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\title{\vspace{-2em}Notes on the Lie Operad}
\author{Todd H. Trimble\\Department of Mathematics\\University of Chicago}
\date{}
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\maketitle
\begin{center}
\footnotesize
\emph{These notes have been re\LaTeX ed by Tim Hosgood, who takes responsibility for any errors.}
\end{center}
The purpose of these notes is to compare two distinct approaches to the Lie operad.
The first approach closely follows work of Joyal \cite{tlo10}, which gives a species-theoretic account of the relationship between the Lie operad and homology of partition lattices.
The second approach is rooted in a paper of Ginzburg-Kapranov \cite{tlo7}, which generalizes within the framework of operads a fundamental duality between commutative algebras and Lie algebras.
Both approaches involve a bar resolution for operads, but in different ways: we explain the sense in which Joyal’s approach involves a “right” resolution and the Ginzburg-Kapranov approach a “left” resolution.
This observation is exploited to yield a precise comparison in the form of a chain map which induces an isomorphism in homology, between the two approaches.
\section{Categorical Generalities}
Let $\mathcal{F}\mathcal{B}$ denote the category of finite sets and bijections. $\mathcal{F}\mathcal{B}$ is equivalent to the permutation category $\mathbf{P}$, whose objects are natural numbers and whose set of morphisms is the disjoint union $\sum_{n\geqslant0} S_n$ of all finite symmetric groups.
$\mathbf{P}$ (and therefore also $\mathcal{F}\mathcal{B}$) satisfies the following universal property: given a symmetric monoidal category $\mathcal{C}$ and an object $A$ of $\mathcal{C}$, there exists a symmetric monoidal functor $\mathbf{P}\to\mathcal{C}$ which sends the object $\mathbf{1}$ of $\mathbf{P}$ to $A$ and this functor is unique up to a unique monoidal isomorphism. (Cf. the corresponding property for the braid category in \cite{tlo11}.)
Let $V$ be a symmetric monoidal closed category, with monoidal product $\oplus$ and unit $1$.
For the time being we assume $V$ is complete and cocomplete: later we will need to relax this condition.
\begin{definition}
A $V$-species is a functor $\mathcal{F}\mathcal{B}\to V$.
The category of $V$-species and their natural transformations is denoted $V^{\mathcal{F}\mathcal{B}}$.
\end{definition}
The category $V^{\mathcal{F}\mathcal{B}}$ carries several monoidal structures.
One is the Day convolution (\cite{tlo3}) induced by the monoidal product $\oplus$ on $\mathcal{F}\mathcal{B}$.
To set this up. we work in the context of $V$-enriched category theory (see \cite{tlo13}), and recall that any locally small category $\mathcal{C}$ (such as $\mathcal{F}\mathcal{B}$) can be regarded as a $V$-category, by composing $\hom\colon\mathcal{C}^\mathrm{op}\times\mathcal{C}\to\mathcal{S}et$ with the symmetric monoidal functor $\mathcal{S}et\to V$ sending a set $U$ to a $U$-indexed coproduct of copies of $1$.
Then Day convolution is given abstractly by the formula
\[
(F\otimes G)[S]
=
\int^{W,X\in\mathcal{F}\mathcal{B}} F[W] \otimes G[X] \otimes \mathcal{F}\mathcal{B}(S,W\oplus X).
\]
Since each finite bijection $\phi\colon S\to W\oplus X$ 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:
\[
(F\otimes G)[S]
=
\sum_{S=T+U} F[T] \otimes G[U].
\]
(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 $y: \mathcal{FB}^\mathrm{op} \to V^{\mathcal{FB}}$ 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
\[
(F \circ G)[S] = \sum_{n \geq 0} F[n] \otimes_{S_n} G^{\otimes n}[S].
\]
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
\[
(F \circ \widehat{X})[0] = \sum_{n \geq 0} F[n] \otimes_{S_n} X^n
\]
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 \textbf{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 $V$ \emph{finitely} cocomplete. For general $n$ we have in that case
\[
G^{\otimes n}[S] = \sum_{S = T_1 + \ldots + T_n} G[T_1] \otimes \ldots \otimes G[T_n]
\]
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
\[
(F \circ G)[S] = \sum_{R \in Eq(S)} F[S/R] \otimes \bigotimes_{x \in S/R} G[\pi^{-1}(x)]
\]
whenever $G[0] = 0$.
\section{Operads}
\begin{definition}
An \textbf{operad} in $V$ is a monoid in the monoidal category $(V^{\mathcal{FB}}, \circ)$.
\end{definition}
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.,
\begin{enumerate}%
\item 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.
\item 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 $S_n$.
\item 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.
\end{enumerate}
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 \emph{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 \emph{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 \emph{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 (\cite{tlo14}) 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''.
\section*{A look ahead}
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.
\section{Poincar\'e{}-Birkhoff-Witt}
\emph{From this section on, we fix a ground field $k$ of characteristic 0, and $V$ henceforth denotes the category of} finite-dimensional \emph{vector spaces over $k$.}
To study the Lie operad $L[-]$, Joyal and others (e.g. \cite{tlo8}) take as starting point the Poincar\'e{}-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 $T_n(L) = L^{\otimes n}$:
\begin{align*}
\pi_n: L^{\otimes n} &\to L^{\otimes n}
\\ v_1 \otimes \ldots \otimes v_n &\mapsto \frac1{n!} \sum_{\sigma \in S_n} v_{\sigma(1)} \otimes \ldots \otimes v_{\sigma(n)}.
\end{align*}
We obtain a composite of maps of filtered spaces
\[
S(L) \to T(L) \to U(L)
\]
and the PBW theorem concerns the application of the associated graded space functor to this composite (denoted $\phi$):
\begin{theorem}
The graded map $\phi_{gr}$ induces an isomorphism between $U_{gr}(L)$ and $S_{gr}(L)$ as graded spaces.
\end{theorem}
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 $Vect_k$:
\[
\frac1{1-X} \cong (\exp \circ L)(X)
\]
which in turn determines a species-isomorphism, which componentwise is an isomorphism of $S_n$-representations:
\[
\frac1{1-X}[n] \cong (\exp \circ L)[n].
