Combinatorial Species and Labelled Structures

The theory of combinatorial species was developed in the 1980s as part of the mathematical subfield of enumerative combinatorics, unifying and putting on a firmer theoretical basis a collection of techniques centered around generating functions. The theory of algebraic data types was developed, around the same time, in functional programming languages such as Hope and Miranda, and is still used today in languages such as Haskell, the ML family, and Scala. Despite their disparate origins, the two theories have striking similarities. In particular, both constitute algebraic frameworks in which to construct structures of interest. Though the similarity has not gone unnoticed, a link between combinatorial species and algebraic data types has never been systematically explored. This dissertation lays the theoretical groundwork for a precise—and, hopefully, useful—bridge bewteen the two theories. One of the key contributions is to port the theory of species from a classical, untyped set theory to a constructive type theory. This porting process is nontrivial, and involves fundamental issues related to equality and finiteness; the recently developed homotopy type theory is put to good use formalizing these issues in a satisfactory way. In conjunction with this port, species as general functor categories are considered, systematically analyzing the categorical properties necessary to define each standard species operation. Another key contribution is to clarify the role of species as labelled shapes, not containing any data, and to use the theory of analytic functors to model labelled data structures, which have both labelled shapes and data associated to the labels. Finally, some novel species variants are considered, which may prove to be of use in explicitly modelling the memory layout used to store labelled data structures.