… abstractly by the formula
Since each finite bijection $\phi: S \to W \oplus X$ induces a decomposition of $S$ as a disjoint union $T+U$ of subsets (setting $T = \phi^{-1}(W)$ and $U = \phi^{-1}(X)$), this coend formula may be simplified:
(For the remainder of this paragraph, categorical terms such as “category”, “functor”, “colimit”, etc. refer to their $V$-enriched analogues.) The product $F \otimes G$ preserves colimits in the separate arguments $F$ and $G$ (i.e., $- \otimes G$ and $F \otimes -$ are cocontinuous for all $F$ and $G$). Since $F$ and $G$ may be canonically presented as colimits of representables, one may define a symmetric monoidal structure on this product, uniquely up to monoidal isomorphism, so that the Yoneda embedding $y: \mathcal{FB}^{op} \to V^{\mathcal{FB}}$ is a symmetric monoidal functor, i.e., so that there is a coherent isomorphism $\hom(W \oplus X, -) \cong \hom(W, -) \otimes \hom(X, -)$. In conjunction with the universal property of $\mathcal{FB}$, we may state a universal property of $V^{\mathcal{FB}}$: let $C$ be a ($V$-)category which is symmetric monoidally cocomplete (meaning its monoidal product is separately cocontinuous), and let $A$ be an object of $C$. Then there exists a cocontinuous symmetric monoidal functor $V^{\mathcal{FB}} \to C$ sending $\hom(\mathbf{1}, -)$ to $A$, and this functor is unique up to monoidal isomorphism.
This universal property may be exploited to yield a second monoidal structure on $V^{\mathcal{FB}}$. Let $[V^{\mathcal{FB}}, C]$ denote the category of cocontinuous symmetric monoidal functors $V^{\mathcal{FB}} \to C$; then the universal property may be better expressed as saying that the functor $[V^{\mathcal{FB}}, C] \to C$ which evaluates a species $F: \mathcal{FB} \to V$ at $\mathbf{1} \in \mathcal{FB}$ is an equivalence. In the case $C = V^{\mathcal{FB}}$, the left-hand side of the equivalence carries a monoidal structure given by endofunctor composition. This monoidal structure transports across the equivalence to yield a monoidal product on $V^{\mathcal{FB}}$, denoted by $\circ$.
An explicit formula for $\circ$ is given as follows. Under one set of conventions, a $V$-species $F$ may be regarded as a right module over the permutation category $\mathbf{P}$, so that the component $F[n]$ carries an action $F[n] \otimes S_n \to F[n]$. The $n$-fold Day convolution $G^{\otimes n}$ carries, under the same conventions, a left $S_n$-action $S_n \otimes G^{\otimes n} \to G^{\otimes n}$. Thn the coend formula for $F \circ G$ may be written as
A special case of this “substitution product” $\circ$ is the analytic functor construction. For each object $X$ in $V$ there is a $V$-species $\widehat{X}$ such that $\widehat{X}[0] = X$ and $\widehat{X}[n] = 0$ otherwise. Letting $X^n$ denote the $n$-fold tensor product in $V$, we have
and $(F \circ \widehat{X})[n] = 0$ otherwise; we write the right-hand side of the above equation as $F(X)$. This defines a functor $F(-): V \to V$, which we call the analytic functor attached to the species $F$. We often commit an abuse of language and write $F(X)$ for the analytic functor, thinking of the $X$ as a variable or placeholder for an argument, much as one abuses language by referring to a function $f(x) = \sin(x)$.
The analytic functor $F(-): V \to V$ determines, up to isomorphism, its generating species $F$; we describe this determination for the category $V = Vect_k$ of vector spaces over a ground field $k$. Let $F_n(X)$ denote the $n^{th}$-degree component $F[n] \otimes_{S_n} X^n$, and let $X$ be a vector space freely generated from a set $\{x_1, \ldots, x_n\}$ whose cardinality equals that degree. Then the species value $F[n]$ can be recovered as the subspace of $F_n(X)$ spanned by equivalence classes of those expressions $\tau \otimes x_{i(1)} \otimes \ldots \otimes x_{i(n)}$ in which each $x_i$ occurs exactly once. We use the notations $F[n]$ and $F(X)[n]$ interchangeably for these species values.