\]
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
\[
\sum_{n \geq 0} \frac{a_n x^n}{n!}
\]
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
\[
(a \cdot b)_n = \sum_p \binom{n}{p} a_p b_{n-p}
\]
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.
\begin{proposition}
(\cite{tlo9}) 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.
\end{proposition}
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 $l[-]$.
\section{Virtual species}
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.
\begin{definition}
$\mathbf{V}$ is the category of $\mathbb{Z}_2$-graded finite-dimensional vector spaces, with monoidal product given by the formula
\[
(V \otimes W)_p = \sum_{m+n = p} V_m \otimes W_n
\]
and symmetry given by the formula
\[
\sigma(x_m \otimes y_n) = (-1)^{m n} y_n \otimes x_m
\]
for $x_m \in V_m$ and $y_n \in W_n$.
\end{definition}
\begin{definition}
Let $F$ and $G$ be $\mathbf{V}$-species. Then $F \sim G$ ($F$ and $G$ are \textbf{virtually equivalent}) if $F_0 \oplus G_1 \cong F_1 \oplus G_0$ as $V$-species.
\end{definition}
\begin{itemize}%
\item \textbf{Remark:} Exact sequences in $V^{\mathcal{FB}}$ split. One can check this assertion componentwise, where exact sequences of $k S_n$-modules split since we assumed $char(k) = 0$.
\end{itemize}
\begin{lemma}
The relation $\sim$ is an equivalence relation.
\end{lemma}
\begin{proof}
Transitivity follows from cancellation: $F \oplus H \cong G \oplus H$ implies $F \cong G$. This in turn follows from the remark.
\end{proof}
It is of course crucial here to work with finite-dimensional vector spaces throughout, in order to avoid the Eilenberg swindle.
\begin{lemma}
The relation $\sim$ is respected by $\oplus$, $\otimes$, and $\circ$.
\end{lemma}
The proof is left to the reader; see also \cite{tlo10} and \cite{tlo16}.
A \textbf{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 \emph{not} come from isomorphisms in $\mathbf{V}^{\mathcal{FB}}$ come about by applying the following remark.
\begin{itemize}%
\item If an object $C$ of $\mathbf{V}^{\mathcal{FB}}$ carries a differential structure (meaning $V$-species maps $\partial: C_0 \to C_1$ and $\partial: C_1: C_0$ satisfying both instances of $\partial^2 = 0$), then $C$ is virtually equivalent to its homology $H(C)$. This observation is tantamount to a structural Euler formula $C_0 - C_1 \sim H_0(C) - H_1(C)$, which obtains because exact sequences in $\mathbf{V}^{\mathcal{FB}}$ split.
\end{itemize}
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 $C \sim H(C)$.
\section{Logarithmic species}
One of our goals is to lift the inversion
\[
\frac1{1-x} = \exp(L(x)) \qquad implies \qquad -\log(1-x) = L(x)
\]
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 $\exp(X)-1$:
\[
\log(1+X) \circ (\exp(X)-1) \sim X \sim (\exp(X)-1) \circ \log(1+X).
\]
\begin{proposition}
If $F$ and $G$ are $\mathbf{V}$-species such that $F[0] = 0 = G[0]$, then $\exp(F \oplus G) \cong \exp(F) \otimes \exp(G)$.
\end{proposition}
\begin{proof}
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.
\end{proof}
Defining $\log((1+F) \otimes (1+G))$ to be
\[
\log(1+X) \circ (F + F + F \otimes G),
\]
it follows that $\log((1+F) \otimes (1+G)) \sim \log(1+F) + \log(1+G)$. In particular,
\[
\log(\frac1{1-X}) \sim -\log(1-X)
\]
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.
\begin{proposition}
$F(-X)[S] \cong (-1)^{|S|} F[S] \otimes \Lambda[S]$, where $\Lambda[S]$ denotes the top exterior power $\Lambda^{|S|}(k S)$ of the vector space $k S$ freely generated from $S$.
\end{proposition}
\begin{proof}
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
\begin{align*}
(F \circ (-X))[S]
&= \sum_{R \in Eq(S)} F[S/R] \otimes \bigotimes_{x \in S/R} (-X)[\pi^{-1}(x)]
\\ &\cong F[S] \otimes (-X)[1]^{\otimes |S|}
\end{align*}
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.
\end{proof}
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
\begin{align*}
\delta_F\colon \mathbf{V}[[x]] &\to \mathbf{V}[[x]]
\\ H &\mapsto H \circ F - H
\end{align*}
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
\[
\delta_F(H)[S] = (\sum_{R \in Eq(S)} H[S/R] \otimes \bigotimes_{x \in S/R} F[\pi^{-1}(x)]) - H[S]
\]
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
\[
\sum_{\widehat{0} < R} H[S/R] \otimes \bigotimes_{x \in S/R} F[\pi^{-1}(x)].
\]
Define the \emph{$\mathbf{V}$-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
\[
\delta_F(H) = \sum_{\widehat{0} < R} H[S/R].
\]
In general, the $n^{th}$ iterate $\delta_{F}^{n}(H)[S]$ is a sum of the form
\[
\sum_{\widehat{0} < R_1 < \ldots < R_n} terms
\]
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
\[
(1 + \delta_F)^{-1}(H) := \sum_{n \geq 0} (-1)^n \delta_{F}^{n}(H)
\]
makes sense as a functor on $\mathbf{V}$-species $H$.
We may now construct the inverse species $F^{-1}(X)$:
\[
F^{-1}(X) = (1 + \delta_F)^{-1}(X) := \sum_{n \geq 0} (-1)^n \delta_{F}^{n}(X)
\]
\begin{proposition}
$(F^{-1} \circ F)(X) \sim X$.
\end{proposition}
\begin{proof}
We have
\begin{align*}
(F^{-1} \circ F)(X)
&\sim (1 + \delta_F)(F^{-1})(X))
&= (1 + \delta_F)(\sum_{n \geq 0} (-1)^n \delta_{F}^{n}(X))
\\ &\sim \sum_{n \geq 0} (-1)^n \delta_{F}^{n}(X) + \sum_{n \geq 0} (-1)^n \delta_{F}^{n+1}(X)
\end{align*}
which telescopes down to $\delta_{F}^{0}(X) = X$.
