# Lie models of homotopy automorphism monoids and classifying fibrations

Yves Félix, Mario Fuentes and Aniceto Murillo***The authors have been
partially supported by a MICINN grant of the Spanish Government.

###### Abstract

Given $X$ a finite nilpotent simplicial set, consider the classifying
fibrations

$X\to B\operatorname{{\rm aut}}_{G}^{*}(X)\to B\operatorname{{\rm
aut}}_{G}(X){\quad\text{and}\quad}X\to Z\to B\operatorname{{\rm
aut}}_{\pi}^{*}(X)$

where $G$ and $\pi$ denote, respectively, subgroups of the free and pointed
homotopy classes of free and pointed self homotopy equivalences of $X$ which
act nilpotently on $H_{*}(X)$ and $\pi_{*}(X)$. We give algebraic models, in
terms of complete differential graded Lie algebras (cdgl’s), of the rational
homotopy type of these fibrations. Explicitly, if $L$ is a cdgl model of $X$,
there are connected sub cdgl’s ${\rm Der}^{\mathcal{G}}L$ and ${\rm
Der}^{\Pi}L$ of the Lie algebra of derivations of $L$ such that the
geometrical realization of the sequences of cdgl morphisms

$L\stackrel{{\scriptstyle\operatorname{{\rm ad}}}}{{\to}}{\rm
Der}^{\mathcal{G}}L\to{\rm
Der}^{\mathcal{G}}L\widetilde{\times}sL{\quad\text{and}\quad}L\to
L\widetilde{\times}{\rm Der}^{\Pi}L\to{\rm Der}^{\Pi}L$

have the rational homotopy type of the above classifying fibrations. Among the
consequences we also describe in cdgl terms the Malcev $\mathbb{Q}$-completion
of $G$ and $\pi$ together with the rational homotopy type of the classifying
spaces $BG$ and $B\pi$.

## Introduction

A foundational result of Stasheff [30] asserts that the classifying space
$B\operatorname{{\rm aut}}(X)$ of the topological monoid $\operatorname{{\rm
aut}}(X)$ of self homotopy equivalences of a given finite complex $X$
classifies fibrations with fiber $X$ by means of a universal fibration over
$B\operatorname{{\rm aut}}(X)$. Furthermore, see [23], certain submonoids of
$\operatorname{{\rm aut}}(X)$ produce also classifying fibrations for
distinguished families of fibrations. The universal cover of
$B\operatorname{{\rm aut}}(X)$ has the homotopy type of the classifying space
$B\operatorname{{\rm aut}}_{1}(X)$ of the monoid of self homotopy equivalences
homotopic to the identity on $X$. Whenever $X$ is simply connected (see [27,
31], [2] for a remarkable extension to the fiberwise setting or [3] for the
relative case) the rational homotopy type of $B\operatorname{{\rm
aut}}_{1}(X)$ is well understood and described in terms of classical algebraic
models of $X$. However, the homotopy type of $B\operatorname{{\rm aut}}(X)$ is
as unmanageable, even for $X$ simply connected, as its fundamental group
${\mathcal{E}}(X)$ of homotopy classes of self homotopy equivalences. For
instance, as we will see, $B\operatorname{{\rm aut}}(S^{n}_{\mathbb{Q}})$ lies
outbound of any known geometric realization of structures shaping the rational
homotopy type of spaces.

Nevertheless, in this paper we are able to algebraically describe the rational
homotopy type of the classifying space of certain non connected distinguished
submonoids of $\operatorname{{\rm aut}}(X)$. Furthermore, we model the
corresponding universal classifying fibrations. As a consequence we also read
in terms of these models the structure of nilpotent subgroups, and their
classifying spaces, of free and pointed homotopy classes of self homotopy
equivalences.

To do so we strongly rely on the homotopy theory developed in the category
$\operatorname{{\bf cdgl}}$ of complete differential graded Lie algebras
(cdgl’s henceforth) by means of the Quillen pair

$\textstyle{\operatorname{{\bf sset}}}$$\textstyle{\operatorname{{\bf
cdgl}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\langle\,\cdot\,\rangle}$$\scriptstyle{{\mathfrak{L}}}$
(1)

given by the global model and realization functor (see next section for a
quick review or [11] for a detailed presentation).

From now on, and throughout the paper, we will often not distinguished a
simplicial set from its realization as a CW-complex. Also, by a fibration
sequence $F\to E\to B$ we understand its more general meaning: the composition
of the two maps is homotopic to a constant $b$ and the induced map from $F$ to
the homotopy fiber of $E\to B$ over $b$ is a weak homotopy equivalence. This
also applies to the category $\operatorname{{\bf cdgl}}$.

Let $X$ be a finite connected complex, let $G\subset{\mathcal{E}}(X)$ be a
subgroup, and consider $\operatorname{{\rm
aut}}_{G}(X)\subset\operatorname{{\rm aut}}(X)$ the submonoid of self homotopy
equivalences whose homotopy classes are elements in $G$. Then, evaluating each
equivalence at the base point induces a fibration sequence

$X\longrightarrow B\operatorname{{\rm aut}}_{G}^{*}(X)\longrightarrow
B\operatorname{{\rm aut}}_{G}(X)$

which classifies fibrations $X\to E\to B$ where the image of the holonomy
action $\pi_{1}(X)\to{\mathcal{E}}(X)$ lies in $G$. Here, $\operatorname{{\rm
aut}}_{G}^{*}(X)\subset\operatorname{{\rm aut}}^{*}(X)$ is the submonoid of
pointed homotopy equivalences whose (free) homotopy classes lie in $G$.

Analogously, given $\pi\subset{\mathcal{E}}^{*}(X)$ a subgroup of pointed
homotopy classes of pointed self equivalences, there is a fibration sequence
of pointed spaces endowed with a homotopy section (pointed fibration from now
on)

$X\longrightarrow Z\longrightarrow B\operatorname{{\rm aut}}_{\pi}^{*}(X)$

which classifies pointed fibrations $X\to E\to B$ where the image of the
holonomy action, this time understood as $\pi_{1}(X)\to{\mathcal{E}}^{*}(X)$,
lies in $\pi$.

We now assume that $X$ is nilpotent and $G\subset{\mathcal{E}}(X)$ acts
nilpotently in $H_{*}(X)$. Take, in $\operatorname{{\bf cdgl}}$, the minimal
model $L$ of $X$. Then, there is a subgroup ${\cal G}$ of homotopy classes of
cdgl automorphisms of $L$ together with an action of $H_{0}(L)$, endowed with
the group structure given by the Baker-Campbell-Hausdorff (BCH) product, such
that ${\cal G}/H_{0}(L)$ is isomorphic to $G_{\mathbb{Q}}$, the
rationalization of $G$. Consider ${\rm Der}^{\mathcal{G}}L$ the connected sub
differential graded Lie algebra of ${\rm Der}L$, the derivations of $L$,
formed by the ${\cal G}$-derivations of $L$. These are all derivations of
positive degrees and, in degree $0$, those derivations $\theta$ whose
exponential $e^{\theta}$ is an automorphism of $L$ in ${\cal G}$. Then, we
prove (see §6 for a thorough and precise preamble of this result):

