I recommend instead the papers listed here.
Table of contents:
{Preface}{iii}
{Introduction}{1}
{Historical remarks}{4}
{PART 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 $\text {\textbf {\textsl {Set}}}[X]$ 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 $\text {\textbf {\textsl {Poly}}}$: only cartesian natural transformations}{60}
{Products in $\text {\textbf {\textsl {Poly}}}$}{63}
{Differentiation in $\text {\textbf {\textsl {Poly}}}$}{65}
{{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}
{{5}Polynomial monads and operads}{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}[Polynomial functors in computer science]}{125}
{{6.1}Data types}{125}
{Shapely types}{133}
{{6.2}Program semantics}{135}
{{7}[Species\relax $\mathsurround \z@ \ldotp \ldotp \ldotp $\ ]}{139}
{{7.1}Introduction to species and analytical functors}{139}
{{7.2}Polynomial functors and species}{140}
{PART II Polynomial functors in many variables}{141}
{{8}Polynomials in many variables}{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}
{{9}Examples}{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}{205}
{{9.5}Polynomial functors on collections and operads}{208}
{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 $\text {\textbf {\textsl {PolyFun}}}(I,J)$: sums and products}{228}
{Misc}{228}
{$\text {\textbf {\textsl {Poly}}}^{\text {c}}(I,J)$: the cartesian fragment}{229}
{Sums and products in $\text {\textbf {\textsl {Poly}}}^{\text {c}}(I,J)$}{229}
{{10.2}Horizontal composition and the bicategory of polynomial functors}{230}
{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 $\text {\textbf {\textsl {Poly}}}$}{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 $\text {\textbf {\textsl {TEmb}}}$}{278}
{{12.4}$\mathsf {P}$-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 and coloured operads}{314}
{{14}Trees (2)}{317}
{{14.1}$\mathsf {P}$-trees and free monads}{319}
{Examples of polynomial monads from trees}{321}
{{14.2}The category $\text {\textbf {\textsl {Tree}}}$}{323}
{{14.3}Trees of trees, constellations, and the Baez-Dolan construction}{331}
{PART III Categorical polynomial functors}{337}
{{14.4}Introduction}{339}
{{15}[Polynomial functors for slices of $\text {\textbf {\textsl {Cat}}}$]}{341}
{$\text {\textbf {\textsl {Cat}}}$ is not locally cartesian closed}{341}
{{15.1}Conduch\'e fibrations}{343}
{{15.2}Polynomial functors in $\text {\textbf {\textsl {Cat}}}$}{351}
{{15.3}The family functor}{353}
{{15.4}Final functors and discrete fibrations}{356}
{{16}[Polynomial functors for presheaf categories]}{357}
{{16.1}Some prelims}{357}
{Kan extensions}{357}
{Categories of elements}{358}
{Nerves}{363}
{Generic morphisms}{365}
{Monads with arities}{369}
{{16.2}Distributors and mixed fibrations}{375}
{{16.3}The free-category monad}{380}
{The free-multicategory monad}{387}
{The free-coloured-operad monad}{388}
{{16.4}Local right adjoints}{389}
{{17}[Generalised species and polynomial functors]}{393}
{{18}Appendices}{395}
{{A}Pullbacks}{395}
{Index}{407}{chapter*.55}