\end{proof}
\begin{itemize}%
\item \textbf{Remark:} Under our hypotheses on $F$, one may prove by induction that $F \circ G \sim F \circ H$ implies $G \sim H$. From this and the proposition, it easily follows that $(F \circ F^{-1})(X) \sim X$.
\end{itemize}
\section{A bar construction}
When $F$ carries an operad structure, this construction of $F^{-1}$ admits a more categorical interpretation. Observe that there is an embedding
\[
\delta_{F}^{n}(X) \hookrightarrow F \circ \ldots \circ F = F^{\circ n}.
\]
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 \emph{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
\[
\partial_i: F^{\circ (n+1)} \stackrel{F^{\circ (n-i)} m F^{\circ (i-1)}}{\to} F^{\circ n}.
\]
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
\[
\delta_{F}^{n+1}(X) \to \delta_{F}^{n}(X)
\]
and the $\delta_{F}^{n}(X)$ form a chain subcomplex. We regard this chain complex as our preferred \textbf{bar construction} for $F$, or more precisely a \emph{right} bar construction $B_r(F)$, as we now explain in more detail.
There is an exact sequence
\[
0 \to \delta_F \to (-) \circ F \to (-) \circ X \to 0
\]
making the functor $\delta_F$, for an operad $F$, analogous to tensoring on the \emph{right} with an augmentation ideal $I G$ of a group ring $\mathbb{Z}G$, sitting in an exact sequence
\[
0 \to I G \to \mathbb{Z}G \to \mathbb{Z} \to 0.
\]
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
\[
X + (-\delta_{F})(X) + (-\delta_F)^{2}(X) + \ldots,
\]
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
\[
\mathbb{Z} + \Sigma(I G) + (\Sigma (I G))^{\otimes 2} + \ldots
\]
originally introduced by Eilenberg-Mac Lane \cite{tlo4}.
\begin{itemize}%
\item \textbf{Remark:} The analogy between the complex $B_r(F)$ and the classical normalized bar resolution is imperfect, however, because the monoidal product $- \circ -$ behaves rather differently from the tensor product $- \otimes -$. In particular, $- \circ -$ does not preserve colimits on each side; only $- \circ F$ preserves colimits. This explains why, in our geometric series construction for $F^{-1}(X)$, we must consistently apply the substitution product $- \circ F$ with $F$ appearing on the right. Also, whereas the classical bar resolution produces a free algebra (or cofree coalgebra) construction on the shifted augmentation ideal via a geometric series, the same geometric series construction in the operadic context does not yield a cofree co-operad, again because of failure of the monoidal product $- \circ -$ to preserve coproducts in each of its separate arguments. We return to this point later.
\end{itemize}
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
\[
\delta_{F}^{n}(X) \circ F \hookrightarrow F^{\circ (n+1)}
\]
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
\[
\partial_i: F^{\circ (n+1)} \circ F \to F^{\circ n} \circ F
\]
restrict to maps
\[
\delta_{F}^{n+1}(X) \circ F \to \delta_{F}^{n}(X) \circ F
\]
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
\[
(F^{-1} \circ F)(X) \sim X
\]
as arising from a homotopy equivalence between dg-species:
\[
\begin{tikzcd}
\ldots \ar[r]
& \delta_F^{n+1}(X) \ar[d] \ar[r]
& \delta_F^n(X) \ar[d] \ar[r]
& \ldots \ar[r]
& \delta_F^0(X) \ar[d,"\mathrm{id}"] \ar[r]
& 0
\\& 0
& 0
&& X
\end{tikzcd}
\]
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
\[
F^{\circ n} \circ u: F^{\circ n} \circ X \to F^{\circ n} \circ F,
\]
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
\section{The Lie species}
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
\[
\delta_F: H \mapsto (S \mapsto \sum_{\widehat{0} < R} H[S/R]),
\]
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
\[
\widehat{0} < R_1 < \ldots R_n = \widehat{1}
\]
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 $|S| > 2$, we may identify the complex $B_r(F)[S]$,
\[
0 \to \delta_{F}^{|S|-1}(X)[S] \to \ldots \to \delta_{F}^{2}(X)[S] \to \delta_{F}^{1}(X)[S] \to X[S] \to 0,
\]
with
\[
0 \to C_{|S|-3}[S] \to \ldots \to C_0[S] \to k \to 0 \to 0
\]
It follows that $\log(1+X)[S]$ is virtually equivalent to \emph{reduced} homology of the simplicial complex $C$ associated with $Eq(S)$.
\begin{definition}
A finite lattice is geometric if every element $x$ is a joint of atoms, if every maximal chain beginning at $0$ and ending at $x$ has the same length $\rho(x)$, and if $\rho(X\vee y)+\rho(x\wedge y) \leq \rho(x)+\rho(y)$.
(A finite lattice is modular iff this last inequality is an equality.)
\end{definition}
Now the lattice of equivalence relations is a \emph{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 (the theorem below) 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 M\"o{}bius function on the lattice $Eq(S)$.
\begin{theorem}
(\cite{tlo5}) If $L$ is a finite geometric lattice, then the reduced homology of $L-\{0,1\}$ is trivial except in top dimension, where the Betti number is the value $\mu(0,1)$ of Rota's M\"o{}bius function on the lattice.
\end{theorem}
\begin{corollary}
$\log(1+X)[S] \sim (-1)^{|S|-1} H_{|S|-1}(B_r(F)[S]).$
\end{corollary}
\begin{proof}
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 $|S|-1$.
\end{proof}
\begin{corollary}
The Lie species $L[S]$ is isomorphic to $H_{|S|-1}(B_r(F)[S]) \otimes \Lambda[S]$.
\end{corollary}
\begin{proof}
Starting from the PBW theorem
\[
\frac1{1-X} \cong \exp(L(X)),
\]
we were led to the virtual equivalence
\[
-\log(1-X) \sim L(X).
\]
Applying Joyal's rule of signs to the previous theorem, $-\log(1-X)[S]$ is virtually equivalent to
\[
-(-1)^{|S|}(-1)^{|S|-1}H_{|S|-1}(B_r(F)[S]) \otimes \Lambda[S]
\]
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.
\end{proof}
This concludes our rendition of Joyal's calculation of $L[S]$.
This calculation does not specify the operad structure of $L[-]$, but in view of the formula just given for $L[n]$, we make the following remark: given an operad in chain complexes with components $C[n]$, the homology operad $H_*(C)$ contains a suboperad with components $H_{n-1}(C[n])$.