######  Theorem 0.1.

The rational homotopy type of the classifying fibration sequence

$X\longrightarrow B\operatorname{{\rm aut}}_{G}^{*}(X)\longrightarrow
B\operatorname{{\rm aut}}_{G}(X)$

is modeled by the cdgl fibration sequence

$L\stackrel{{\scriptstyle\operatorname{{\rm ad}}}}{{\longrightarrow}}{\rm
Der}^{\mathcal{G}}L\longrightarrow{\rm
Der}^{\mathcal{G}}L\widetilde{\times}sL.$

In fact, see Corollary 6.9, this cdgl sequence is part of a larger one

$\operatorname{\text{\rm
Hom}}(\overline{\mathscr{C}}(L),L)\to\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L),L)\to L\stackrel{{\scriptstyle\operatorname{{\rm
ad}}}}{{\to}}{\rm Der}^{\mathcal{G}}L\to{\rm
Der}^{\mathcal{G}}L\widetilde{\times}sL,$

whose suitable restriction to certain components of $\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L),L)$ and $\operatorname{\text{\rm
Hom}}(\overline{\mathscr{C}}(L),L)$ models the fibration sequence

$\operatorname{{\rm
aut}}_{G_{\mathbb{Q}}}^{*}(X_{\mathbb{Q}})\to\operatorname{{\rm
aut}}_{G_{\mathbb{Q}}}(X_{\mathbb{Q}})\stackrel{{\scriptstyle\operatorname{{\rm
ev}}}}{{\to}}X_{\mathbb{Q}}\to B\operatorname{{\rm
aut}}_{G_{\mathbb{Q}}}^{*}(X_{\mathbb{Q}})\to B\operatorname{{\rm
aut}}_{G_{\mathbb{Q}}}(X_{\mathbb{Q}})$

Whenever $X$ is simply connected, choosing universal covers in Theorem 0.1
recovers the classical result as the pair (1) extends, in the homotopy
categories, the classical Quillen model and realization functors [25] in the
simply connected setting.

On the other hand, let $\pi\subset{\mathcal{E}}^{*}(X)$ acting nilpotently on
$\pi_{*}(X)$. Then, there is a subgroup $\Pi$ of homotopy classes of
automorphisms of $L$ naturally isomorphic to $\pi_{\mathbb{Q}}$. As before,
let ${\rm Der}^{\Pi}L$ the connected dgl of $\Pi$-derivations of $L$. Then
(see again §6 for a detailed statement):

######  Theorem 0.2.

The rational homotopy type of the pointed classifying fibration sequence

$X\longrightarrow Z\longrightarrow B\operatorname{{\rm aut}}_{\pi}^{*}(X)$

is modeled by the cdgl fibration sequence

$L{\longrightarrow}L\widetilde{\times}{\rm Der}^{\Pi}L\longrightarrow{\rm
Der}^{\Pi}L.$

Under the assumptions and notation of the above results, immediate
consequences are:

######  Corollary 0.3.

$\pi_{\mathbb{Q}}\cong H_{0}({\rm Der}^{\Pi}L)$ and $G_{\mathbb{Q}}\cong
H_{0}({\rm Der}^{\mathcal{G}}L)/H_{0}(\operatorname{{\rm ad}})$.

In both cases $H_{0}$ is considered as a group with the BCH product.

######  Corollary 0.4.

$BG$ and $B\pi$ have the rational homotopy type of the realization of the
cdgl’s ${\rm Der}^{\mathcal{G}}_{0}L\oplus R$ and ${\rm Der}^{\Pi}_{0}L\oplus
S$ respectively.

Here, $R$ and $S$ denote a complement of the $1$-cycles of ${\rm
Der}^{\mathcal{G}}L\widetilde{\times}sL$ and ${\rm Der}^{\Pi}L$ respectively.

Along the way to the above results we must translate to $\operatorname{{\bf
cdgl}}$ the needed geometric phenomena in each situation, mostly arising from
the non triviality of the fundamental group. Distinguishing free and pointed
homotopy classes from the cdgl point of view, or describing evaluation
fibrations as cdgl’s morphisms of derivations, are examples of this conveyance
which enlarges the dictionary between the homotopy categories of simplicial
sets and cdgl’s. This is precisely and briefly outlined in the following
summary of the content.

In §1 we give a short digest of the main facts concerning the homotopy theory
of cdgl’s and its connection with that of simplicial sets by means of the pair
(1). From this point on, we refer to this part for general notation in this
matter.

In the second section we describe the subtle but important difference between
free and pointed homotopy classes in $\operatorname{{\bf cdgl}}$: given cdgl’s
$L$ and $L^{\prime}$, the group $H_{0}(L)$ endowed with the BCH product acts
on the set of cdgl homotopy classes $\\{L^{\prime},L\\}$ by means of the
exponential map $[x]\bullet[\varphi]=e^{\operatorname{{\rm
ad}}_{x}}\circ\varphi$. Whenever $L$ is connected and $L^{\prime}$ is the
minimal model of the connected simplicial $X$, $\\{L^{\prime},L\\}$ is in
bijective correspondence with the set of pointed homotopy classes
$\\{X,\langle L\rangle\\}^{*}$ (Corollary 2.2) and the mentioned action
corresponds to that of $\pi_{1}(X)$ on $\\{X,\langle L\rangle\\}^{*}$ (Theorem
2.4). That is, $\\{X,\langle L\rangle\\}\cong\\{L^{\prime},L\\}/H_{0}(L)$.

Again by the presence of non trivial fundamental groups, we need to analyze
distinguished subgroups of degree $0$ elements of particular cdgl’s, as always
endowed with the BCH product. To this end we show in Section 3 that any
complete subgroup of a complete Lie algebra concentrated in degree $0$ is in
fact a sub Lie algebra (Theorem 3.2).

In §4 we describe cdgl models of evaluation fibrations $\operatorname{{\rm
map}}^{*}(X,Y)\to\operatorname{{\rm map}}(X,Y)\to Y$ (Theorem 4.2) and any of
its components (Proposition 4.6) by refining existing Lie models of mapping
spaces. Later on, in Section 5 we transform these models in terms of
derivations of the models of the involved simplicial sets (Theorem 5.1 and
Corollary 5.2). Of special interest is the description of the long homotopy
exact sequence of the evaluation fibration as the long homology sequence of a
short exact sequence of chain complexes of derivations (Theorem 5.3).

Section 6 contains a detailed and precise statement of our main results,
together with the first consequences.

Although $\operatorname{{\bf cdgl}}$ is complete and cocomplete from the
categorical point of view, essential manipulations of cdgl’s may lie outbound
of this category. The twisted product $L\widetilde{\times}M$ of cdgl’s, or
even the dgl ${\rm Der}L$ of derivations of a given Lie algebra are examples
of this malfunction, as they fail to be complete even if $L$ and $M$ are. In
§7 we fix these issues for distinguished sub dgl’s of ${\rm Der}L$
(Proposition 7.6) and specific twisted products we use (Propositions 7.9, 7.10
and 7.12).