If $G[0] = 0$, we have $G^{\otimes n}[S] = 0$ whenever $n$ exceeds the cardinality $|S|$, in which case $F \circ G$ makes sense for $V$ finitely cocomplete. For general $n$ we have in that case
where the sum is indexed over ordered partitions of $S$ into $n$ nonempty subsets $T_i$. The group $S_n$ permutes such ordered partitions in such a way that the orbits correspond to unordered partitions, which are tantamount to equivalence relations on $S$. Let $Eq(S)$ denote the set of such equivalence relations, and let $\pi: S \to S/R$ denote the canonical projection of $S$ onto the set of $R$-equivalence classes. Then the substitution product may be rewritten as
whenever $G[0] = 0$.
An operad in $V$ is a monoid in the monoidal category $(V^{\mathcal{FB}}, \circ)$.
The unit for the monoidal product $\circ$ will be denoted $X$; it is defined by $X[n] = 0$ if $n \geq 1$, and $X[1] = I$ where $I$ is the monoidal unit of $V$. (We repeat that we also use $X$ as abusive notation for a placeholder or variable with values ranging over objects of $V$.)
Clearly an operad $M$ induces a monad $M \circ -: V^{\mathcal{FB}} \to V^{\mathcal{FB}}$, which in turn restricts to the analytic monad $M(-): V \to V$ along the embedding $\widehat{(-)}: V \to V^{\mathcal{FB}}$ if $V$ is cocomplete. Many algebraic structures arising in practice are algebras of analytic monads. For $V = Vect_k$, we have, e.g.,
The tensor algebra $T(X) = 1 + X + X^{\otimes 2} + \ldots$, denoted $\frac1{1-X}$. The algebras of $T(-)$ are associative algebras. The species value $T[n]$ is the space freely generated from the set of linear orders on an $n$-element set, with the evident $S_n$-action.
The symmetric algebra $S(X) = 1 + X + X^{\otimes 2}/S_2 + \ldots$, denoted $\exp(X)$. Algebras of $S(-)$ are commutative associative algebras. The species value $\exp[n]$ is the trivial 1-dimensional representation of $S_n$.
The Lie operad $L[-]$ may be presented as an operad generated by a binary operation $[-, -] \in L[2]$, subject to the Jacobi relation $[-, [-, -]] + [-, [-, -]]\sigma + [-, [-, -]]\sigma^2 = 0$ and the alternating relation $[-, -] + [-, -]\tau = 0$, where $\sigma$ is a 3-cycle and $\tau$ is a 2-cycle. The algebras of the analytic monad $L(-)$ are Lie algebras.
By way of contrast, Boolean algebras are not algebras of an analytic monad, since the equation $x \wedge x = x$ inevitably involves the use of a diagonal map not available in $\mathcal{FB}$.
Although analytic monads are obviously important, we stress that they are simply restrictions of monads $V^{\mathcal{FB}} \to V^{\mathcal{FB}}$, and that it is often more flexible to work in the latter setting. For example, if $V$ is only finitely cocomplete, then analytic monads cannot be defined in general; for example, the free commutative monoid construction does not define an analytic monad on finite-dimensional vector spaces. However, if $V$ is finitely cocomplete and $M$ is a $V$-operad such that $M[0] = 0$, then there is a monad $M \circ -$ acting on the orthogonal complement $V^\perp \hookrightarrow V^{\mathcal{FB}}$, i.e., the full subcategory of $V$-species $G$ such that $G[0] = 0$.
This situation occurs often. For example, consider the operad $M(X) = \exp(X)-1$, whose $Vect_k$-algebras are commutative monoids without unit. This operad induces a monad $M \circ -$ on $V^\perp$ where $V$ is the category of finite-dimensional vector spaces; the algebras are again commutative ($V^\perp$-)monoids without unit. The operad itself can be regarded as the free commutative algebra without unit, $M \circ X$, generated by the monoidal unit $X$ considered as living in the subcategory $V^\perp$. For other reasons, Markl (see references) has also considered algebras over monads $M \circ -$ and $- \circ M$, more general than algebras of analytic monads; he refers to the former as “$M$-modules”.
