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Notes on Polynomial Functors Joachim Kock Notes on Polynomial Functors Very preliminary version: 2009-10-15 20:38 but with some mistakes fixed on 2016-08-16 15:09 PLEASE CHECK IF A NEWER VERSION IS AVAILABLE! http://mat.uab.cat/~kock/cat/polynomial.html PLEASE DO NOT WASTE PAPER PRINTING THIS! Joachim Kock Departament de Matemàtiques Universitat Autònoma de Barcelona 08193 Bellaterra (Barcelona) SPAIN e-mail: [email protected] VERSION 2009-10-15 20:38 Available from http://mat.uab.cat/~kock This text was written in alpha. It was typeset in LATEX in standard book style, with mathpazo and fancyheadings. The figures were coded with the texdraw 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. There is a lot of redundancy and bad organisation. There are whole sections that have 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 I am especially indebted to André Joyal. 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 place to dump 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 research papers that have seen the light in the interim: • Polynomial functors and polynomial monads [39] with Gambino • Polynomial functors and trees [63] (single-authored) [Rough draft, version 2009-10-15 20:38.] [Input file: polynomial.tex, 2009-10-15 20:38] iv • Polynomial functors and opetopes [64] with Joyal, Batanin, and Mascari, • Local fibered right adjoints are polynomial with Anders Kock, • Polynomials in categories with pullbacks by Mark Weber, • Categorification of Hopf algebras of rooted trees (by me), • Data types with symmetries and polynomial functors over groupoids (by me), (I will gradually incorporate the material from 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. This will require of course that I learn more about these connections and applications, and I am gradually picking up these subjects. I have benefited from conversations and email correspondence with Anders Kock, Andy Tonks, Clemens Berger, David Gepner, Ieke Moerdijk, Imma Gálvez, Jiri Adámek, Juan Climent Vidal, Krzysztof Kapulkin, Marek Zawadowski, Mark Weber, Martin Hyland, Michael Batanin, Neil Ghani, Nicola Gambino, Pierre Hyvernat, Tom Fiore, Tom Hirschowitz... I am very thankful to Jesús Hernando Pérez Alcázar for organising 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 2009-10-15 20:38.] [Input file: polynomial.tex, 2009-10-15 20:38] Contents Preface iii Introduction 1 Historical remarks ........................ 5 I Polynomial functors in one variable 9 0.0 Prologue: natural numbers and finite sets ........... 11 1 Basic theory of polynomials in one variable 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 2 Categories of polynomial functors in one variable 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: only cartesian natural transformations 60 Products in Poly .......................... 63 Differentiation in Poly ...................... 65 [Rough draft, version 2009-10-15 20:38.] [Input file: polynomial.tex, 2009-10-15 20:38] vi CONTENTS 3 Aside: Polynomial functors and negative sets 67 3.1 Negative sets ............................ 67 3.2 The geometric series revisited .................. 74 3.3 Moduli of punctured Riemann spheres ............. 77 4 Algebras 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 Fixpoint construction via bar-cobar duality ..........104 5 Polynomial monads and operads 111 5.1 Polynomial monads ........................111 Cartesian monads .........................111 The free monad on a polynomial endofunctor (one variable) 116 Examples ..............................118 5.2 Classical definition of operads ..................120 5.3 The monoidal category of collections ..............121 5.4 Finitary polynomial functors and collections .........124 Equivalence of monoidal categories ..............126 5.5 The free operad on a collection .................127 5.6 P-operads .............................128 6 [Polynomial functors in computer science] 131 6.1 Data types .............................131 Shapely types ...........................139 6.2 Program semantics ........................141 7 [Species... ] 145 7.1 Introduction to species and analytical functors ........145 7.2 Polynomial functors and species ................146 [Rough draft, version 2009-10-15 20:38.] [Input file: polynomial.tex, 2009-10-15 20:38] CONTENTS vii II Polynomial functors in many variables 147 8 Polynomials in many variables 149 8.1 Introductory discussion .....................149 8.2 The pullback functor and its adjoints ..............152 Coherence .............................162 8.3 Beck-Chevalley and distributivity ................163 8.4 Further Beck-Chevalley gymnastics ...............173 The twelve ways of a square ...................173 The six ways of a pair of squares ................175 One more lemma .........................176 8.5 Composition ............................178 Rewrite systems and coherence .................183 8.6 Basic properties ..........................186 8.7 Examples ..............................188 The free-category functor on graphs with fixed object set . 190 9 Examples 193 9.1 Linear functors (matrices) ....................193 9.2 Finite polynomials: the Lawvere theory of comm. semirings 199 Lawvere theories .........................200 Proof of Tambara’s theorem ...................205 9.3 Differential calculus of polynomial functors ..........208 Introduction ............................208 Partial derivatives .........................208 Homogeneous functors and Euler’s Lemma ..........209 9.4 Classical combinatorics ......................214 9.5 Polynomial functors on collections and operads .......217 The free-operad functor .....................217 Linear differential operators are linear .............218 9.6 Bell polynomials ..........................222 10 Categories and bicategories of polynomial functors 229 10.1 Natural transformations between polynomial functors . 229 Basic properties of PolyFun(I, J): sums and products . 238 Misc .................................238 Polyc(I, J): the cartesian fragment ...............239 Sums and products in Polyc(I, J) ................239 [Rough draft, version 2009-10-15 20:38.] [Input file: polynomial.tex, 2009-10-15 20:38] viii CONTENTS 10.2 Horizontal composition and the bicategory of polynomial functors240 Some preliminary exercises in the cartesian fragment . 241 Horizontal composition of 2-cells ................244 11 Double categories of polynomial functors 251 11.1 Summary ..............................251 Reminder on double categories .................253 The double category of polynomial functors ..........254 Lifts .................................260 old calculations ..........................261 11.2 Horizontal composition .....................267 11.3 Cartesian ..............................270 Horizontal composition of cartesian 2-cells ..........271 Misc issues in the cartesian fragment ..............273 Surjection-injection factorisation in Poly ............273 Sums and products in the variable-type categories ......274 Coherence problems .......................275 12 Trees (1) 277 12.1
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