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\begin{document}

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\section*{species}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{combinatorics}{}\paragraph*{{Combinatorics}}\label{combinatorics}

[[!include combinatorics - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{1categorical}{1-categorical}\dotfill \pageref*{1categorical} \linebreak
\noindent\hyperlink{2categorical}{2-categorical}\dotfill \pageref*{2categorical} \linebreak
\noindent\hyperlink{categorical}{$(\infty,1)$-categorical}\dotfill \pageref*{categorical} \linebreak
\noindent\hyperlink{operations_on_species}{Operations on species}\dotfill \pageref*{operations_on_species} \linebreak
\noindent\hyperlink{sum}{Sum}\dotfill \pageref*{sum} \linebreak
\noindent\hyperlink{cauchy_product}{Cauchy product}\dotfill \pageref*{cauchy_product} \linebreak
\noindent\hyperlink{hadamard_product}{Hadamard product}\dotfill \pageref*{hadamard_product} \linebreak
\noindent\hyperlink{dirichlet_product}{Dirichlet product}\dotfill \pageref*{dirichlet_product} \linebreak
\noindent\hyperlink{composition_product}{Composition product}\dotfill \pageref*{composition_product} \linebreak
\noindent\hyperlink{InHomotopyTypeTheory}{In Homotopy Type Theory}\dotfill \pageref*{InHomotopyTypeTheory} \linebreak
\noindent\hyperlink{operations_on_species_2}{Operations on species}\dotfill \pageref*{operations_on_species_2} \linebreak
\noindent\hyperlink{coproduct}{Coproduct}\dotfill \pageref*{coproduct} \linebreak
\noindent\hyperlink{hadamard_product_2}{Hadamard product}\dotfill \pageref*{hadamard_product_2} \linebreak
\noindent\hyperlink{HoTTCauchyProduct}{Cauchy product}\dotfill \pageref*{HoTTCauchyProduct} \linebreak
\noindent\hyperlink{composition}{Composition}\dotfill \pageref*{composition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{cardinality}{Cardinality}\dotfill \pageref*{cardinality} \linebreak
\noindent\hyperlink{variants}{Variants}\dotfill \pageref*{variants} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \textbf{(combinatorial) species} is a [[presheaf]] or [[higher category theory|higher categorical]] presheaf on the [[groupoid]] [[core]]([[FinSet]]), the [[permutation groupoid]].

A species is a \emph{[[symmetric sequence]]} by another name. Meaning: they are categorically equivalent notions.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{1categorical}{}\subsubsection*{{1-categorical}}\label{1categorical}

The [[category]] of \textbf{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]]:

\begin{displaymath}
\hat \mathbb{P} := PSh(core(FinSet))
  \,.
\end{displaymath}
For more, see [[structure type]].

\hypertarget{2categorical}{}\subsubsection*{{2-categorical}}\label{2categorical}

A \textbf{2-species} (usually called a [[stuff type]]) is a [[2-presheaf]] on [[core]]([[FinSet]]), i.e. a [[pseudofunctor]]

\begin{displaymath}
core(FinSet) \to Grpd
\end{displaymath}
to the [[2-category]] [[Grpd]].

\hypertarget{categorical}{}\subsubsection*{{$(\infty,1)$-categorical}}\label{categorical}

Generally, an \textbf{$\infty$-species} or \textbf{homotopical species} is an [[(∞,1)-presheaf]] on $core(FinSet)$, i.e. an [[(∞,1)-functor]]

\begin{displaymath}
core(FinSet)^{op} \to \infty Grpd
\end{displaymath}
to the [[(∞,1)-category]] [[∞Grpd]].

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

\begin{displaymath}
\infty Species := PSh_{(\infty,1)}(core(FinSet))
  \,,
\end{displaymath}
For more on this see also below the discussion \emph{\hyperlink{InHomotopyTypeTheory}{In homotopy type theory}}.

\hypertarget{operations_on_species}{}\subsubsection*{{Operations on species}}\label{operations_on_species}

There are in fact 5 important monoidal structures on the category of species.

\hypertarget{sum}{}\paragraph*{{Sum}}\label{sum}

The \textbf{sum} $A + B$ of two species $A$, $B$ is their [[coproduct]] $A \coprod B$. Since [[colimit]]s of [[presheaves]] are computed objectwise, we have

\begin{displaymath}
(A + B)_n \simeq A_n \coprod B_n
  \,.
\end{displaymath}
The generating function of the sum of species is the sum of their generating functions.

\hypertarget{cauchy_product}{}\paragraph*{{Cauchy product}}\label{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 \textbf{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]]

\begin{displaymath}
(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)
  \,.
\end{displaymath}
\hypertarget{hadamard_product}{}\paragraph*{{Hadamard product}}\label{hadamard_product}

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

\begin{displaymath}
(A \times B)_n \simeq A_n \times B_n
  \,.
\end{displaymath}
This operation is often called the \textbf{Hadamard product} of species.