In the next few sections, we will reprise the beautiful work of Joyal which leads up to a computation of the Lie species $L$. Our general methodology is to reinterpret Joyal’s approach via virtual species by appeal to dg-structures on finite-dimensional super vector spaces; ultimately we feel that a proper approach to virtual species should draw on the standard model category structure on this category.
A second point of our approach is to place Joyal’s calculations within the context of a particular bar construction. This will better enable us to compare these calculations with those of Ginzburg-Kapranov, which involve a slightly different bar construction.
From this section on, we fix a ground field $k$ of characteristic 0, and $V$ henceforth denotes the category of finite-dimensional vector spaces over $k$.
To study the Lie operad $L[-]$, Joyal and others (see reference to Hinich and Schechtman) take as starting point the Poincaré-Birkhoff-Witt (PBW) theorem. If $L$ is a Lie algebra, then its universal enveloping algebra $U(L)$ carries a canonical filtration, inherited as a quotient of the tensor algebra $T(L)$ equipped with the degree filtration. Embedded in $T(L)$ as a filtered subspace is the symmetric algebra $S(L)$, whose homogeneous components $S_n(L)$ may be realized as images of symmetrizing operators acting on components $T_n(L) = L^{\otimes n}$:
We obtain a composite of maps of filtered spaces
and the PBW theorem concerns the application of the associated graded space functor to this composite (denoted $\phi$):
The graded map $\phi_{gr}$ induces an isomorphism between $U_{gr}(L)$ and $S_{gr}(L)$ as graded spaces.
If $L(X)$ is the free Lie algebra on $X$, then $U(L(X)) \cong T(X)$ (as algebras even) by an adjoint functor argument. Assembling some prior notation, it follows from PBW that there exists an isomorphism of analytic functors on $Vect_k$:
which in turn determines a species-isomorphism, which componentwise is an isomorphism of $S_n$-representations:
Notice both sides makes sense as $V$-species. A guiding idea behind the species methodology is that the components of such species are structural analogues of coefficients of formal power series. This analogy can be made precise. Let $V[ [x]]$ denote the rig (ring without additive inverses) of isomorphism classes of $V$-species, with $+$ given by coproduct and $\cdot$ given by $\otimes$. Let $\mathbb{N}[ [x]]$ denote the rig of formal power series
where the coefficients $a_n$ are natural numbers. This may be equivalently defined to be the rig of sequences $a_n$ of natural numbers where two sequences are multiplied according to the rule
The rigs $V[ [x]]$ and $\mathbb{N}[ [x]]$ also have a partially defined composition operation $\circ$, where $f \circ g$ is defined whenever $g(0) = 0$. In the case of $V[ [x]]$, it is of course the operation which is descended from the substitution product by passing to isomorphism classes.
(See reference to Joyal’s first paper on species) The function $\dim: V[ [x]] \to \mathbb{N}[ [x]]$, sending $F$ to the sequence of coefficients $a_n = \dim(F[n])$, is a rig homomorphism which preserves the $\circ$ operation.
It follows from the proposition and the preceding species isomorphism that $\dim(L[n]) = (n-1)!$, since $\dim(L)(x)$ is the formal power series expansion of $-\log(1-x)$. We are interested in finding an appropriate lift of $-\log(1-x) \in \mathbb{N}[ [x]]$ to a species $-\log(1-X)$ in $V[ [x]]$, and hence an identification between $-\log(1-X)$ and the Lie species $l[-]$.
Before we construct the species $\log(1-X)$, it is convenient to complete the rig $V[ [x]]$ to a ring. One proceeds exactly as in K-theory, where one passes from vector bundles to virtual bundles.
$\mathbf{V}$ is the category of $\mathbb{Z}_2$-graded finite-dimensional vector spaces, with monoidal product given by the formula
and symmetry given by the formula
for $x_m \in V_m$ and $y_n \in W_n$.
Let $F$ and $G$ be $\mathbf{V}$-species. Then $F \sim G$ ($F$ and $G$ are virtually equivalent) if $F_0 \oplus G_1 \cong F_1 \oplus G_0$ as $V$-species.
The realtion $\sim$ is an equivalence relation.