Section 8 contains the proofs of Theorems 0.1 and 0.2 and finally, in §9, we
present several consequences of these results, of which Corollary 0.3 is an
immediate one. Also (see Theorem 9.6 and Propositions 9.7, 9.8), we model the
rational homotopy type of the fibration sequences,

$\operatorname{{\rm aut}}_{G}(X)\to G\to B\operatorname{{\rm aut}}_{1}(X)\to
B\operatorname{{\rm aut}}_{G}(X)\to BG$

and

$\operatorname{{\rm aut}}_{\pi}^{*}(X)\to\pi\to B\operatorname{{\rm
aut}}_{1}^{*}(X)\to B\operatorname{{\rm aut}}_{\pi}^{*}(X)\to B\pi,$

obtaining in particular Corollary 0.4.

Acknowledgement. We thank Alexander Berglund for helpful conversations.

##  1 Homotopy theory of complete Lie algebras

In this section we recall the basics for complete differential graded Lie
algebras, from the homotopy point of view. For it, original references are [7,
8, 9, 10] whose main results are developed in the complete and detailed
monograph [11]. Sometimes we will also use classical facts from the Sullivan
commutative approach to rational homotopy theory. These are not included here
and we refer to [14, 15] as standard references.

For any category ${\mathcal{C}}$, we denote by $\operatorname{\text{\rm
Hom}}_{\mathcal{C}}$ its set of morphisms, except for that of (graded) vector
spaces whose morphisms will be denoted by the unadorned
$\operatorname{\text{\rm Hom}}$.

All considered chain (or cochain) complexes, with possibly extra structures,
are unbounded and have $\mathbb{Q}$ as ground field. The suspension of such a
complex $V$ is denoted by $sV$ where $(sV)_{n}=V_{n-1}$, $n\in\mathbb{Z}$ and
$dsv=-sdv$. The desuspension complex is denoted by ${s^{-1}}V$.

We denote by $\operatorname{{\bf sset}}$ the category of simplicial sets. For
any $n\geq 0$, we denote by
$\underline{\Delta}^{n}=\\{\underline{\Delta}^{n}_{p}\\}_{p\geq 0}$ the
simplicial set in which $\underline{\Delta}^{n}_{p}=\\{(j_{0},\dots,j_{p})\mid
0\leq j_{0}\leq\dots\leq j_{p}\leq n\\}$ with the usual faces and
degeneracies. Given $X$ and $Y$ simplicial sets we denote by
$\operatorname{{\rm map}}(X,Y)$ the simplicial mapping space, that is,

$\operatorname{{\rm map}}(X,Y)_{\bullet}=\operatorname{\text{\rm
Hom}}_{\operatorname{{\bf sset}}}(X\times\underline{\Delta}^{\bullet},Y).$

Analogously, we denote by $\operatorname{{\rm map}}^{*}(X,Y)$ the pointed
mapping space $\operatorname{\text{\rm Hom}}_{\operatorname{{\bf
sset}}^{*}}(X\times\underline{\Delta}^{\bullet},Y)$ In the category
$\operatorname{{\bf sset}}^{*}$ of pointed simplicial sets.

Let $\operatorname{{\bf dgl}}$ denote the category of differential graded Lie
algebras (dgl’s henceforth). A dgl $L$, or $(L,d)$ if we want to specify the
differential, is $n$-connected, for $n\geq 0$, if $L=L_{\geq n}$. As usual,
connected means $0$-connected. The $n$-connected cover of a dgl $L$ is the
$n$-connected sub dgl ${L}^{(n)}$ given by

$L^{(n)}_{p}=\begin{cases}\ker d&\text{if $p=n$},\\\ \,\,\,L_{p}&\text{if
$p>n$}.\end{cases}$

We denote by $\widetilde{L}=L^{(1)}$ the $1$-connected cover.

Given a dgl $L$ we denote by ${\rm Der}L$ the dgl of its derivations in which,
for each $n\in\mathbb{Z}$, ${\rm Der}_{n}L$ are linear maps $\theta\colon L\to
L$ of degree $n$ such that

$\theta[a,b]=[\theta(a),b]+(-1)^{|a|n}[a,\theta(b)],{\quad\text{for}\quad}a,b\in
L.$

The Lie bracket and differential are given by,

$[\theta,\eta]=\theta\circ\eta-(-1)^{|\theta||\eta|}\eta\circ\theta,\quad
D\theta=d\circ\theta-(-1)^{|\theta|}\theta\circ d.$

On the other hand, given a dgl morphism $f\colon L\to M$ we denote by ${\rm
Der}_{f}(L,M)$ the chain complex of $f$-derivations in which ${\rm
Der}_{f}(L,M)_{n}$ are linear maps $\theta\colon L\to M$ of degree $n$ such
that

$\theta[a,b]=[\theta(a),f(b)]+(-1)^{n|a|}[f(a),\theta(b)],\quad a,b\in L.$

The differential is given as before.

Following [31, §7.2], or more generally [2, §3.5], a twisted product
$L\widetilde{\times}M$ of the dgl’s $L$ and $M$ is a dgl structure on the
underline vector space $L\times M$ so that

$0\to L\longrightarrow L\widetilde{\times}M{\longrightarrow}M\to 0$

is an exact dgl sequence. In particular $L$ is a sub dgl of the twisted
product.

A filtration of a dgl $L$ is a decreasing sequence of differential Lie ideals,

$L=F^{1}\supset\dots\supset F^{n}\supset F^{n+1}\supset\dots$

such that $[F^{p},F^{q}]\subset F^{p+q}$ for $p,q\geq 1$. In particular, if

$L^{1}\supset\dots\supset L^{n}\supset L^{n+1}\supset\dots$

denote the lower central series of $L$, that is, $L^{1}=L$ and
$L^{n}=[L,L^{n-1}]$ for $n>1$, then $L^{n}\subset F^{n}$ for all $n$.

A complete differential graded Lie algebra, cdgl henceforth, is a dgl $L$
equipped with a filtration $\\{F^{n}\\}_{n\geq 1}$ for which the natural map

$L\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\varprojlim_{n}L/F^{n}$

is a dgl isomorphism. A morphism $f\colon L\to L^{\prime}$ between cdgl’s,
associated to filtrations $\\{F^{n}\\}_{n\geq 1}$ and $\\{{G}^{n}\\}_{n\geq
1}$ respectively, is a dgl morphism which preserves the filtrations, that is,
$f(F^{n})\subset G^{n}$ for each $n\geq 1$. We denote by $\operatorname{{\bf
cdgl}}$ the corresponding category which is complete and cocomplete [11,
Proposition 3.5].

Given a dgl $L$ filtered by $\\{F^{n}\\}_{n\geq 1}$, its completion is the dgl

$\widehat{L}=\varprojlim_{n}L/F^{n}.$

which is always complete with respect to the filtration

$\widehat{F}^{n}=\ker(\widehat{L}\to L/F^{n})$

as $\widehat{L}/\widehat{F}^{n}=L/F^{n}$. Unless a particular filtration is
explicitly mentioned, the completion of a given dgl is taken over the lower
central series. This is the case, in particular, for any cdgl
$L=({\widehat{\mathbb{L}}}(V),d)$ where

${\widehat{\mathbb{L}}}(V)=\varprojlim_{n}\mathbb{L}(V)/\mathbb{L}(V)^{n}$

is the completion of the free Lie algebra $\mathbb{L}(V)$ generated by the
graded vector space $V$, see [11, §3.2] for an exhaustive treatment of this
cdgl. In this instance,

$\widehat{F}^{n}=\widehat{\mathbb{L}}^{\geq n}(V)=\prod_{q\geq
n}\mathbb{L}^{q}(V),{\quad\text{for}\quad}n\geq 1.$

By an abuse of notation, we write $\widehat{F}^{n}=L^{n}$ so that
$L=\varprojlim_{n}L/L^{n}$.