\hypertarget{dirichlet_product}{}\paragraph*{{Dirichlet product}}\label{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 could be called the \textbf{Dirichlet product} of species, or sometimes simply the product, since the Dirichlet series of this product of species is the product of their Dirichlet series. (See \href{https://ncatlab.org/johnbaez/show/Dirichlet+species+and+the+Hasse-Weil+zeta+function}{Dirichlet series and the Hasse--Weil zeta function}.)

\hypertarget{composition_product}{}\paragraph*{{Composition product}}\label{composition_product}

Finally, there is a monoidal structure on the category of species known as \textbf{plethysm} or \textbf{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]].

\hypertarget{InHomotopyTypeTheory}{}\subsection*{{In Homotopy Type Theory}}\label{InHomotopyTypeTheory}

The following discusses formalization of the concept of species in [[homotopy type theory]].

Let [[FinSet]] be the [[type]] of [[finite sets]] (see at \emph{[[hSet]]}). Notice that in [[homotopy type theory]] this automatically comes out as the [[groupoid]] of finite sets (see also at \emph{[[universe in type theory]]}), so we may (and do) suppress writing ``[[core]]'' here.

\begin{defn}
\label{HoTTSpecies}\hypertarget{HoTTSpecies}{}
A \emph{species} is a type $X \colon Type$ equipped with a [[function]] into $FinSet$. Hence the type of all species is the [[dependent sum]]

\begin{displaymath}
Species \coloneqq \sum_{X : Type} (X \to FinSet)
\end{displaymath}
of all [[function types]] $X \to FinSet$ as $X$ ranges over the [[type of types]].

\end{defn}
Write

\begin{displaymath}
FinSet \longrightarrow {\|FinSet\|_{0}} \simeq \mathbb{N}
\end{displaymath}
for the [[unit of a monad|unit]] of the [[n-truncation modality|0-truncation modality]] on [[FinSet]], going into the [[type of natural numbers]].

Given a species $f \colon X \to FinSet$ as in def. \ref{HoTTSpecies}, its ``[[decategorification]]'' is the composite

\begin{displaymath}
f \circ card \colon X \stackrel{f}{\longrightarrow} FinSet \to {\|FinSet\|_{0}} \simeq \mathbb{N}
\end{displaymath}
\begin{defn}
\label{HoTTGeneratinFunction}\hypertarget{HoTTGeneratinFunction}{}
The \emph{[[generating function]]} of a species $f \colon X \to FinSet$, def. \ref{HoTTSpecies}, is

\begin{displaymath}
{|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!}
  \,,
\end{displaymath}
where $fib(-)$ denotes the [[homotopy fiber]] and $Fin(n)$ denotes a [[0-type]] with exactly $n$ elements.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
If $\Phi : FinSet \to Type$ is a family of types, then we obtain the species

\begin{displaymath}
pr_{1} : \bigg(\sum_{A : FinSet} \Phi(A)\bigg) \to FinSet
\end{displaymath}
with generating function

\begin{displaymath}
{|\Phi|}(z)
  =
  \sum_{n=0}^{\infty} {|\Phi(Fin(n))|} \frac{z^{n}}{n!}
\end{displaymath}
\end{defn}
\begin{example}
\label{}\hypertarget{}{}
If $\Phi$ is a [[stuff, structure, property|stuff, structure, or property]] on finite sets, then we immediately know something about its generating function, considered as an exponential generating function:

\begin{itemize}%
\item If $\Phi(A)$ is [[contractible type|contractible]] for all $A$, then $\Phi(Fin(n)) = 1$, so its generating function is the [[exponential function]] $e^{z}$.


\item 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.


\item 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]] $\mathbb{N}$.


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



\end{itemize}
\end{example}
\begin{remark}
\label{}\hypertarget{}{}
Since any species $f : X \to Type$ is equal to

\begin{displaymath}
pr_{1} : \bigg(\sum_{x:X}fib_{f}(x)\bigg) \to Type
\end{displaymath}
every species is equal to one obtained from a [[stuff type]].

\end{remark}
\hypertarget{operations_on_species_2}{}\subsubsection*{{Operations on species}}\label{operations_on_species_2}

The five monoidal structures mentioned above under \hyperlink{operations_on_species}{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 (\hyperlink{Dougherty15}{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.

