nLab species

Idea

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

A species is a symmetric sequence by another name. Meaning: they are categorically equivalent notions.

Definition

1-categorical

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.

2-categorical

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.

$(\infty,1)$-categorical

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 (Gepner-Haugseng-Kock 17, Def. 3.2.9).

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

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

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.

Sum

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 generating function of the sum of species is the sum of their generating functions.

Cauchy product

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 $Species := PSh(core(FinSet))$. This is often called the Cauchy product of species, or sometimes (confusingly) just the “product”, since the generating function of the Cauchy product of species is the product of their generating functions.

Explicitly, the Cauchy product of species $A$ and $B$ is given by the coend

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

Hadamard product

The category of species also has cartesian products. Since products of presheaves are computed objectwise, we have

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

This operation is often called the Hadamard product of species.

Dirichlet product

The category $core(FinSet)$ also becomes a monoidal category under cartesian product of finite sets. This monoidal structure induces another Day convolution monoidal structure on $Species := PSh(core(FinSet))$. This monoidal structure on species seems to be what Maia and Méndez call the arithmetic product, but here we call it the Dirichlet product of species, since the Dirichlet series of this product of species is the product of their Dirichlet series. (See Dirichlet series and the Hasse–Weil zeta function.)

Composition product

Finally, there is a monoidal structure on the category of species known as plethysm or composition. The generating function of a composite of two species is the composite of their generating functions (in the sense of formal power series). A monoid with respect to the plethysm tensor product is called an operad.

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 type of finite types), so we may (and do) suppress writing “core” here.

Definition

A species is a type $X \colon Type$ equipped with a function into $FinSet$. Hence the type of all species is the dependent sum

$$Species \coloneqq \sum_{X : Type} (X \to FinSet)$$

of all function types $X \to FinSet$ as $X$ ranges over the type of types.

Write

$$FinSet \longrightarrow {\|FinSet\|_{0}} \simeq \mathbb{N}$$

for the unit of the 0-truncation modality on FinSet, going into the type of natural numbers.

Given a species $f \colon X \to FinSet$ as in def. , its “decategorification” is the composite

$$f \circ card \colon X \stackrel{f}{\longrightarrow} FinSet \to {\|FinSet\|_{0}} \simeq \mathbb{N}$$

Definition

The generating function of a species $f \colon X \to FinSet$, def. , is

$${|X|}(z) \coloneqq \sum_{n=0}^{\infty} {\big| fib_{f \circ card}(n) \big|}\, z^{n} = \sum_{n=0}^{\infty} {\big| fib_{f}(Fin(n)) \big|}\, \frac{z^{n}}{n!} \,,$$

where $fib(-)$ denotes the homotopy fiber and $Fin(n)$ denotes a 0-type with exactly $n$ elements.

Definition

If $\Phi : FinSet \to Type$ is a family of types, then we obtain the species

$$pr_{1} : \bigg(\sum_{A : FinSet} \Phi(A)\bigg) \to FinSet$$

with generating function

$${|\Phi|}(z) = \sum_{n=0}^{\infty} {|\Phi(Fin(n))|} \frac{z^{n}}{n!}$$

Example

If $\Phi$ 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 $\Phi(A)$ is contractible for all $A$, then $\Phi(Fin(n)) = 1$, so its generating function is the exponential function $e^{z}$.

• If $\Phi(A)$ is a family of mere propositions, then every coefficient is either $0$ or $1$, according to whether $A$ can be equipped with a $\Phi$-structure or not.

• If $\Phi(A)$ is a family of sets (i.e., a structure type), then every coefficient is an element of the type of natural numbers $\mathbb{N}$.

• If $\Phi(A)$ has a higher homotopy level (i.e., is a stuff type), then the coefficients are in $[0, \infty)$.

Remark

Since any species $f : X \to Type$ is equal to

$$pr_{1} : \bigg(\sum_{x:X}fib_{f}(x)\bigg) \to Type$$

every species is equal to one obtained from a stuff type.

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 $f : X \to \Finset$ and $g : Y \to Finset$ are two species, and $\Phi, \Psi : FinSet \to Type$ are two stuff types.