If $A$ is a commutative differential graded algebra (cdga henceforth) and $L$
is a cdgl with respect to the filtration $\\{F^{n}\\}_{n\geq 1}$, we define
their complete tensor product as the cdgl

$A\widehat{\otimes}L=\varprojlim_{n}A\otimes(L/F^{n})$

where the differential and the bracket in $A\otimes(L/F^{n})$ are defined as
usual by $d(a\otimes x)=da\otimes x+(-1)^{|a|}a\otimes dx$ and $[a\otimes
x,a^{\prime}\otimes
x^{\prime}]=(-1)^{|a^{\prime}||x|}aa^{\prime}\otimes[x,x^{\prime}]$.

A Maurer-Cartan element, or simply MC element, of a given dgl $L$ is an
element $a\in L_{-1}$ satisfying the Maurer-Cartan equation
$da=-\frac{1}{2}[a,a]$. We denote by ${\operatorname{{\rm MC}}}(L)$ the set of
MC elements in $L$. Given $a\in{\operatorname{{\rm MC}}}(L)$, we denote by
$d_{a}=d+\operatorname{{\rm ad}}_{a}$ the perturbed differential. This is a
new differential on $L$ where $d$ is the original one and $\operatorname{{\rm
ad}}$ is the usual adjoint operator. The component of $L$ at $a$ is the
connected sub dgl $L^{a}$ of $(L,d_{a})$ given by

$L^{a}_{p}=\begin{cases}\ker d_{a}&\text{if $p=0$},\\\ \,\,\,L_{p}&\text{if
$p>0$}.\end{cases}$

In other terms, $L^{a}=(L,d_{a})^{0}$. Whenever $L$ is complete
${\operatorname{{\rm MC}}}(L)$ acts on the group $L_{0}$, endowed with the
Baker-Campbell-Hausdorff product (see §3 for a precise definition in our
context) by

$x\,{\mathscr{G}}\,a=e^{\operatorname{{\rm
ad}}_{x}}(a)-\frac{e^{\operatorname{{\rm ad}}_{x}}-1}{\operatorname{{\rm
ad}}_{x}}(dx)=\sum_{i\geq 0}\frac{\operatorname{{\rm
ad}}_{x}^{i}(a)}{i!}-\sum_{i\geq 0}\frac{\operatorname{{\rm
ad}}_{x}^{i}(dx)}{(i+1)!},\quad x\in L_{0},\,a\in{\operatorname{{\rm
MC}}}(L).$

This is the gauge action and we denote by $\widetilde{\operatorname{{\rm
MC}}}(L)={\operatorname{{\rm MC}}}(L)/{\mathscr{G}}$ the corresponding orbit
set. This is thoroughly studied in [11, §4.3] and a particular homotopically
friendly description of the gauge action can be found in [11, §5.3].

There is a pair of adjoint functors, (global) model and realization,

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
11.11111pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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35.11111pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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cdgl}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
13.97227pt\raise-12.30554pt\hbox{{}\hbox{\kern 0.0pt\raise
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0.0pt\raise-2.5pt\hbox{$\scriptstyle{\langle\,\cdot\,\rangle}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
11.11111pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
18.3125pt\raise 9.6972pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{{\mathfrak{L}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 35.11111pt\raise
4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$
(2)

which are deeply studied in [11, Chapter 7] and are based in the cosimplicial
cdgl

${\mathfrak{L}}_{\bullet}=\\{{\mathfrak{L}}_{n}\\}_{n\geq 0}.$

For each $n\geq 0$, see [11, Chapter 6],

${\mathfrak{L}}_{n}=\bigl{(}{\widehat{\mathbb{L}}}(s^{-1}\Delta^{n}),d)$

where $s^{-1}\Delta^{n}$ denotes the desuspension
$s^{-1}N_{*}(\underline{\Delta}^{n})$ of the non-degenerate simplicial chains
on $\underline{\Delta}^{n}$. That is, for any $p\geq 0$, a generator of degree
$p-1$ of $s^{-1}\Delta^{n}$ can be written as $a_{i_{0}\dots i_{p}}$ with
$0\leq i_{0}<\dots<i_{p}\leq n$. The cofaces and codegeneracies in
${\mathfrak{L}}_{\bullet}$ are induced by those on the cosimplicial chain
complex ${s^{-1}}N_{*}\underline{\Delta}^{\bullet}$, and the differential $d$
on each ${\mathfrak{L}}_{n}$ is the only one (up to cdgl isomorphism)
satisfying:

  * (1)

For each $i=0,\dots,n$, the generators of $s^{-1}\Delta^{n}$, corresponding to
vertices, are MC elements.

  * (2)

The linear part of $d$ is induced by the boundary operator of
$s^{-1}\Delta^{n}$.

  * (3)

The cofaces and degeneracies are cdgl morphisms.

In particular, ${\mathfrak{L}}_{0}$ is the free Lie algebra $\mathbb{L}(a)$
generated by a MC element and ${\mathfrak{L}}_{1}$ is the Lawrence-Sullivan
interval [20] (see also [12]).

The realization of a given cdgl $L$ is the simplicial set

$\langle L\rangle=\operatorname{\text{\rm Hom}}_{\operatorname{{\bf
cdgl}}}({\mathfrak{L}}_{\bullet},L).$

In particular, the set $\langle L\rangle_{0}$ of $0$-simplices coincides with
${\operatorname{{\rm MC}}}(L)$. Moreover, see [11, §7.2], if $\langle
L\rangle^{a}$ denotes the path component of $\langle L\rangle$ containing $a$,
we have:

$\langle L\rangle^{a}\simeq\langle L^{a}\rangle,\quad\langle
L\rangle\simeq\amalg_{a\in\widetilde{{\operatorname{{\rm MC}}}}(L)}\,\,\langle
L^{a}\rangle.$ (3)

However, it is important to observe that the realization of a cdgl is
invariant under perturbations. That is, for any cdgl $L$ and any
$a\in{\operatorname{{\rm MC}}}(L)$,

$\langle L\rangle\cong\langle(L,d_{a})\rangle$ (4)

Indeed, by [11, Proposition 4.28] there is a bijection
$\widetilde{\operatorname{{\rm MC}}}(L)\cong\widetilde{\operatorname{{\rm
MC}}}(L,d_{a})$ which sends the gauge class of an MC element $z$ of $L$ to the
gauge class of the MC element $z-a$ of $(L,d_{a})$. On the other hand, observe
that $L^{z}=(L,d_{a})^{a-z}$. Finally, apply (3).

Also, if $L$ is connected, and for any $n\geq 1$, the map

$\rho_{n}\colon\pi_{n}\langle
L\rangle\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}H_{n-1}(L),\quad\rho[\varphi]=[\varphi(a_{0\dots
n})],$

is a group isomorphism [11, Theorem 7.18]. Here, the group structure in
$H_{0}(L)$ is considered with the Baker-Campbell-Hausdorff (BCH) product. We
also remark that, for each $n\geq 1$, the $n$th connected cover $\langle
L\rangle^{(n)}$ of the realization of a connected cdgl is homotopically
equivalent to the realization $\langle L^{(n)}\rangle$ of its $n$th connected
cover [11, §12.5.3].

