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
For more on this see also below the discussion In homotopy type theory.
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
In Homotopy Type Theory
The following discusses formalization of the concept of species in homotopy type theory.
Let FinSet be the type of finite sets (see at hSet). Notice that in homotopy type theory this automatically comes out as the groupoid of finite sets (see also at universe in type theory), so we may (and do) suppress writing “core” here.
A species is a type equipped with a function into . Hence the type of all species is the dependent sum
of all function types as ranges over the type of types.
for the unit of the 0-truncation modality on FinSet, going into the type of natural numbers.
Given a species as in def. 1, its “decategorification” is the composite
The generating function of a species , def. 1, is
where denotes the homotopy fiber and denotes a 0-type with exactly elements.
If is a family of types, then we obtain the species
with generating function
If is a stuff, structure, or property on finite sets, then we immediately know something about its generating function, considered as an exponential generating function:
If is contractible for all , then , so its generating function is the exponential function .
If is a family of mere propositions, then every coefficient is either or , according to whether can be equipped with a -structure or not.
If is a family of sets (i.e., a structure type), then every coefficient is an element of the type of natural numbers .
If has a higher homotopy level (i.e., is a stuff type), then the coefficients are in .
Operations on species
The five monoidal structures mentioned above under Operations on species can be represented in HoTT, along with a couple other useful operations. We give four of the five monoidal structures here. For more operations, see (Dougherty15). We assume throughout this subsection that and are two species, and are two stuff types.
The recursion principle for the coproduct gives the species
If and are obtained from and , respectively, then the coproduct is equal to
The generating function for the coproduct is
The Hadamard product of species is
which for stuff types is equal to
The generating functions are related by
where and are the th coefficients of the functions and , respectively.
The name “Hadamard product” is used in (Aguiar-Mahajan10) and (Bergeron-Labelle-Leroux08). In (Baez-Dolan01) it is called the “inner product” of stuff types, because equipping the category of stuff types with this operation makes it a categorified version of the Hilbert space of a quantum harmonic oscillator.
The Cauchy product of species is given by sending two species as in def. 1 to the species given by
For stuff types , , their Cauchy product, def. 4, is equal to
In this form the Cauchy product is discussed in (Aguiar-Mahajan 10, def. 8.5, Aguiar-Mahajan 12, section 2.2). Monoids and comonoid in the resulting monoidal category of species are discussed in (Aguiar-Mahajan 10, section 8.2) and Hopf monoids in this category in (Aguiar-Mahajan 12, section 2.2).
This means that stuff on a finite set is the stuff of “being chopped in two, with stuff on one part and stuff on the other”.
The generating function is
The composition of two species is
where is the -ary direct sum of the finite sets . For stuff types, this amounts to
That is, stuff on a finite set is a partition of the set into subsets, with stuff on the partition and stuff on each of the subsets.
The generating function of the composition is
This species is also called the plethysm product.
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.
Instead of , we may choose as domain for a small category , with objects , for in , i.e., finite families of objects of . Also, instead of valued presheaves, we may consider those valued in presheaves on a small category . This joint generalisation yields what are called generalised species. These generalised species can be collected in a cartesian closed 2-category (FGHW).
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)
Marcelo Aguiar, Swapneel Mahajan, Hopf monoids in the category of species (arXiv:1210.3120)
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
Formalization in homotopy type theory:
An application in statistical mechanics:
- W. Faris, Combinatorial species and cluster expansions, Mosc. Math. J. 10:4 (2010), 713–727 pdf, MR2791054
Discussion in relation to Feynman diagrams:
- John Baez and James Dolan, From finite sets to Feynman diagrams, in Mathematics Unlimited - 2001 and Beyond, vol. 1, eds. Björn Engquist and Wilfried Schmid, Springer, Berlin, 2001, pp. 29-50. (arXiv)
Discussion of generalised species:
- M. Fiore, N. Gambino, M. Hyland, G. Winskel, The cartesian closed bicategory of generalised species of structures, (pdf)