Unfortunately, this remark does not apply to $C[n] = B_r(F)[n] \otimes \Lambda[n]$, since the bar construction $B_r(F)[n]$ carries no obvious operad structure.
A different bar construction $B_l(F)$, or rather its dual cobar construction, is better suited for applying the present remark.
\section{Free operads}
Let $F$ be a $V$—operad such that $F[O] = 0$ and $F[1] = 1$. Our earlier construction
for $F^{-1}(X)$, namely $\sum_{n\geq0}(-\delta_F)^n(X)$ is not a free operad on the suspension
$-\delta_F(X)$ of the augmentation ideal $\delta_F(X)$, due to the failure of $-\circ-$ to preserve separate colimits.
To construct free operads, it is convenient to use the language of trees \cite{tlo2}.
\begin{definition}
A rooted tree is an (undirected) acyclic connected finite graph together with a distinguished node called the root.
\end{definition}
Let $N$ be the set of nodes of a rooted tree $T$ with root $r$, and $E$ the set of edges.
There is a bijection $\phi_-\colon E \xrightarrow{\sim} N-\{r\}$, uniquely determined by the requirement
that the node $\phi_-(e)$ lie on the edge $e$. Let $\phi_+(e)$ denote the other node lying
on $e$. One may give a rooted tree a canonical directed graph structure, with
source—target map $\langle\phi_-,\phi_+\rangle\colon E\to N\times N$. The free category on a directed
graph has a terminal object $r$ iff the directed graph comes from a tree rooted at
$r$. The free category on a rooted tree $T$ defines a partial order on its set of nodes:
an atomic element in this poset is called a \emph{leaf} of $T$. A directed subgraph of $T$
whose free category possesses a terminal object is called a \emph{subtree}. In particular,
the slice over an object $x$ comes from a subtree $T_x$ called the \emph{tree over $x$}.
A rooted tree $T$ may be characterized up to isomorphism by a function
$f_T\colon N-\{r\} \to N$, sending a node to its unique successor in the induced partial
order. Identifying functions of this form with based endofunctions $(N,r) \to (N,r)$, the endofunctions which arise in this way are precisely those whose iterates
converge to the constant function at $r$. If $x$ is a node in $T$, the set
$\sigma_x = f_T^{-1}(x)$ is called the \emph{sprout over $x$}.
Up to isomorphism, a rooted tree whose set of leaves is $S$ is characterized in
terms of $S$ by the following data: an element $C$ of the free commutative monoid
$\exp(\mathcal{P}S)$ on the power set of S (i.e. a multiset of subsets of $S$), together with a
linear order on each set of repetitions in $C$ of a given subset. To obtain such a
structure $C_T$ from a rooted tree $T$, assign to each node $x$ the set $\lambda(x)$ of leaves
$\leq x$ and let $C_T$ be the multiset $\{\lambda(x) : x\text{ is a node of }T\}$. There is a repetition
$\lambda(y) = \lambda(x)$ whenever $\sigma_x = \{y\}$; in that case impose the order $\lambda(y) < \lambda(x)$.
if on the other hand $\{y\}$ is a proper subset of $\sigma_x$, there is a proper inclusion
$\lambda{y} \subset \lambda(x)$. The set $C_T$ contains $S$ as $\lambda(r)$, and contains each singleton $\{s\}$
in $S$ as $\lambda(s)$. The absence of cycles in the tree $T$ implies a \emph{trichotomy law}: for
all $U,V\in C_T$, either $U\subseteq V$ or $V\subseteq U$ or $U\cap V = \varnothing$. In view of the essential
equivalence between trees and such multisets, we make the following definition.
\begin{definition}
A tree on a finite set $S$ is a finite multiset $C$ of subsets of $S$ containing $S$ and each singleton of $S$, and satisfying the trichotomy law, together with a linear ordering on each set of repetitions.
The set of trees on $S$ is denoted by $\mathcal{T}[S]$.
For $T\in\mathcal{T}[S]$, the set of singletons $\{s\}\in T$ is identified with $S$.
\end{definition}
$\mathcal{T}[-]$ carries a structure of $\mathcal{S}et$-species: to each bijection $\phi\colon S \to T$ we form a
map $\mathcal{T}[S] \to \mathcal{T}[T]$ by relabelling along it.
Next, we introduce a grafting operation on trees. Suppose given a tree
$T \in \mathcal{T}[X]$ and for each $x\in X$ a tree $T_x$ rooted at $x$. Then the discrete graph
on $X$ (having no edges and denoted again by $X$) is obviously embedded in $T$
and in the disjoint union of the $T_x$; the pushout of these two embeddings in the
category of undirected graphs gives an acyclic connected graph. Defining the
root of the pushout as the image of the root of $T$, the result is a rooted tree
called a \emph{grafting}, denoted by $m(T; T_x : x\in X)$. Let $R$ be an equivalence relation
on $S$; put $X = S/R$. and let $\pi\colon S \to X$ be the canonical quotient. Then the
grafting operation just described induces a map
\[
m[S]\colon
\sum_{R\in Eq(s)} \mathcal{T}[S/R] \otimes \bigotimes_{x\in S/R} \mathcal{T}[\pi^{-1}(x)]
\longrightarrow
\mathcal{T}[S]
\]
There is a map $u\colon X \to T$ which sends the element $r$ of $X[1]$ to the rooted
tree consisting only of $r$. The triple $(\mathcal{T},m,u)$ defines a $\mathcal{S}et$-operad: satisfaction
of the operad axioms follows easily from universal properties of pushouts. In
fact, it is not difficult to show that $\mathcal{T}$ is the free operad generated by the $\mathcal{S}et$-
species $\exp(X)-1$. The unit map $\eta\colon (\exp(X)-1) \to \mathcal{T}$ which generates it
is given componentwise by maps $(\exp(X)-1)[S] \to \mathcal{T}[S]$ each of which, for
$|S| > 0$, sends the unique element in the domain to the “sprout“ on $S$: the
multiset $S + \sum_{s\in S}\{s\}$. It is clear that every structure of species $\mathcal{T}$ is obtained
by grafting together a collection of such sprouts and instances of $u[1]$.