Coproduct

The recursion principle for the coproduct gives the species

$$\array{ (f + g) &:& X + Y \to Finset \\ (f + g) &:& rec_{X + Y}(FinSet, f, g) = z \mapsto \left\{\array{ f(x) & \text{if}\; z \equiv inl x \\ g(y) & \text {if}\; z \equiv inr y .}\right. }$$

If $X$ and $Y$ are obtained from $\Phi$ and $\Psi$, respectively, then the coproduct is equal to

$$\sum_{A : FinSet} (\Phi(A) + \Psi(A))$$

The generating function for the coproduct is

$${|X + Y|}(z) = {|X|}(z) + {|Y|}(z)$$

Hadamard product

The Hadamard product of species is

$$\array{ \langle X, Y \rangle &:& X \times_{FinSet} Y \to FinSet \\ \langle X, Y \rangle &:& (x, y, p) \mapsto f(x) }$$

which for stuff types is equal to

$$\array{ \sum_{A : FinSet} (\Phi(A) \times \Psi(A)) }$$

The generating functions are related by

$${|\langle X, Y \rangle|}(z) = \sum_{n=0}^{\infty} n! X_{n} Y_{n} z^{n}$$

where $X_{n}$ and $Y_{n}$ are the $n$th coefficients of the functions ${|X|}(z)$ and ${|Y|}(z)$, respectively.

The name “Hadamard product” is used in (Aguiar-Mahajan10) and (Bergeron-Labelle-Leroux08). In (Baez-Dolan01) ${|\langle X, Y \rangle|}(1)$ 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.

Cauchy product

Definition

The Cauchy product of species is given by sending two species $f,g$ as in def. to the species given by

$$\array{ (f \cdot g) &:& X \times Y \to FinSet \\ (f \cdot g) &:& (x, y) \mapsto f(x) + g(y) }$$

Remark

The Cauchy product, def. , is eqivalently the phased tensor product on the slice over the symmetric monoidal groupoid $(FinSet, \coprod)$.

Example

For stuff types $\Phi$, $\Psi$, their Cauchy product, def. , is equal to

$$\sum_{A, U, V : FinSet} \sum_{p : U + V = A} (\Phi(U) \times \Psi(V))$$

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 $\Phi\Psi$ stuff on a finite set is the stuff of “being chopped in two, with $\Phi$ stuff on one part and $\Psi$ stuff on the other”.

The generating function is

$${|X \cdot Y|}(z) = {|X|}(z) \cdot {|Y|}(z) \,.$$

Composition

The composition of two species is

$$\array{ (f \tilde{\circ} g) &:& \bigg(\sum_{x:X}(Fin(card(f(x)) \to Y)\bigg) \to FinSet \\ (f \tilde{\circ} g) &:& (x, P, p) \mapsto \bigoplus_{i=1}^{card(f(x))}g(P(i)) }$$

where $\bigoplus_{i=1}^{n}g(P(i))$ is the $n$-ary direct sum of the finite sets $g(P(i))$. For stuff types, this amounts to

$$\sum_{(A, C : FinSet)}\sum_{(B : Fin(card(C)) \to FinSet)} \bigg( (B\vdash_{card(C)}A) \times \Phi(C) \times \prod_{k : Fin(card(C))} \Psi(B(k)) \bigg)$$

That is, $\Phi \tilde{\circ} \Psi$ stuff on a finite set is a partition of the set into $card(C)$ subsets, with $\Phi$ stuff on the partition and $\Psi$ stuff on each of the subsets.

The generating function of the composition is

$${|X \tilde{\circ} Y|}(z) = {|X|}({|Y|}(z))$$

This species is also called the plethysm product.

Properties

Cardinality

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:

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

Variants

• If in the definition of combinatorial species the domain core(FinSet) is replaced with FinDimVect and also the presheaves are take with values in FinDimVect then one obtains the notion of Schur functor.

• Instead of $core(FinSet)$, we may choose as domain $\mathbf{B} A$ for a small category $A$, with objects $\langle a_i \rangle_{i \in I}$, for $I$ in $core(FinSet)$, i.e., finite families of objects of $A$. Also, instead of $Set$ valued presheaves, we may consider those valued in presheaves on a small category $B$. This joint generalisation yields what are called generalised species. These generalised species can be collected in a cartesian closed 2-category (FGHW).

References

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 expository 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. (author pdf)

• Marcelo Aguiar, Swapneel Mahajan, Hopf monoids in the category of species (arXiv:1210.3120)

The Dirichlet product of species seems to match what Maia and Méndez call the arithmetic product:

• Manuel Maia and Miguel Méndez, On the arithmetic product of combinatorial species, Discrete Mathematics 308, (2008), 5407–5427. (web)

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

Homotopical species are defined in (Def. 3.2.9):

• David Gepner, Rune Haugseng, Joachim Kock, ∞-Operads as Analytic Monads, (arXiv:1712.06469)

Formalization in homotopy type theory:

• Brent Yorgey, Combinatorial Species and labelled structures. (pdf)

• John Dougherty, Species in HoTT (pdf) (formalization in Coq)

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)