A final and fundamental property of the realization functor is that it
coincides with any known geometrical realization of dgl’s: if $L$ is a
$1$-connected dgl of finite type then $\langle L\rangle\simeq\langle
L\rangle^{Q}$, see [11, Corollary 11.17], where $\langle\,\cdot\,\rangle^{Q}$
stand for the classical Quillen realization functors [25]. Furthermore, see
[7, Theorem 0.1], [26, Theorem 3.2] and [11, Theorem 11.13], for any cdgl $L$,
$\langle L\rangle$ is a strong deformation retract of ${\operatorname{{\rm
MC}}}_{\bullet}(L)$, the Deligne-Getzler-Hinich simplicial realization of $L$
[17, 19].

On the other hand, the global model of a simplicial set $X$ is the cdgl

${\mathfrak{L}}_{X}=\varinjlim_{\sigma\in X}{\mathfrak{L}}_{|\sigma|}.$

It can be checked, see [11, Proposition 7.8], that as complete Lie algebra,
${\mathfrak{L}}_{X}={\widehat{\mathbb{L}}}(s^{-1}X)$ where $s^{-1}X$ denotes
the desuspension $s^{-1}N_{*}(X)$ of the chain complex of non-degenerate
simplicial chains on $X$. Moreover, the differential $d$ on
${\mathfrak{L}}_{X}$ is completely determined by the following:

  * (1)

The $0$-simplices of $X$ are Maurer-Cartan elements.

  * (2)

The linear part of $d$ is the desuspension of the differential in $N_{*}(X)$.

  * (3)

If $j\colon Y\subset X$ is a subsimplicial set, then
${\mathfrak{L}}_{j}={\widehat{\mathbb{L}}}\bigl{(}s^{-1}N_{*}(j)\bigr{)}$.

If $X$ is a simply connected simplicial set of finite type and $a$ is any of
its vertices, then [11, Theorem 10.2], ${\mathfrak{L}}_{X}^{a}$ is quasi-
isomorphic to $\lambda(X)$ where $\lambda$ is the classical Quillen dgl model
functor [25]. Moreover, see [11, Theorem 11.14], for any connected simplicial
set $X$ of finite type, $\langle{\mathfrak{L}}_{X}^{a}\rangle$ is weakly
homotopy equivalent to $\mathbb{Q}_{\infty}X$ the Bousfield-Kan
$\mathbb{Q}$-completion of $X$ [4]. Recall that, whenever $X$ is nilpotent,
its $\mathbb{Q}$-completion coincides, up to homotopy, with its classical
rationalization, which we denote by $X_{\mathbb{Q}}$.

The category $\operatorname{{\bf cdgl}}$ has a cofibrantly generated model
structure, see [11, Chapter 8], for which the functors in (2) become a Quillen
pair with the classical model structure on $\operatorname{{\bf sset}}$: a cdgl
morphism $f\colon L\to M$ is a fibration if it is surjective in non-negative
degrees and is a weak equivalence if $\widetilde{\operatorname{{\rm
MC}}}(f)\colon\widetilde{\operatorname{{\rm
MC}}}(L)\stackrel{{\scriptstyle\cong}}{{\to}}\widetilde{\operatorname{{\rm
MC}}}(M)$ is a bijection, and $f^{a}\colon
L^{a}\stackrel{{\scriptstyle\simeq}}{{\to}}M^{a}$ is a quasi-isomorphism for
every $a\in{\operatorname{{\rm MC}}}(L)$.

Although quasi-isomorphisms are weak equivalence only in the connected case,
the generalized Goldman–Milson Theorem, see [11, Theorem 4.33], provides a
criterium for a quasi-isomorphism to be a weak equivalence in the general
case: let $f\colon L\to L^{\prime}$ be a cdgl morphism such that map induced
by the corresponding filtrations
${F}^{n}/F^{n+1}\xrightarrow[]{\simeq}{G}^{n}/G^{n+1}$ is a quasi-isomorphism
for any $n\geq 1$. Then, $f$ is a weak equivalence.

A quasi-isomorphism of connected cdgl’s of the form

$({\widehat{\mathbb{L}}}(V),d)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}L$

makes of $({\widehat{\mathbb{L}}}(V),d)$ a cofibrant replacement of $L$ and we
say that it is a model of $L$. If $d$ has no linear term we say that
$({\widehat{\mathbb{L}}}(V),d)$ is minimal and is unique up to cdgl
isomorphism. A fundamental object in this theory is:

######  Definition 1.1.

Let $X$ be a connected simplicial set and $a$ any of its vertices. The minimal
model $({\widehat{\mathbb{L}}}(V),d)$ of ${\mathfrak{L}}_{X}^{a}$ is called
the minimal model of $X$ [11, Definition 8.32].

If $({\widehat{\mathbb{L}}}(V),d)$ is the minimal model of $X$ then, see [11,
Proposition 8.35], $sV\cong\widetilde{H}_{*}(X;\mathbb{Q})$ and, provided $X$
of finite type,
$sH_{*}({\widehat{\mathbb{L}}}(V),d)\cong\pi_{*}(\mathbb{Q}_{\infty}X)$. Here,
the group $H_{0}({\widehat{\mathbb{L}}}(V),d)$ is again considered with the
BCH product. Furthermore, from all of the above, if $X$ is simply connected,
the minimal model of $X$ is isomorphic to its classical Quillen minimal model.

Let $\operatorname{{\bf cdgc}}$ denote the category of cocommutative
differential graded coalgebras (cdgc’s from now on). Every cdgc $C$ is always
assumed to have a counit $\varepsilon\colon C\to\mathbb{Q}$ and a
coaugmentation $\eta\colon\mathbb{Q}\to C$. We write
$\overline{C}=\ker\varepsilon$, so that $C\cong\overline{C}\oplus\mathbb{Q}$.
The map
$\overline{\Delta}\colon\overline{C}\to\overline{C}\otimes\overline{C}$ is
defined by $\overline{\Delta}x=\Delta x-(u\otimes x+x\otimes u)$, with
$u=\eta(1)$. A cdgc $C$ is $n$-connected, for $n\geq 0$, if
$\overline{C}=C_{>n}$.