More generally, letting $\mathcal{C}$ be a complete cocomplete symmetric monoidal
closed category, we describe the free operad $\mathcal{O}(G)$ on a $\mathcal{C}$-species $G$ such that
$G[0] = 0$. Let $N(T)$ denote the set of nodes in a tree $\mathcal{T} \in \mathcal{T}[S]$. As a species,
\[
\mathcal{O}(G)[S]
=
\sum_{T\in\mathcal{T}[S]} \bigotimes_{x\in N(T)-S} G[\sigma_x]
\]
for $|S| > 0$ and $\mathcal{O}(G)[0] = 0$. If in addition $G[1] = 0$, a summand corresponding
to a tree $T$ is $0$ whenever there are nodes $x\in N(T)-S$ such that $\sigma_x$ is a singleton:
i.e., when there are repetitions of subsets. So if $G[1] = 0$, the summation can
be restricted to multisets without repetitions, i.e., to rooted trees where each
node is the join of the leaves below it. These shall be called \emph{proper trees}.
\begin{itemize}
\item \textbf{Remark:} The description we have given of $\mathcal{O}(G)$ can be formulated as
a ``wreath product'' $T\smallint G$, as in unpublished notes of Kelly, similar to
the wreath product constructions pertaining to Kelly’s ``clubs'' \cite{tlo12}. The
essential point is that the multiplication on the operad $\mathcal{O}(G)$ is induced
from the grafting multiplication on $\mathcal{}$. The unit map $u\colon X \to \mathcal{O}(G)$ is
also induced from that of $\mathcal{T}$; the map $u[1]$ factors through the summand
where the iterated tensor product is indexed over an empty set (empty
products are interpreted as the unit $I$ for the tensor).
\end{itemize}
There is a familiar free—forgetful adjunction arising from the free operad
construction: it will be useful to describe the counit of this adjunction using a
kind of term-rewrite system. To begin, let $T\in\mathcal{T}[S]$ be a tree described as a
multiset, and $U$ a subset of $S$ occurring in $T$ neither as $\lambda(r)$ nor as $\lambda(s)$ for
$s \in S$. Then we may delete $U$ from $T$ to obtain another tree $T/U \in \mathcal{T}[S]$, called
the contraction of $T$ along $U$ (deletion corresponds to contraction of an internal
edge in the tree). Contractions may be continued until one reaches a sprout,
i.e., the element in the image of the unit $\eta\colon(\exp(X)-1)[S] \to \mathcal{T}[S]$. This
iterated contraction reproduces the unique $\mathcal{S}et$-species map $\mathcal{T} \to \exp(X)-1$,
necessarily a morphism of operads.
If a tree is regarded as the result of grafting together a collection of sprouts.
then each contraction may be regarded as a tree surgery, where a subtree $T'$
obtained by grafting together two sprouts is replaced by a sprout having the
same leaves as the subtree. More formally. suppose given two subsprouts with
leaf—sets of the form $\sigma_x$ and $\sigma_y$, where $y\in\sigma_x$ and $\lambda(y) = U$, the deleted set,
$T’$, and the sprout which replaces it in $T/U$, both have $x$ as root: we use the
notation $x/y$ to denote the root. in the latter case. Both have the same set of
leaves. which we may write as $\sigma_{x/y}$. Observe there is an inclusion $\sigma_y \hookrightarrow \sigma_{x/y}$
and a surjection $\pi_y\colon \sigma_{x/y} \to \sigma_x$: indeed. defining an equivalence relation $R$ on
the set $\sigma_{x/y}$ by $zRw$ if $z=w$ or $z,w\in\sigma_y$, the set $\sigma_x$ is canonically identified
with the set of $R$—equivalence classes on $\sigma_{x/y}$.
Now suppose $G$ carries an operad structure. In $(G\circ G)[\sigma_{x/y}]$ there is a
summand of the form
\[
G[\sigma_x] \otimes \bigotimes_{z\in\sigma_x} G[\pi_y^{-1}(z)]
\]
where $|\pi_y^{-1}(z)|=1$ if $z\neq y$.
There is a unit map $I \cong X[\pi_y^{-1}(z)] \to G[\pi_y^{-1}(z)]$ for $z\neq y$, and of course $\pi_y^{-1}(y) = \sigma_y$:
therefore there is an induced map
\[
G[\sigma_x] \otimes G[\sigma_y]
\longrightarrow
(G\circ G)[\sigma_{x/y}]
\]
whence, after composing with the multiplication on $G$, a map
\[
G[\sigma_x] \otimes G[\sigma_y]
\longrightarrow
G[\sigma_{x/y}]
\]
Expand this by tensoring with identity maps to obtain a ``$G$—contraction'' map
\[
m_{T/U}\colon
\bigotimes_{x\in N(T)-S} G[\sigma_x]
\longrightarrow
\bigotimes_{x\in N(T/U)-S} G[\sigma_x]
\]
on the $T$-summand of $\mathcal{O}(G)[S]$. Such $G$—contractions may be iterated until a
sprout is reached, leading to a map
\[
\bigotimes_{x\in N(T)-S} G[\sigma_x]
\longrightarrow
G[S]
\]
and by the operad axioms, this map is independent of the order in which contractions are performed. Finally, all of these maps assemble to give a map
\[
\epsilon_G[S]\colon
\mathcal{O}(G)[S]
\longrightarrow
G[S]
\]
which is the component at $S$ of the counit $\epsilon_G$ of the adjuiiction.
We remark that if $F$ is an operad and $F[1] = 1$, then $\epsilon_F$ restricts to a map
$\mathcal{O}(\delta_F(X))-X \to \delta_F(X)$. In particular there are contraction maps
\[
\mu_{T/U}\colon
\bigotimes_{x\in N(T)-S} \delta_F(X)[\sigma_x]
\longrightarrow
\bigotimes_{x\in N(T/U)-S} \delta_F(X)[\sigma_x].
\]
\section{The left bar construction}
\emph{From this section on, $F$ denotes a $\mathcal{V}$—operad with $F[0] = 0$ and $F[1] = 1$.} We
recall (see \cite{tlo6,tlo7}) a chain complex structure $B_l(F)$ on the free operad generated
from the suspension $G=-\delta_F(X)$ (In this case $\mathcal{O}(G)[S]$ is a coproduct over
\emph{proper} trees on $S$, so that both its $\mathbb{Z}_2$—graded components are finite-dimensional.)
