A (combinatorial) species is a presheaf or higher categorical presheaf on the groupoid core(FinSet).
The category of species, , is the category of presheaves on the groupoid of finite sets and bijections, the core of FinSet:
For more, see structure type.
A 2-species (usually called a stuff type) is a 2-presheaf on core(FinSet), i.e. a pseudofunctor
to the 2-category Grpd.
Generally, an -species or homotopical species is an (∞,1)-presheaf on , i.e. an (∞,1)-functor
to the (∞,1)-category ∞Grpd.
The (∞,1)-category of -species is the (∞,1)-category of (∞,1)-presheaves
Operations on species
There are in fact 5 important monoidal structures on the category of species. For a discussion of all five, you’ll currently have to read about Schur functors, where these operations are discussed in the context of -valued species, i.e. -valued presheaves on the groupoid of finite sets. But here are two:
The sum of two species , is their coproduct . Since colimits of presheaves are computed objectwise, we have
The category becomes a monoidal category under disjoint union of finite sets. This monoidal structure induces canonically the Day convolution monoidal structure on .
For and two combinatorial species, their product is given by the coend
Under groupoid cardinality
every (tame) ∞-groupoid is mapped to a real number
A species assigns an ∞-groupoid to each natural number . Therefore under groupoid cardinality we may naturally think of a tame species as mapping to a power series
This cardinality operation maps the above addition and multiplication of combinatorial species to addition and multiplication of power series.
That coproduct of species maps to sum of their cardinalities is trivial. That Day convolution of species maps under cardinality to the product of their cardinality series depends a little bit more subtly on the combinatorial prefactors:
If in the definition of combinatorial species the domain core(FinSet) is replaced with FinVect and also the presheaves are take with values in FinVect then one obtains the notion of Schur functor.
The notion of species goes back to
An expositional discussion can be found at
- Todd Trimble, Exponential Generating Function and Introduction to Species (blog) (scroll down a bit).
See also wikipedia: combinatorial species and
- François Bergeron, Gilbert Labelle, Pierre Leroux, Théorie des espèces et combinatoire des structures arborescentes , LaCIM, Montréal (1994). English version: Combinatorial species and tree-like structures, Cambridge University Press (1998).
- F. Bergeron, G. Labelle, P. Leroux, Introduction to the theory of species of structures, 2008, pdf
- François Bergeron, Species and variations on the theme of species, invited talk at Category Theory and Computer Science ‘04, Copenhagen (2004). Slides (pdf).
- G. Labelle, video intro into combinatorial species at Newton Institute, Cambridge 2008
- Marcelo Aguiar, Swapneel Mahajan, Monoidal Functors, Species and Hopf Algebras, With forewords by Kenneth Brown, Stephen Chase, André Joyal. CRM Monograph Series 29 Amer. Math. Soc. 2010. lii+784 pp. (pdf draft)
An application in computer science:
- Brent Yorgey, Random testing and beyond! with combinatorial species, slides pdf; Species and functors and types, oh my, article, pdf
An application in statistical mechanics:
- W. Faris, Combinatorial species and cluster expansions, Mosc. Math. J. 10:4 (2010), 713–727 pdf, MR2791054