Recall the classical pair of adjoint functors

$\textstyle{{\rm\bf
cdgc}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathscr{L}}$$\textstyle{\mathbf{dgl}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathscr{C}}}$
(5)

defined as follows:

Given $C$ a cdgc, ${\mathscr{L}}(C)$ is $(\mathbb{L}(s^{-1}\overline{C}),d)$,
the free dgl generated by the desuspension of $\overline{C}$ and
$d=d_{1}+d_{2}$ where

$\displaystyle d_{1}(s^{-1}c)$ $\displaystyle=-s^{-1}dc,$ $\displaystyle
d_{2}(s^{-1}c)$
$\displaystyle=\frac{1}{2}\sum_{i}(-1)^{|a_{i}|}[s^{-1}a_{i},s^{-1}b_{i}]\quad\text{with}\quad\overline{\Delta}c=\sum_{i}a_{i}\otimes
b_{i}.$

On the other hand, ${\mathscr{C}}(L)$ is $(\land(sL),d)$, the free
cocommutative coalgebra cogenerated by the suspension of $L$ and in which
$d=d_{1}+d_{2}$ where

$\displaystyle d_{1}(sv_{1}\land\dots\land sv_{n})$
$\displaystyle=-\sum_{i=1}^{n}(-1)^{n_{i}}sv_{1}\land\dots\land
s(dv_{i})\land\dots\land sv_{n},$ $\displaystyle d_{2}(sv_{1}\land\dots\land
sv_{n})$ $\displaystyle=\sum_{1\leq i<j\leq
k}(-1)^{|sv_{i}|}\,\rho_{ij}\,s[v_{i},v_{j}]\land
sv_{1}\land\dots\land\widehat{sv_{i}}\land\dots\land\widehat{sv_{j}}\land\dots\land
sv_{n},$

$n_{i}=\sum_{j<i}|sv_{j}|$ and $\rho_{ij}$ is the Koszul sign of the
permutation

$sv_{1}\land\dots\land sv_{n}\mapsto sv_{i}\land sv_{j}\land
sv_{1}\land\dots\land\widehat{sv_{i}}\land\dots\land\widehat{sv_{j}}\land\dots\land
sv_{n}.$

In particular, $d_{1}(sv)=-sdv$ and $d_{2}(sv\land sw)=(-1)^{|sv|}s[v,w]$.

In general, ${\mathscr{C}}$ preserves quasi-isomorphisms [24, Proposition
4.4]. However, ${\mathscr{L}}$ preserves quasi-isomorphisms between
$1$-connected cdgc’s [24, Proposition 6.4] and only between connected fibrant
cdgc’s of finite type [11, Proposition 2.4(2)].

The _cochain algebra_ ${\mathscr{C}}^{*}(L)$ on the dgl $L$ is the cdga dual
of ${\mathscr{C}}(L)$. Whenever $L$ connected and of finite type, this cdga
has a well known description:

${\mathscr{C}}^{*}(L)\cong(\land(sL)^{\\#},d)$

in which $d=d_{1}+d_{2}$ where

$\displaystyle\langle d_{1}v,sx\rangle$ $\displaystyle=(-1)^{|v|}\langle
v,sdx\rangle,$ $\displaystyle\langle d_{2}v,sx\land sy\rangle$
$\displaystyle=(-1)^{|y|+1}\langle v,s[x,y]\rangle,$

The adjunction map $\alpha{\colon}{\mathscr{L}}{\mathscr{C}}(L)\to L$ is the
unique dgl morphism $(\mathbb{L}(s^{-1}{\land^{+}sL}),d)\to L$ extending the
projection $s^{-1}{\land^{+}sL}\to s^{-1}{\land^{+}sL}/\left(s^{-1}\land^{\geq
2}sL\right)\cong L$. The other adjunction
$\beta{\colon}C\to{\mathscr{C}}{\mathscr{L}}(C)$ is the unique cdgc morphism
$C\to(\land s{\mathscr{L}}(C),d)$ lifting the inclusion
$\overline{C}=ss^{-1}\overline{C}\subset s\mathbb{L}(s^{-1}\overline{C})$. By
[11, Proposition 2.3] $\alpha$ is always a quasi-isomorphism while $\beta$ is
a quasi-isomorphism if $C$ is connected.

##  2 Free and pointed homotopy classes in $\operatorname{{\bf cdgl}}$

Throughout the next sections it will be essential to make a clear distinction
between the sets of free and pointed homotopy classes of maps, from the cdgl
point of view. This is a subtle but crucial point that we settle here. Note
that this discussion is meaningless in the classical Quillen theory as, in the
simply connected category, both sets coincide.

Recall from [11, §8.3] that two cdgl morphisms $\varphi,\psi\colon M\to L$ are
homotopic if there is a cdgl morphism $\Phi\colon M\to
L\widehat{\otimes}\land(t,dt)$ such that $\varepsilon_{0}\circ\Phi=\varphi$
and $\varepsilon_{1}\circ\Phi=\psi$. Here, $\land(t,dt)$ is the free
commutative graded algebra generated by the element $t$ of degree $0$ and its
differential $dt$. Also, for $i=0,1$, $\varepsilon_{i}\colon
L\widehat{\otimes}\land(t,dt)\to L$ is the cdgl morphism obtained by sending
$t$ to $i$. We denote by $\\{M,L\\}$ the set of homotopy classes of cdgl
morphisms from $M$ to $L$.

For the rest of this section we fix a connected cdgl $L$ and a connected (non
necessarily reduced) simplicial set $X$, pointed by the $0$-simplex $b\in
X_{0}$. Analogously, we denote respectively by $\\{X,\langle L\rangle\\}$ and
$\\{X,\langle L\rangle\\}^{*}$ the set of free and pointed homotopy classes of
simplicial maps. Note that, since $\langle L\rangle$ is reduced, being
$0\in{\operatorname{{\rm MC}}}(L)$ the only $0$-simplex, every map
$X\to\langle L\rangle$ is pointed. Then:

######  Proposition 2.1.

$\\{X,\langle L\rangle\\}\cong\\{{\mathfrak{L}}_{X},L\\}$ and $\\{X,\langle
L\rangle\\}^{*}\cong\\{{\mathfrak{L}}_{X}/(b),L\\}$.

Here $(b)$ denotes the Lie ideal generated by the Maurer-Cartan element $b$.

###### Proof.

As the model and realization functors constitute a Quillen pair, the first
identity is obvious [11, Corollary 8.2(iv)]. Nevertheless, an explicit
description of this identity will lead to the second:

The adjunction (2) provides a bijection

$\operatorname{\text{\rm Hom}}_{\operatorname{{\bf sset}}}(X,\langle
L\rangle)\cong\operatorname{\text{\rm Hom}}_{\operatorname{{\bf
cdgl}}}({\mathfrak{L}}_{X},L)$

which sends any simplicial map $f\colon X\to\langle L\rangle$ to the cdgl
morphism $\varphi_{f}\colon{\mathfrak{L}}_{X}\to L$ defined as follows: recall
that, for $p\geq 0$, a generator of ${\mathfrak{L}}_{X}$ of degree $p-1$ is
identified with a non-degenerate $p$-simplex $\sigma\in X_{p}$. Then,
$\varphi_{f}(\sigma)=f(\sigma)(a_{0\dots p})$, being $a_{0\dots p}\in
s^{-1}\Delta^{p}$ the top generator. Note that $\varphi_{f}(b)=0$ so it
induces a map $\overline{\varphi}_{f}\colon{\mathfrak{L}}_{X}/(b)\to L$.