The grading on $B_l(F)[S]$ is determined from the construction of $\mathcal{O}(-\delta_F(X))$
as a $\mathcal{V}$—species: for example. if F is the $V$—operad $\exp(X)-1$, and $T$ is a proper
tree then the summand
\[
\bigotimes_{x\in N(T)-S} (-\delta_F(X))[\sigma_x]
\]
is concentrated in degree given by the number $n$ of sprouts $\sigma_x$ (taken modulo
2). Taking $F(X) = \exp(X)-1$ as basic, we define differentials for $B_l(F)$. Let
$\mathcal{T}_j[S]$ denote those proper trees on $S$ consisting of exactly $j$ subsets of $S$ aside
from $S$ and $\{s\}$ for $s\in S$. Arguing as we did in the proof of proposition 3,
$\mathcal{O}(-\delta_F(X))$ lifts to a $\mathbb{Z}$-graded species $\mathcal{O}(\Sigma\delta_F(X))$ whose $(j+1)$-st component
is given by
\[
\sum_{T\in\mathcal{T}_j[S]} \Lambda[T].
\]
If $\mathcal{P}S$ denotes the set $\mathcal{P}S - \{S\} - \cup_{s\in S}\{\{s\}\}$, and $\Lambda^j[S]$ the $j$-th exterior power
of a space freely generated from $S$, there is a monomorphism
\[
\sum_{T\in\mathcal{T}_j[S]} \Lambda[T] \hookrightarrow \Lambda^j[\mathcal{P}S].
\]
To see this, first observe that every tree on $S$ contains $S$ and all singletons $\{s\}$,
so no essential information is lost if we ignore these: thus there is a species
isomorphism
\[
\sum_{T\in\mathcal{T}_j[S]} \Lambda[T]
\cong
\sum_{T\in\mathcal{T}_j[S]} \Lambda[T - \{S\} - \cup_{s\in S}\{\{s\}\}].
\]
\begin{center}
\textbf{MISSING PAGE 16 OF THE ORIGINAL}
\end{center}
to work with species valued in the category of $\mathbb{Z}_2$-graded spaces (rather than
just the finite—dimensional ones). We have inclusions
\[
\begin{tikzcd}
-F \circ \mathcal{O}(-\delta_F(X))
\ar[r,hook,"\eta\circ i"]
& \mathcal{O}(-F) \circ \mathcal{O}(-F)
\ar[r,hook,"m"]
& \mathcal{O}(-F)
\end{tikzcd}
\]
and a differential structure on $\mathcal{O}(-F)$ from the preceding discussion: by restriction we get a differential structure on $-F \circ \mathcal{O}(-\delta_F(X))$, and hence a complex
$E_l(F)$. To show $E_l(F) \to X$ is a homotopy equivalence. it suffices to exhibit
a contracting homotopy for $E_l(F)[S]$ when $|S| > 1$. We have
\[
-F -\circ \mathcal{O}(-\delta_F(X))
\cong
(-\delta_F(X) \circ \mathcal{O}(-\delta_F(X)))
\oplus
(-X \circ \mathcal{O}(-\delta_F(X)))
\]
and now we write down the restriction of the contracting homotopy $h$ to each
summand. For the first summand we have
\[
-\delta_F(X) \circ \mathcal{O}(-\delta_F(X))
\xrightarrow{\eta\circ1}
\mathcal{O}(-\delta_F(X)) \circ \mathcal{O}(-\delta_F(X))
\xrightarrow{m}
\mathcal{O}(-\delta_F(X))
\]
which we further compose with
\[
\mathcal{O}(-\delta_F(X))
\xrightarrow{i}
-X \circ \mathcal{O}(-\delta_F(X))
\xrightarrow{-u\circ1}
-F \circ \mathcal{O}(-\delta_F(X))
\]
to obtain the first restriction $h_1$; here the map $i$ is a degree 1 shift of the
canonical isomorphism $\mathcal{O}(-\delta_F(X)) \cong X\circ\mathcal{O}(-\delta_F(X))$. The restriction $h_2$ of
the contracting homotopy to the second summand $-X\circ\mathcal{O}(-\delta_F(X))$ is just a
zero map. The proof that $h = h_1+h_2$ is a contracting homotopy (which only
uses simplicial identities) is standard and left to the reader.
The virtual equality $F\circ\mathcal{O}(-\delta_F(X)) \sim X$ follows from the fact that $E_l(F) \to X$
induces an isomorphism in homology. Thus $\mathcal{O}(-\delta_F(X))$ and $\sum_{n\geq0}(-\delta_F)^n(X)$
both represent the virtual species $F^{-1}\in \mathcal{V}[[x]]$. In the next section we exhibit
a chain map between their associated bar constructions $B_r(F)$, $B_l(F)$ which is
a quasi-isomorphism.
\section{A quasi-isomorphism}
Under the hypotheses on an operad $F$ used in the previous section, we describe
a chain map $B_l(F) \to B_r(F)$ which induces an isomorphism] in homology.
As usual, it is easiest to begin with the case $F(X) = \exp(X)-1$. First we
exhibit a map
\[
\mathcal{O}(-\delta_F(X))
\longrightarrow
\sum_{n\geq0} (-\delta_F)^n(X)
\]
which comes from a map of $\mathbb{Z}$-graded species with components
\[
\sum_{T\in\mathcal{T}_j[S]} \Lambda[T]
\longrightarrow
(\delta_F)^{j+1}(X)[S].
\]
Let $T$ be a proper tree on $S$ whose elements, aside from $S$ and the singleton
sets, are subsets $U_1,\ldots,U_j$ of $S$. To each non-empty subset $U\subseteq S$ we may
associate an equivalence relation $R_U$ on $S$ defined by $xR_Uy$ if $x=y$ or $x,y\in U$.
If $R,R'\in Eq(S)$, let the sum $R + R'$ denote their join in this lattice. Suppose
the subsets $U_i$ are arranged so that $U_i\subseteq U_j$ implies $i\leq j$, and let $R^{(i)}$ denote
$\sum_{k\leq i}R_{U_k}$. Then there are strict inclusions $R^{(i)} < R^{(j)}$ when $i < j$, and the
trichotomy law guarantees that $0 < R^{(i)} < i$ for all $i$. We thus obtain a chain
in $\delta_F^{j+1}(X)[S]$ of the form
\[
0
<
R^{(1)}
<
\ldots
<
R^{(j)}
<
R^{(j+1)} := 1.
