous functors and Euler’s Lemma ..........201 9.4 Classical combinatorics ......................206 9.5 Polynomial functors on collections and operads .......209 The free-operad functor .....................209 Linear differential operators are linear .............210 9.6 Bell polynomials ..........................214 10 Categories and bicategories of polynomial functors 219 10.1 Natural transformations between polynomial functors ...219 Basic properties of PolyFun(I, J):sumsandproducts ....228 Misc .................................228 Polyc(I, J):thecartesianfragment ...............229 Sums and products in Polyc(I, J) ................229 [Rough draft, version 2009-08-05 23:56.] [Input file: polynomial.tex, 2009-08-05 23:56] viii CONTENTS 10.2 Horizontal composition and the bicategory of polynomial functors230 Some preliminary exercises in the cartesian fragment ....231 Horizontal composition of 2-cells ................234 11 Double categories of polynomial functors 241 11.1 Summary ..............................241 Reminder on double categories .................243 The double category of polynomial functors ..........244 Lifts .................................250 old calculations ..........................251 11.2 Horizontal composition .....................257 11.3 Cartesian ..............................260 Horizontal composition of cartesian 2-cells ..........261 Misc issues in the cartesian fragment ..............263 Surjection-injection factorisation in Poly ............263 Sums and products in the variable-type categories ......264 Coherence problems .......................265 12 Trees (1) 267 12.1 Trees ................................270 12.2 From trees to polynomial endofunctors ............271 Examples of trees .........................275 12.3 The category TEmb ........................278 12.4 P-trees ...............................286 13 Polynomial monads 291 13.1 The free polynomial monad on a polynomial endofunctor .291 13.2 Monads in the double category setting .............295 relative ...............................299 13.3 Coloured operads and generalised operads ..........299 13.4 P-spans, P-multicategories, and P-operads ..........299 Coloured operads .........................313 Polynomial monads
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