species

A **(combinatorial) species** is a presheaf or higher categorical presheaf on the groupoid core(FinSet).

The category of **species**, $Species := PSh(core(FinSet))$, is the category of presheaves on the groupoid $\mathbb{P} := core(FinSet)$ of finite sets and bijections, the core of FinSet:

$\hat \mathbb{P} := PSh(core(FinSet))
\,.$

For more, see structure type.

A **2-species** (usually called a stuff type) is a 2-presheaf on core(FinSet), i.e. a pseudofunctor

$core(FinSet) \to Grpd$

to the 2-category Grpd.

Generally, an **$\infty$-species** or **homotopical species** is an (∞,1)-presheaf on $core(FinSet)$, i.e. an (∞,1)-functor

$core(FinSet)^{op} \to \infty Grpd$

to the (∞,1)-category ∞Grpd.

The (∞,1)-category of $\infty$-species is the (∞,1)-category of (∞,1)-presheaves

$\infty Species := PSh_{(\infty,1)}(core(FinSet))
\,,$

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 $Fin\Vect$-valued species, i.e. $Fin\Vect$-valued presheaves on the groupoid of finite sets. But here are two:

The **sum** $A + B$ of two species $A$, $B$ is their coproduct $A \coprod B$. Since colimits of presheaves are computed objectwise, we have

$(A + B)_n \simeq A_n \coprod B_n
\,.$

The category $core(FinSet)$ becomes a monoidal category under disjoint union of finite sets. This monoidal structure $(core(FinSet), \coprod)$ induces canonically the Day convolution monoidal structure on $Species := PSh(core(FinSet))$.

For $A$ and $B$ two combinatorial species, their product is given by the coend

$(A \otimes B)_n
\simeq
\int^{n \in FinSet} A_k \times B_l \times Hom_{core(FinSet)}(n,k+l)
\simeq
\coprod_{k+l = n} \prod_{\frac{(k+l)!}{k! + l!}} (A_k \times B_l)
\,.$

Under groupoid cardinality

${|-|} : \infty Grpd_{tame} \to \mathbb{R}$

every (tame) ∞-groupoid is mapped to a real number

$X \mapsto {|X|} := \sum_{[x] \in \pi_0(X)}\prod_{i = 1}^{\infty} (\pi_i(X,x)^{(-1)^{i}})
\,.$

A species $\mathbf{X}$ assigns an ∞-groupoid $\mathbf{X}_n$ to each natural number $n \in \mathbb{Z}$. Therefore under groupoid cardinality we may naturally think of a tame species as mapping to a power series

$\mathbf{X} \mapsto {|\mathbf{X}|}
:=
\sum_{n = 0}^{\infty} \frac{1}{n!} {|\mathbf{X}_n|} z^n
\in
\mathbb{R}[ [ z ] ]
\,.$

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:

${| A \otimes B |}
=
{|A|} \cdot {|B|}
=
\sum_{n=0}^\infty
\frac{1}{n!}
\sum_{k+l = n} \frac{n!}{k! l!}
{|A_k|} \cdot {|B_l|}
\,.$

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

- André Joyal,
*Une théorie combinatoire des séries formelles*, Advances in Mathematics 42:1-82 (1981) doi, MR84d:05025

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

Revised on September 19, 2012 11:38:58
by Urs Schreiber
(82.169.65.155)