Now, let $H\colon X\times\underline{\Delta}^{1}\to Y$ be a homotopy between
two simplicial maps $f,g\colon X\to\langle L\rangle$. That is, $H$ is a
simplicial map for which $H|_{X\times(0)}=f$ and $H|_{X\times(1)}=g$, being
$(0)$ and $(1)$ the subsimplicial sets of $\underline{\Delta}^{1}$ generated
by the $0$-simplices $0$ and $1$ respectively. Taking into account the
bijection

$\operatorname{\text{\rm Hom}}_{\operatorname{{\bf
sset}}}(X\times\underline{\Delta}^{1},\langle
L\rangle)\cong\operatorname{\text{\rm Hom}}_{\operatorname{{\bf
sset}}}\bigl{(}X,\operatorname{{\rm map}}(\underline{\Delta}^{1},\langle
L\rangle)\bigr{)},$

$H$ corresponds to a simplicial map $X\to\operatorname{{\rm
map}}(\underline{\Delta}^{1},\langle L\rangle)$. But, see [11, Theorem 12.18]
which is precisely [1, Theorem 6.6] in our context, there is a homotopy
equivalence

$\operatorname{{\rm map}}(\underline{\Delta}^{1},\langle
L\rangle)\simeq\langle L\widehat{\otimes}\land(t,dt)\rangle.$

Hence, we identify $H$ with a map $X\to\langle
L\widehat{\otimes}\land(t,dt)\rangle$ which, again by adjunction, corresponds
to a cdgl morphism $\Phi\colon{\mathfrak{L}}_{X}\to
L\widehat{\otimes}\land(t,dt)$. The fact that $H$ is a homotopy between $f$
and $g$ translates to $\varepsilon_{0}\circ\Phi=\varphi_{f}$ and
$\varepsilon_{1}\circ\Phi=\varphi_{g}$ so that $\Phi$ is a homotopy between
$\varphi_{f}$ and $\varphi_{g}$. This shows again that $\\{X,\langle
L\rangle\\}\cong\\{{\mathfrak{L}}_{X},L\\}$

Now, assume that $H$ is a pointed homotopy. That is, $H$ is constant (to the
null and only $0$-simplex of $\langle L\rangle$) on the subsimplicial set
$(b)\times\underline{\Delta}^{1}$ of $X\times\underline{\Delta}^{1}$. Then, a
careful inspection along this identification shows that the homotopy $\Phi$
sends $b$ to $0$ (which is not the case in general) and thus it induces a
homotopy $\overline{\Phi}\colon{\mathfrak{L}}_{X}/(b)\to
L\widehat{\otimes}\land(t,dt)$ between $\overline{\varphi}_{f}$ and
$\overline{\varphi}_{g}$. This proves that $\\{X,\langle
L\rangle\\}^{*}\cong\\{{\mathfrak{L}}_{X}/(b),L\\}$.

∎

######  Corollary 2.2.

If $L^{\prime}$ is the minimal Lie model of $X$, $\\{X,\langle
L\rangle\\}^{*}\cong\\{L^{\prime},L\\}$. In particular, if $L$ is a Lie model
of the finite type simplicial set $Y$,
$\\{X,\mathbb{Q}_{\infty}Y\\}^{*}\cong\\{L^{\prime},L\\}$.

###### Proof.

It is known , see [11, Proposition 8.7], that the composition

${\mathfrak{L}}_{X}^{b}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}({\mathfrak{L}}_{X},d_{b})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}{\mathfrak{L}}_{X}/(b)$

is not only an injective quasi-isomorphism but a weak equivalence in
$\operatorname{{\bf cdgl}}$. Hence, $L^{\prime}$ is also weakly equivalent to
${\mathfrak{L}}_{X}/(b)$ and thus
$\\{L^{\prime},L\\}\cong\\{{\mathfrak{L}}_{X}/(b),L\\}\cong\\{X,\langle
L\rangle\\}^{*}$. To finish, recall that being $L$ a Lie model of $Y$,
$\langle L\rangle\simeq\mathbb{Q}_{\infty}Y$. ∎

We now translate to $\operatorname{{\bf cdgl}}$ the action of $\pi_{1}\langle
L\rangle$ on $\\{X,\langle L\rangle\\}^{*}$. For it, and for any given two
connected simplicial sets $X$ and $Y$, the action ${\scriptstyle\,\bullet\,}$
of $\pi_{1}(Y)$ on $\\{X,Y\\}^{*}$ can be described as follows: Consider the
wedge $[a,b]\vee X$ of an interval with $X$, and let $r{\colon}[a,b]\vee X\to
X$ be the retraction obtained by contracting $[a,b]$ into $b$. As
$b\hookrightarrow X$ is a cofibration, there exists a homotopy equivalence
$g{\colon}X\to[a,b]\vee X$ such that $g(b)=a$ and $r\circ
g\simeq\operatorname{{\rm id}}_{X}$. Let $\widetilde{g}=c\circ g\colon X\to
S^{1}\vee X$ where $c$ glues $a$ with $b$. Then, given $\alpha\colon S^{1}\to
Y$ and $f\colon X\to Y$, $[\alpha]^{*}{\scriptstyle\,\bullet\,}[f]^{*}$ is
represented by $(\alpha\vee f)\circ\widetilde{g}$.

We define now an analogous action of $H_{0}(L)$ on
$\\{{\mathfrak{L}}_{X}/(b),L\\}$. For it, recall from [11, Proposition 4.10]
that, for any given element $y\in L_{0}$, the exponential

$e^{\operatorname{{\rm ad}}_{y}}=\sum_{n\geq 0}\frac{\operatorname{{\rm
ad}}_{y}^{n}}{n!}$

is a well defined automorphism of $L$. In fact, this is a particular instance
of a more general setting covered by Remark 3.1 below. Furthermore, if $y*z$
denotes the BCH product of elements $y,z\in L_{0}$, then
$e^{\operatorname{{\rm ad}}_{y*z}}=e^{\operatorname{{\rm ad}}_{y}}\circ
e^{\operatorname{{\rm ad}}_{z}}$, see [11, Corollary 4.12].

######  Definition 2.3.

Given any cdgl $L^{\prime}$, the group $(L_{0},*)$ acts on
$\operatorname{\text{\rm Hom}}_{\operatorname{{\bf cdgl}}}(L^{\prime},L)$ by
$y{\scriptstyle\,\bullet\,}\varphi=e^{\operatorname{{\rm
ad}}_{y}}\circ\varphi$. We denote in the same way the induced action,

$H_{0}(L)\times\\{L^{\prime},L\\}\stackrel{{\scriptstyle\bullet}}{{\longrightarrow}}\\{L^{\prime},L\\},\qquad[y]{\scriptstyle\,\bullet\,}[\varphi]=[e^{\operatorname{{\rm
ad}}_{y}}\circ\varphi].$

Then, if we denote by $\gamma\colon\\{X,\langle
L\rangle\\}^{*}\cong\\{{\mathfrak{L}}_{X}/(b),L\\}$ the explicit bijection
described in Proposition 2.1, we prove the following which extends [11,
§12.5.1]:

######  Theorem 2.4.

The following diagram commutes,

$\textstyle{\pi_{1}\langle L\rangle\times\\{X,\langle
L\rangle\\}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\scriptstyle{\rho_{1}\times\gamma}$$\scriptstyle{\cong}$$\textstyle{\\{X,\langle
L\rangle\\}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\gamma}$$\textstyle{H_{0}(L)\times\\{{\mathfrak{L}}_{X}/(b),L\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\textstyle{\\{{\mathfrak{L}}_{X}/(b),L\\}.}$

###### Proof.

Note that, by definition, the following commutes:

$\textstyle{\pi_{1}\langle L\rangle\times\\{X,\langle
L\rangle\\}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\scriptstyle{\cong}$$\textstyle{\\{S^{1}\vee
X,\langle
L\rangle\\}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\widetilde{g}}^{*}}$$\textstyle{\\{X,\langle
L\rangle\\}^{*}.}$ (6)

