Combinatorial species
Combinatorial species are presheaves (or equivalently copresheaves) on the category of finite sets and bijections. Species were invented by Joyal as a categorification of generating functions in enumerative combinatorics.
Literature
Books
- Bergeron, Labelle, Leroux, 1998: Combinatorial species and tree-like structures (doi)
- Bergeron, Labelle, Leroux, 2008: Introduction to the theory of species of structures (pdf)
- Aguiar & Mahajan, 2010: Monoidal functors, species and Hopf algebras (doi, pdf)
Applications to computer science
There are connections between combinatorial species, algebraic data types, and polynomial functors.
- Kock, 2012: Data types with symmetries and polynomial functors over groupoids (doi, arxiv)
- Yorgey, 2014, PhD thesis: Combinatorial species and labelled structures (pdf)
Restriction species
Restriction species, introduced by William Schmitt, are presheaves on the category of finite sets and injections. Because every injection factors as a bijection followed by an inclusion, a restriction species can be seen as an ordinary species equipped with a functorial family of restriction maps.