\hypertarget{coproduct}{}\paragraph*{{Coproduct}}\label{coproduct}

The recursion principle for the coproduct gives the species

\begin{displaymath}
\itexarray{
  (f + g) &:& X + Y \to Finset \\
  (f + g) &:& rec_{X + Y}(FinSet, f, g) = z \mapsto \left\{\itexarray{ f(x) & \text{if}\; z \equiv inl x \\ g(y) & \text {if}\; z \equiv inr y .}\right.
}
\end{displaymath}
If $X$ and $Y$ are obtained from $\Phi$ and $\Psi$, respectively, then the coproduct is equal to

\begin{displaymath}
\sum_{A : FinSet} (\Phi(A) + \Psi(A))
\end{displaymath}
The generating function for the coproduct is

\begin{displaymath}
{|X + Y|}(z) = {|X|}(z) + {|Y|}(z)
\end{displaymath}
\hypertarget{hadamard_product_2}{}\paragraph*{{Hadamard product}}\label{hadamard_product_2}

The Hadamard product of species is

\begin{displaymath}
\itexarray{
  \langle X, Y \rangle &:& X \times_{FinSet} Y \to FinSet \\
  \langle X, Y \rangle &:& (x, y, p) \mapsto f(x)
}
\end{displaymath}
which for stuff types is equal to

\begin{displaymath}
\itexarray{
  \sum_{A : FinSet} (\Phi(A) \times \Psi(A))
}
\end{displaymath}
The generating functions are related by

\begin{displaymath}
{|\langle X, Y \rangle|}(z)
  =
  \sum_{n=0}^{\infty} n! X_{n} Y_{n} z^{n}
\end{displaymath}
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 (\hyperlink{AguiarMahajan10}{Aguiar-Mahajan10}) and (\hyperlink{Bergeron08}{Bergeron-Labelle-Leroux08}). In (\hyperlink{Baez01}{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.

\hypertarget{HoTTCauchyProduct}{}\paragraph*{{Cauchy product}}\label{HoTTCauchyProduct}

\begin{defn}
\label{HoTTCauchyProduct}\hypertarget{HoTTCauchyProduct}{}
The \emph{Cauchy product} of species is given by sending two species $f,g$ as in def. \ref{HoTTSpecies} to the species given by

\begin{displaymath}
\itexarray{
  (f \cdot g) &:& X \times Y \to FinSet \\
  (f \cdot g) &:& (x, y) \mapsto f(x) + g(y) 
}
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The Cauchy product, def. \ref{HoTTCauchyProduct}, is eqivalently the \href{%28infinity,n%29-category+of+correspondences#PhasedTensorProduct}{phased tensor product} on the slice over the [[symmetric monoidal category|symmetric monoidal]] groupoid $(FinSet, \coprod)$.

\end{remark}
\begin{example}
\label{}\hypertarget{}{}
For [[stuff types]] $\Phi$, $\Psi$, their Cauchy product, def. \ref{HoTTCauchyProduct}, is equal to

\begin{displaymath}
\sum_{A, U, V : FinSet} \sum_{p : U + V = A} (\Phi(U) \times \Psi(V))
\end{displaymath}
In this form the Cauchy product is discussed in (\hyperlink{AguiarMahajan10}{Aguiar-Mahajan 10, def. 8.5}, \hyperlink{AguiarMahajan12}{Aguiar-Mahajan 12, section 2.2}). [[monoids|Monoids]] and [[comonoid]] in the resulting [[monoidal category]] of species are discussed in (\hyperlink{AguiarMahajan10}{Aguiar-Mahajan 10, section 8.2}) and [[Hopf monoids]] in this category in (\hyperlink{AguiarMahajan12}{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

\begin{displaymath}
{|X \cdot Y|}(z) = {|X|}(z) \cdot {|Y|}(z)
  \,.
\end{displaymath}
\end{example}
\hypertarget{composition}{}\paragraph*{{Composition}}\label{composition}

The composition of two species is

\begin{displaymath}
\itexarray{
  (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))
}
\end{displaymath}
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

\begin{displaymath}
\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)
\end{displaymath}
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

\begin{displaymath}
{|X \tilde{\circ} Y|}(z) = {|X|}({|Y|}(z))
\end{displaymath}
This species is also called the [[plethysm]] product.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{cardinality}{}\subsubsection*{{Cardinality}}\label{cardinality}

Under [[groupoid cardinality]]

\begin{displaymath}
{|-|} : \infty Grpd_{tame} \to \mathbb{R}
\end{displaymath}
every (tame) [[∞-groupoid]] is mapped to a [[real number]]

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

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

\begin{displaymath}
{| 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|}
  \,.
\end{displaymath}
\hypertarget{variants}{}\subsection*{{Variants}}\label{variants}

\begin{itemize}%
\item 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 \textbf{[[Schur functor]]}.


\item 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 \textbf{generalised species}. These generalised species can be collected in a cartesian closed 2-category (\hyperlink{FGHW}{FGHW}).