\]
It is sometimes convenient to omit the last member from this chain, which
carries no essential information.
The space $\Lambda[T]$ may be regarded as being generated by a $j$-fold exterior
product $U_1\wedge \ldots \wedge U_j$. Let $T$ denote the poset whose elements are the sets $U_i$
ordered by inclusion, and let $[j]$ denote the poset $\{1 < \ldots < j\}$. Consider the
set $\mathrm{Ord}_b(T,[j])$ of order-preserving bijections $T\to[j]$: to each $\phi\in\mathrm{Ord}_b(T,[j])$
we define a sign $\mathrm{sgn}(\phi)$ as the sign of the permutation $[j]\to[j]$ which sends
the element $i$ to $\phi(U_i)$, and define equivalence relations
\[
R_\phi^{(i)}
=
\sum_{k\leq i} R_{\phi^{-1}(k)}
\]
for $1\leq i\leq j$, which form a chain
\[
R_\phi
=
[0 < R_\phi^{(1)} < \ldots < R_\phi^{(j)} < 1].
\]
Finally. define a map
\begin{align*}
\iota_j[S]\colon
\sum_{T\in\mathcal{T}_j[S]} \Lambda[T]
&\longrightarrow
(\delta_F)^{j+1}(X)[S]
\\ U_1 \wedge \ldots \wedge U_j
&\longmapsto
\sum_{\phi\in\mathrm{Ord}_b(T,[j])} \mathrm{sgn}(\phi)\cdot R_\phi.
\end{align*}
This is clearly well-defined and natural in $S\in\mathcal{F}\mathcal{B}$.
\begin{proposition}
The map $\iota\colon B_l(F) \to B_r(F)$ is a monomorphism of chain complexes.
\end{proposition}
\begin{proof}
First we remark that chains of the form $R_\phi$ for $\phi\in\mathrm{Ord}_b(T,[j])$ are
chains $[0 < R_1 < \ldots < R_j < 1]$ such that, for all $i$, the equivalence relation
$R_{i+1}/R_i$ on $S/R_i$ (given by the kernel pair of the quotient $S/R_i \to S/R_{i+1}$
with $R_0 = 0$ and $R_{j+1} = 1$ for convenience) is an equivalence relation $R_{V_i}$
associated with a subset $V_i\subseteq S/R_i$. Let us call chains which satisfy this property
``good''. To prove that $\iota$ is monic, it is enough to see that we can retrieve $T$ and
$\phi\colon T \to [j]$ from the data of a good chain. This is easy: letting $\pi_i\colon S \to S/R_i$
be the canonical quotient, $T$ is the set of subsets $U_i = \pi_i^{-1}(V_i)$, and $\phi$ is defined
by $\phi(U_i) = i$. To show that $\iota$ preserves differentials, consider the differential of
$R_\phi$:
\[
\sum_{i=1}^j (-1)^i [\ldots < R_\phi^{(i-1)} < R_\phi^{(i+1)} < \ldots]
\]
By the trichotomy property for trees, either the $i$-th summand is a good chain,
or $\phi^{-1}(i) \cap \phi^{-1}(i+1)$ is empty. In the latter case, consider $\phi'\in\mathrm{Ord}_b(T,[j])$
obtained by composing $\phi$ with the transposition $(i i+1)$. Clearly $\mathrm{sgn}(\phi') = -\mathrm{sgn}(\phi)$,
and one easily checks that the $i$-th summands in $\partial(R_\phi)$ and $\partial(R_{\phi'})$
are the same. Thus, in computing $\partial(\iota(U_1 \wedge \ldots \wedge U_j))$, the ``bad'' chains cancel:
what remains is a linear combination of good chains $R_{\phi_i}$, where $\phi_i$ denotes the
restriction of $\phi$ to $T-\phi^{-1}(i)$. Specifically, we have
\[
\partial(\iota(U_1 \wedge \ldots \wedge U_j))
=
\sum_{\phi\in\mathrm{Ord}_b(T,[j])} \mathrm{sgn}(\phi) \cdot \sum_{i=1}^j (-1)^i R_{\phi_i}
\]
and some painstaking care with signs shows this equals
\[
\sum_{i=1}^j (-1)^i \sum_{\phi\in\mathrm{Ord}_b(\widehat{T/U_i},[j-1])} \mathrm{sgn}(\phi) \cdot R_\phi
=
\iota(\partial(U_1 \wedge \ldots \wedge U_j))
\]
so that $\iota$ preserves differentials. This completes the proof.
\end{proof}
\begin{theorem}
For $F(X) = \exp(X)-1$, $B_r(F)$ and $B_l(F)$ are quasi-isomorphic.
\end{theorem}
\begin{proof}
From theorem 2, $H_*(B_r(F)[n])$ is concentrated in the top degree $n-1$,
and from the results of \cite{tlo7}, so is $H_*(B_l(F)[n])$. Since this is top degree, both of
these homologies are given by the cycle groups in that degree, and by proposition
5, the chain map $\iota$ restricts to a monomorphism
\[
Z_{n-1}(B_l(F)[n])
\longrightarrow
Z_{n-1}(B_r(F)[n])
\]
between these cycle groups. On the other hand, both of these homologies are
virtually equivalent since both represent the $S_n$—character $\log(1+X)[n]$. Since
both are concentrated in the same degree, this virtual equivalence implies they
are isomorphic. Therefore the monomorphism above must itself be an isomor—
phism. This completes the proof.