Transitivity follows from cancellation: $F \oplus H \cong G \oplus H$ implies $F \cong G$. This in turn follows from the remark.
It is of course crucial here to work with finite-dimensional vector spaces throughout, in order to avoid the Eilenberg swindle.
The relation $\sim$ is respected by $\oplus$, $\otimes$, and $\circ$.
The proof is left to the reader; see also references to Joyal and Yeh.
A virtual species is a virtual equivalence class of $\mathbf{V}$-species. The ring of equivalence classes is denoted $\mathbf{V}[ [x]]$.
Many of our calculations refer to manipulations in the ring $\mathbf{V}[ [x]]$ of virtual species, but methodologically it is useful to distinguish the various ways in which virtual equivalences arise. Part of the philosophy behind species is that clarity is promoted and calculations are under good combinatorial control when power series operations $+$, $\cdot$, and $\circ$ can be viewed as arising from categorified functorial operations $\oplus$, $\otimes$, and $\circ$. Put differently, passage from $\mathbf{V}^{\mathcal{FB}}$ to $\mathbf{V}[ [x]]$ loses categorical information, and it helps to recognize when a virtual equivalence $F \sim G$ comes from an isomorphism $F \cong G$ in $\mathbf{V}^{\mathcal{FB}}$.
In practice, many virtual equivalences which do not come from isomorphisms in $\mathbf{V}^{\mathcal{FB}}$ come about by applying the following remark.
This principle can be quite powerful. Its application does not commit one to any particular choice of differential structure on $C$, so that one is enabled to choose differentials to suit the local occasion. The downside is that because there is no canonical way to split exact sequences, it is sometimes harder to give precise formulas that exhibit such virtual equivalences $C \sim H(C)$.
One of our goals is to lift the inversion
from the ring $\mathbb{Z}[ [x]]$ to the ring $\mathbf{V}[ [x]]$. In either ring, a necessary condition for $F(x)$ to have an inverse $F^{-1}(x)$ (with respect to $\circ$) is that the $0^{th}$ coefficient $F[0]$ be $0$. Thus, instead of inverting $\exp(X)$, we invert $\exp(X)-1$. Suppose then that $\log(1+X)$ is a virtual inverse of $\exp(X)-1$:
If $F$ and $G$ are $\mathbf{V}$-species such that $F[0] = 0 = G[0]$, then $\exp(F \oplus G) \cong \exp(F) \otimes \exp(G)$.
It is immediate that $\exp(F) = \sum_{n \geq 0} F^{\otimes n}/S_n$ is the free commutative monoid in $(\mathbf{V}^{\mathcal{FB}}, \otimes)$ generated from $F$, and the assertion says that the left adjoint $\exp$ preserves coproducts.
Defining $\log((1+F) \otimes (1+G))$ to be
it follows that $\log((1+F) \otimes (1+G)) \sim \log(1+F) + \log(1+G)$. In particular,
where of course $-(F_0, F_1)$ is defined to be $(F_1, F_0)$. The species $\log(1-X)$ is easily obtained from $\log(1+X)$ by the following result.
$F(-X)[S] \cong (-1)^{|S|} F[S] \otimes \Lambda[S]$, where $\Lambda[S]$ denotes the top exterior power $\Lambda^{|S|}(k S)$ of the vector space $k S$ freely generated from $S$.
The $\mathbf{V}$-species $X$, which by definition is the unit with respect to $\circ$, is given by $(X[1]_0, X[1]_1) = (k, 0)$ and $X[n] = 0$ otherwise. Thus $(-X[1]_0, -X[1]_1) = (0, k)$ and $-X[n] = 0$ otherwise. Hence
Now $(-X)[1]^{\otimes n}$ is 1-dimensional and is concentrated in degree $n \pmod 2$ (whence the sign $(-1)^{|S|}$). A transposition in $S_n$ induces a sign change in $(-X[1]^{\otimes n})_{n \pmod 2}$, by definition of symmetry: $\sigma(x_1 \otimes y_1) = - y_1 \otimes x_1$. This proves the claim.
We proceed to compute the inverse $\log(1+X)$ to $\exp(X)-1$. Recalling an earlier remark, $F(X) = \exp(X)-1$ is the operad such that algebras of the monad $F \circ -$ (acting on $\mathbf{V}$-species $G$ such that $G[0] = 0$) are commutative algebras without unit.
