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Brent A. Yorgey, Species and Functors and Types, Oh My!, pdf, Abstract:
‘’The theory of combinatorial species, although invented as a purely mathematical formalism to unify much of combinatorics, can also serve as a powerful and expressive language for talking about data types. With potential applications to automatic test generation, generic programming, and language design, the theory deserves to be much better known in the functional programming community. This paper aims to teach the basic theory of combinatorial species using motivation and examples from the world of functional pro- gramming. It also introduces the species library, available on Hack- age, which is used to illustrate the concepts introduced and can serve as a platform for continued study and research.’‘
‘’The theory described in this book is at the crossroads between category theory, algebra, and combinatorics. Its main goal is to unify and clarify a large number of constructions of Hopf algebras found in the literature and to reveal many new connections between them. The book contains a systematic description of the rel- evant aspects of category theory together with many examples of Hopf algebras. It includes the tensor algebra, the shuffle algebra, the symmetric algebra, the ex- terior algebra, the divided power algebra, the algebra of symmetric functions, the algebra of permutations introduced by Malvenuto and Reutenauer, the algebra of quasi-symmetric functions introduced by Gessel and the algebra of noncommuta- tive symmetric functions introduced by Gelfand, Krob, Lascoux, Leclerc, Retakh and Thibon. It also includes q-deformations of these Hopf algebras as well as multi-parameters deformations and decorated versions. The book contains other interesting material which is not discussed in this foreword.’‘