\end{proof}
Now consider more generally an operad $F$ such that $F[0] = 0$, $F[1] = 1$. To
construct a chain map $\iota\colon B_l(F) \to B_r(F)$, we couple the construction for the
special case $F(X) = \exp(X)-1$ with a few simple observations. In this special
case there are inclusions
\[
\delta_F^{j+1}(X)
\hookrightarrow
F^{\circ(j+1)}
\hookrightarrow
\mathcal{O}(F)
\]
so that each chain $0 < R_1 < \ldots < R_j < 1$ of equivalence relations on $S$ may be
regarded as a (non-proper) tree on $S$. Abbreviating $\delta_{\exp(X)-1}$ to $\delta$, we have for
general $F$
\[
\delta_F^{j+1}(X)[S]
=
\sum_{T\in\delta^{j+1}(X)[S]} \bigotimes_{x\in N(T)-S} F[\sigma_x]
\]
Let $T\in\mathcal{T}_j[S]$ be a proper tree and consider a map $\phi\in\mathrm{Ord}_b(T,[j])$; earlier
we produced a chain $R_\phi$ of equivalence relations, which we now regard as a tree.
Observe that there is an isomorphism
\[
\bigotimes_{x\in N(T)-S} \delta_F(X)[\sigma_x]
\cong
\bigotimes_{x\in N(R_\phi)-S} F[\sigma_x]
\]
since we assumed $F[1] = 1$. From this isomorphism, it is clear that there is an
induced chain map
\[
\sum_{T\in\mathcal{T}_j[S]} \Lambda[T] \otimes \bigotimes_{x\in N(T)-S} \delta_F(X)[\sigma_x]
\longrightarrow
\sum_{T\in\delta^{j+1}(X)[S]} \bigotimes_{x\in N(T)-S} F[\sigma_x]
\]
which gives the desired map $\iota\colon B_l(F) \to B_r(F)$.
It is not difficult to show, using theorem 3, that $\iota$ induces an isomorphism
in homology. This may suggest new techniques for deciding whether a given
quadratic operad satisfies Koszul duality (\cite{tlo7}). For example, in certain cases
one can establish Koszul duality through an appeal to theorem 2.
\section{The Lie operad}
Let $F(X) = \exp(X)-1$, and let $B_l(F)'$ denote the dual of the complex $B_l(F)$,
i.e., the cochain complex with differentials
\[
\sum_{T\in\mathcal{T}_{j-1}[S]} \Lambda[T]'
\longrightarrow
\sum_{T\in\mathcal{T}_j[S]} \Lambda[T]'
\]
obtained by transposing those of $B_l(F)[S]$. In \cite{tlo7} it is shown that the operad
multiplication and unit on $\mathcal{O}(-\delta_F(X))$, which is the graded species underlying
$B_l(F)'$, respect these differentials, so that $B_l(F)'$ carries a structure of an operad
valued in the category of $\mathbb{Z}_2$—graded cochain complexes. It is called the \emph{cobar}
construction of $F$, and may be regarded as a lift of the virtual species $F^{-1}(X) = \log(1+X)$ to an operad.
Since the operation of additive inverse on $\mathcal{V}[[x]]$ lifts to the $\mathbb{Z}_2$—graded suspension
(as an involution on $\mathcal{V}^{\mathcal{F}\mathcal{B}}$ or on species valued in $\mathbb{Z}_2$-graded complexes),
we can transport the monad $B_l(F)' \circ (-)$ across the suspension to lift the virtual
species $-F^{-1}(-X) = -\log(1-X)$ to an operad. We can describe this operad
using proposition 3. First, there is a $\mathcal{V}$—operad with components
\[
(-1)^{|S|-1} \Lambda[S],
\]
obtained by transporting the monad $\exp$ for free commutative monoids across
the $\mathbb{Z}_2$-graded suspension. This is described in detail in \cite{tlo7}, where it is called the
determinantal operad. Next, if $F$ and $G$ are operads, then there is an obvious
operad structure on the species with components $F[S] \otimes G[S]$. Putting this
together, there is an identification between operads given componentwise as
\[
-B_l(F)'(-X)[S]
\cong
B_l(F)'[S] \otimes (-1)^{|S|-1} \Lambda[S]
\]
and since cohomology with coefficients in a field preserves coproducts and tensor
products, there is an operad whose components are
\[
H^{n-1}(B_l(F)[n]) \otimes \Lambda[n]
\]
in degree 0 (compare with corollary 2 and the remarks which follow it).
\begin{theorem}
(\cite{tlot})
The Lie operad $L$ is isomorphic to this operad.
\end{theorem}
\begin{proof}
\emph{(What follows is a sketch of the proof.)}
The cohomology space above is a cokernel of a space generated
by the set of binary trees in $\mathcal{T}[n]$, so that the operad aboVe is an operad
generated by a single binary operation (represented by a sprout with two leaves.
called a 2-sprout). The kernel of this cokernel is the space of trees in $\mathcal{T}[n]$
obtained by grafting a collection of 2—sprouts with a 3-sprout (which represents
a ternary operation), so that the operad $H^*(B_l(F))$ is generated by a binary
operation and subject to a single equation in which the tertiary operation is set
equal to zero. Upon twisting with the determinantal operad in degree 3, one
may verify by hand that this equation is the Jacobi identity. Twisting with the
determinantal operad in degree 2 gives the alternating identity, and these give
a complete set of identities for the operad. But these are exactly the identities
governing the Lie operad.
\end{proof}
\begin{itemize}
\item \textbf{Remark:} In \cite{tlo1}, an explicit description is given for the correspondence
between certain elements in free Lie algebras (Lyndon basis elements)
and homology classes for partition lattices. Our notes here provide a
conceptual framework for this description, which can be extracted from
theorems 3 and 4 (compare with corollary 2).
\end{itemize}
\begin{definition}
A ($\mathbb{Z}_2$-graded) homotopy Lie algebra is an algebra over the operad
\[
(-1)^{n-1} B_l(F)'[n] \otimes \Lambda[n]
\]
(which is valued in the category of $\mathbb{Z}_2$-graded complexes).
\end{definition}
\begin{itemize}
\item \textbf{Remark:} If $F(X) = \frac{X}{1-X}$ is the operad governing associative algebras
without unit, then there is a virtual equivalence $F(X) \sim -F^{-1}(-X)$. The
right-hand side is represented by an operad with components
\[
(-1)^{|S|-1} B_l(F)'[S] \otimes \Lambda[S],
\]
which is easily (and classically) shown to be homotopy-equivalent to the
operad $\frac{X}{1-X}$. Algebras over this (right-hand) operad are called homotopy
associative algebras: they are equivalent to the $A_\infty$—algebras of \cite{tlo15}.
\end{itemize}
\nocite{*}
\printbibliography
\end{document}