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The notion of species goes back to

\begin{itemize}%
\item [[André Joyal]], \emph{Une th\'e{}orie combinatoire des s\'e{}ries formelles} , Advances in Mathematics 42:1-82 (1981) , \href{http://www.ams.org/mathscinet-getitem?mr=633783}{MR84d:05025}

\end{itemize}
An expositional discussion can be found at

\begin{itemize}%
\item [[Todd Trimble]], \emph{Exponential Generating Function and Introduction to Species} (\href{http://topologicalmusings.wordpress.com/2009/03/28/number-of-idempotent-endofunctions/}{blog}) (scroll down a bit).

\end{itemize}
See also wikipedia: \href{http://en.wikipedia.org/wiki/Combinatorial_species}{combinatorial species} and

\begin{itemize}%
\item Fran\c{c}ois Bergeron, Gilbert Labelle, Pierre Leroux, \emph{Th\'e{}orie des esp\`e{}ces et combinatoire des structures arborescentes} , LaCIM, Montr\'e{}al (1994). English version: [[Combinatorial species and tree-like structures]], Cambridge University Press (1998).


\item F. Bergeron, G. Labelle, P. Leroux, \emph{Introduction to the theory of species of structures}, 2008, \href{http://bergeron.math.uqam.ca/Site/bergeron_anglais_files/livre_combinatoire.pdf}{pdf}


\item Fran\c{c}ois Bergeron, \emph{Species and variations on the theme of species}, invited talk at \href{http://www.itu.dk/research/theory/ctcs2004/}{Category Theory and Computer Science `04}, Copenhagen (2004). Slides (\href{http://bergeron.math.uqam.ca/Especes_trans.pdf}{pdf}).


\item G. Labelle, \href{http://www.sms.cam.ac.uk/media/937;jsessionid=9540827CB40AC9F1E61BF944127EBAF4}{video} intro into combinatorial species at Newton Institute, Cambridge 2008


\item [[Marcelo Aguiar]], [[Swapneel Mahajan]], \emph{[[Monoidal Functors, Species and Hopf Algebras]]}, With forewords by [[Kenneth Brown]], [[Stephen Chase]], [[André Joyal]]. CRM Monograph Series \textbf{29} Amer. Math. Soc. 2010. lii+784 pp. (\href{http://www.math.cornell.edu/~maguiar/a.pdf}{author pdf})


\item [[Marcelo Aguiar]], [[Swapneel Mahajan]], \emph{Hopf monoids in the category of species} (\href{http://arxiv.org/abs/1210.3120}{arXiv:1210.3120})



\end{itemize}
An application in computer science:

\begin{itemize}%
\item [[Brent Yorgey]], \emph{Random testing and beyond! with combinatorial species}, slides \href{http://www.cis.upenn.edu/~byorgey/talks/combtesting-ku-20091124.pdf}{pdf}; \emph{Species and functors and types, oh my}, article, \href{http://www.cis.upenn.edu/~byorgey/papers/species-pearl.pdf}{pdf}

\end{itemize}
Formalization in [[homotopy type theory]]:

\begin{itemize}%
\item [[Brent Yorgey]], \emph{Combinatorial Species and labelled structures.} (\href{http://ozark.hendrix.edu/~yorgey/pub/thesis.pdf}{pdf})


\item [[John Dougherty]], \emph{Species in HoTT} (\href{https://github.com/jdoughertyii/hott-species/blob/master/species.pdf?raw=true}{pdf}) (\href{https://github.com/jdoughertyii/hott-species}{formalization in Coq})



\end{itemize}
An application in statistical mechanics:

\begin{itemize}%
\item W. Faris, \emph{Combinatorial species and cluster expansions}, Mosc. Math. J. \textbf{10}:4 (2010), 713--727 \href{http://www.ams.org/distribution/mmj/vol10-4-2010/faris.pdf}{pdf}, \href{http://www.ams.org/mathscinet-getitem?mr=2791054}{MR2791054}

\end{itemize}
Discussion in relation to [[Feynman diagrams]]:

\begin{itemize}%
\item [[John Baez]] and [[James Dolan]], From finite sets to Feynman diagrams, in \emph{Mathematics Unlimited - 2001 and Beyond}, vol. 1, eds. Bj\"o{}rn Engquist and Wilfried Schmid, Springer, Berlin, 2001, pp. 29-50. (\href{http://arxiv.org/abs/math.QA/0004133}{arXiv})

\end{itemize}
Discussion of generalised species:

\begin{itemize}%
\item M. Fiore, N. Gambino, M. Hyland, G. Winskel, \emph{The cartesian closed bicategory of generalised species of structures}, (\href{http://www1.maths.leeds.ac.uk/~pmtng/Publications/generalised-species.pdf}{pdf})

\end{itemize}
[[!redirects homotopical species]] [[!redirects (infinity,1) species]] [[!redirects ∞-species]] [[!redirects combinatorial species]] [[!redirects species of structure]]



\end{document}