The underlying species $F$ satisfies $F[0] = 0$, $F[1]= 1$ (i.e., $= (k, 0)$). Joyal gives a general method due to G. Labelle for inverting such species. Introduce an operator
so that $(1 + \delta_F)(H) = H \circ F$; here $1$ denotes the identity functor. Observe that $\delta_F$ preserves sums, because $- \circ F: \mathbf{V}^{\mathcal{FB}} \to \mathbf{V}^{\mathcal{FB}}$ is the restriction of a cocontinuous monoidal functor which thus preserves coproducts.
Let $\widehat{0}$ denote the bottom element of the lattice $eq(S)$ ordered by inclusion of equivalence relations. This $\widehat{0}$ is the discrete equivalence relation on $S$, so that $S/\widehat{0} \cong S$. We have
and since $H[S] \cong H[S/\widehat{0}] \otimes \bigotimes_{x \in S/\widehat{0}} F[1]$ by our assumptions on $F$, we may rewrite the right-hand side (up to virtual equivalence) as
Define the $\mathbf{V}$-species $\delta_F(H)$ by the above expression, so that $\delta_F$ will be used to denote a functor on $\mathbf{V}$-species $H$, in addition to an operator on $\mathbf{V}[ [x]]$. In particular, when $F(X) = \exp(X)-1$, we have
In general, the $n^{th}$ iterate $\delta_{F}^{n}(H)[S]$ is a sum of the form
where terms are indexed by strictly increasing chains of equivalence relations on $S$. As soon as $n \geq |S|$, there are no chains of that length, so this sum will be empty. In this way, for each finite $S$, $\delta_{F}^{n}(H)[S] = 0$ for all sufficiently large $n$, and so the expression
makes sense as a functor on $\mathbf{V}$-species $H$.
We may now construct the inverse species $F^{-1}(X)$:
$(F^{-1} \circ F)(X) \sim X$.
We have
which telescopes down to $\delta_{F}^{0}(X) = X$.
When $F$ carries an operad structure, this construction of $F^{-1}$ admits a more categorical interpretation. Observe that there is an embedding
Let us regard the operad $F$ as a monoid with multiplication $m: F \circ F \to F$ and unit $u: X \to F$. There is a (necessarily unique) operad map $\varepsilon: F \to X$, called an augmentation, and this may be used to turn $X$ into a left $F$-module and also into a right $F$-module, in the usual way. We may thus form a two-sided bar construction $B(X, F, X)$, whose component in dimension $n$ is isomorphic to $F^{\circ n}$.
The bar construction $B(X, F, X)$ is a simplicial object in an additive category, and hence gives rise to a $\mathbb{Z}$-graded chain complex, where each differential is a signed sum of face maps of the form
By reduction of the grading, we may regard $B(X, F, X)$ as a $\mathbb{Z}_2$-graded chain complex, provided that the two components are taken as species valued in the category of (possibly infinite-dimensional) vector spaces.
However, since we are dealing with virtual species, we want to cut back to $\mathbf{V}$-valued species, where the components are finite-dimensional. To this end, notice that each of the maps $\partial_i$ restricts to a map
and the $\delta_{F}^{n}(X)$ form a chain subcomplex. We regard this chain complex as our preferred bar construction for $F$, or more precisely a right bar construction $B_r(F)$, as we now explain in more detail.
There is an exact sequence
making the functor $\delta_F$, for an operad $F$, analogous to tensoring on the right with an augmentation ideal $I G$ of a group ring $\mathbb{Z}G$, sitting in an exact sequence
Here $\delta_F$ (resp. $- \otimes I G$) is regarded as a formal or virtual difference between $- \circ F$ and $- \circ X$ (resp. $- \otimes \mathbb{Z}G$ and $- \otimes \mathbb{Z}$). In forming $F^{-1}(X)$ as
we interpret the additive inverse $-\delta_F$ as the result of composing $\delta_F$ with the degree 1 shift operator $\Sigma$ acting on $\mathbb{Z}_2$-graded species, which plays the structural role of additive inverse. Hence the construction for $F^{-1}(X)$ above is analogous to the normalized bar resolution
originally introduced by Eilenberg-Mac Lane (reference).
To give further weight to the sense in which $B_r(F)$ is a bar construction, we assemble the components $\delta_{F}^{n}(X) \circ F$ of $F^{-1} \circ F$ into an acyclic chain complex $E_r(F)$ of $\mathbf{V}$-species that will be a (right) $F$-free resolution of the unit operad $X$. Indeed, there is an embedding
where now $F^{\circ (n+1)}$ is the degree $n$ component of the two-sided bar resolution $B(X, F, F)$ (as simplicial object). Again, the face maps
restrict to maps
so that the $\delta_{F}^{n}(X) \circ F$ are components of a subcomplex $E_r(F)$. This endows the $\mathbb{Z}_2$-graded species $F^{-1} \circ F$ with a dg-structure.
Then, we may regard the virtual equivalence
as arising from a homotopy equivalence between dg-species:
Indeed, we have an augmentation map $B(X, F, F) \to X$ between simplicial objects (regarding $X$ on the right as a constant simplicial object). This restricts to an augmentation map $E_r(F) \to X$ which, by the Dold-Kan correspondence, corresponds to a map between chain complexes as displayed above. To check that the augmentation $E_r(F) \to X$ is a homotopy equivalence, one checks that the standard contracting homotopy on $B(X, F, F)$, with components
restricts to a contracting homotopy on $E_r(F)$. In more detail, each component $\delta_{F}^{n}(X) \circ F$ breaks up as a coproduct
We now return to our example $F(X) = \exp(X)-1$ and the construction of the inverse $F^{-1}(X) = \log(1+X)$. By iterating the functor
we derive an expression of $\delta_{F}^{n}(X)[S]$ as the space whose basis elements are $n$-fold chains of strict inclusions of equivalence relations
where $\widehat{1}$ denotes the top element of the lattice $Eq(S)$, namely the indiscrete equivalence relation with one equivalence class. Each such chain may be identified with an $(n-1)$-fold chain in the poset $Eq(S) - \{\widehat{0}, \widehat{1}\}$, in other words as cells of dimension $n-2$ in the simplicial complex underlying the nerve of this poset. Let $C_i[S]$ denote the set of cells of dimension $i$ (or rather the vector space it generates). Then for $|S| \gt 2$, we may identify the complex $B_r(F)[S]$,
with
It follows that $\log(1+X)[S]$ is virtually equivalent to reduced homology of the simplicial complex $C$ associated with $Eq(S)$.
Now the lattice of equivalence relations is a geometric lattice, and a result of Folkman is that the reduced homology of such a lattice is trivial except in top degree; more exactly, Folkman’s result implies that the nerve of the poset $Eq(S) - \{\widehat{0}, \widehat{1}\}$ is a bouquet of spheres of dimension $|S|-3$, and the number of spheres is given by the value $\mu(\widehat{0}, \widehat{1})$ of Rota’s Möbius function on the lattice $Eq(S)$.
$\log(1+X)[S] \sim (-1)^{|S|-1} H_{|S|-1}(B_r(F)[S]).$
By the structural Euler formula, $F^{-1}(X)[S] = \log(1+X)$ as given by the $\mathbb{Z}_2$-graded chain complex $B_r(F)[S]$ is virtually equivalent to its homology, which in turn is equivalent to the reduced homology of $C[S]$. By Folkman’s theorem, the reduced homology of $C[S]$ is concentrated in top degree $|S|-3$, corresponding to the homology of $B_r(F)[S]$ sitting in degree $|S|-1$.
The Lie species $L[S]$ is isomorphic to $H_{|S|-1}(B_r(F)[S]) \otimes \Lambda[S]$.
Starting from the PBW theorem
we were led to the virtual equivalence
Applying Joyal’s rule of signs to the previous theorem, $-\log(1-X)[S]$ is virtually equivalent to
which boils down to $H_{|S|-1}(B_r(F)[S]) \otimes \Lambda[S]$. As virtual species, both this and the Lie species value $L[S]$ are concentrated in degree $0 \pmod 2$. But if two $\mathbf{V}$-species concentrated in degree 0 are virtually equivalent, they are isomorphic.