By the naturality of the bijections in Proposition 2.1 the following also
commutes:

$\textstyle{\\{S^{1}\vee X,\langle
L\rangle\\}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\widetilde{g}}^{*}}$$\scriptstyle{\cong}$$\textstyle{\\{X,\langle
L\rangle\\}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\\{{\mathfrak{L}}_{S^{1}\vee
X}/(b),L\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{{\mathfrak{L}}_{\widetilde{g}}}^{*}}$$\textstyle{\\{{\mathfrak{L}}_{X}/(b),L\\}.}$
(7)

Next, as any connected simplicial set is weakly equivalent to a reduced one,
we lose no generality by assuming $X$ reduced with $b$ the only $0$-simplex.
Hence we may write,

${\mathfrak{L}}_{X}=({\widehat{\mathbb{L}}}(V\oplus\langle
b\rangle),d){\quad\text{with}\quad}V=V_{\geq 0}.$

Then, [11, Proposition 7.21] guarantees that $d$ may be chosen so that, for
the perturbed differential,

$({\mathfrak{L}}_{X},d_{b})=({\widehat{\mathbb{L}}}(V),d_{b})\,\widehat{\amalg}\,\,(\mathbb{L}(b),d_{b}).$

Here, $\,\widehat{\amalg}\,\,$ denotes the coproduct in $\operatorname{{\bf
cdgl}}$. On the other hand, ${\mathfrak{L}}_{[a,b]}={\mathfrak{L}}_{1}$ is the
cdgl $({\widehat{\mathbb{L}}}(a,b,x),d)$ in which $a$ and $b$ are MC elements,
and

$dx=\operatorname{{\rm
ad}}_{x}b+\sum_{n=0}^{\infty}\frac{B_{n}}{n!}\,\operatorname{{\rm
ad}}_{x}^{n}(b-a)$

where $B_{n}$ denotes the $n$th Bernoulli number. Thus, as the realization
functor preserves colimits,

${\mathfrak{L}}_{[a,b]\vee X}=({\widehat{\mathbb{L}}}(V\oplus\langle
a,b,x\rangle),d).$

We define,

$h\colon({\mathfrak{L}}_{X},d_{b})\longrightarrow({\mathfrak{L}}_{[a,b]\vee
X},d_{a}),\qquad h(b)=a,\qquad h(v)=e^{\operatorname{{\rm ad}}_{x}}(v),\quad
v\in V,$

and check that it is a cdgl morphism: obviously $h(d_{b}b)=d_{a}h(b)$ while,
applying [11, Proposition 4.24] and the fact that
$d_{b}(v)\in{\widehat{\mathbb{L}}}(V)$ for any $v\in V$, we get

$h(d_{b}v)=e^{\operatorname{{\rm ad}}_{x}}(d_{b}v)=d_{a}e^{\operatorname{{\rm
ad}}_{x}}(v)=d_{a}h(v).$

On the other hand,

${\mathfrak{L}}_{r}\colon({\widehat{\mathbb{L}}}(V\oplus\langle
a,b,x\rangle),d)\to({\widehat{\mathbb{L}}}(V\oplus\langle b\rangle),d)$

is the identity on $V\oplus\langle b\rangle$, and sends $a$ to $b$ and $x$ to
$0$. Therefore, as ${\mathfrak{L}}_{r}\circ h=\operatorname{{\rm
id}}_{{\mathfrak{L}}_{X}}$, ${\mathfrak{L}}_{g}$ is necessarily homotopic to
$h$ and thus, also up to homotopy,

${\mathfrak{L}}_{\widetilde{g}}\colon({\widehat{\mathbb{L}}}(V\oplus\langle
b\rangle),d)\longrightarrow({\widehat{\mathbb{L}}}(V\oplus\langle
b,x\rangle),d),\quad{\mathfrak{L}}_{\widetilde{g}}(b)=b,\quad{\mathfrak{L}}_{\widetilde{g}}(v)=e^{\operatorname{{\rm
ad}}_{x}}(v).$

Hence, the induced map on the quotients
${\mathfrak{L}}_{\widetilde{g}}\colon{\mathfrak{L}}_{X}/(b)\to{\mathfrak{L}}_{S^{1}\vee
X}/(b)$ is the morphism

$({\widehat{\mathbb{L}}}(V),d)\longrightarrow({\widehat{\mathbb{L}}}(V),d)\,\widehat{\amalg}\,\,(\mathbb{L}(x),0),\quad
v\mapsto e^{\operatorname{{\rm ad}}_{x}}(v),$

and therefore, the bottom row of diagram (7) becomes

$\\{(\mathbb{L}(x),0)\,\widehat{\amalg}\,\,({\widehat{\mathbb{L}}}(V),d),L\\}\stackrel{{\scriptstyle{{\mathfrak{L}}_{\widetilde{g}}}^{*}}}{{\longrightarrow}}\\{({\widehat{\mathbb{L}}}(V),d),L\\}.$

But
$\\{(\mathbb{L}(x),0)\,\widehat{\amalg}\,\,({\widehat{\mathbb{L}}}(V),d),L\\}\cong
H_{0}(L)\times\\{{\mathfrak{L}}_{X}/(b)\\}$ and thus, the following commutes,

$\textstyle{\\{{\mathfrak{L}}_{S^{1}\vee
X}/(b),L\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{{\mathfrak{L}}_{\widetilde{g}}}_{*}}$$\scriptstyle{\cong}$$\textstyle{\\{{\mathfrak{L}}_{X}/(b),L\\}}$$\textstyle{H_{0}(L)\times\\{{\mathfrak{L}}_{X}/(b)\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\scriptstyle\,\bullet\,}}$
(8)

Joining diagrams (6), (7) and (8) produces the diagram of the statement. ∎

As $\\{X,\langle L\rangle\\}\cong\\{X,\langle L\rangle\\}^{*}/\pi_{1}\langle
L\rangle$, we deduce:

######  Corollary 2.5.

If $L^{\prime}$ is a Lie model of $X$, $\\{X,\langle
L\rangle\\}\cong\\{L^{\prime},L\\}/H_{0}(L)$. $\square$

######  Example 2.6.

Let $L=(\widehat{\mathbb{L}}(u,v),0)$ with $|u|=|v|=0$, which is the minimal
model of $S^{1}\vee S^{1}$, and recall that the model of the circle is
${\mathfrak{L}}_{S^{1}}=(\widehat{\mathbb{L}}(b,x),d)$ with $|b|=-1$,
$db=-\frac{1}{2}[b,b]$ and $dx=[x,b]$. Denote by
$f,g\colon{\mathfrak{L}}_{S^{1}}\to L$ the morphisms defined by $f(x)=v$ and
$g(x)=e^{ad_{u}}(v)$. The induced morphisms
$\overline{f},\overline{g}\colon{\mathfrak{L}}_{S^{1}}/(b)=(\mathbb{L}(x),0)\to
L$ are clearly not homotopic but
$u{\scriptstyle\,\bullet\,}\overline{f}=\overline{g}$ which means that $f$ and
$g$ are homotopic. An explicit homotopy is given by
$\Psi\colon{\mathfrak{L}}_{S^{1}}\to L\widehat{\otimes}\land(t,dt)$,
$\psi(x)=e^{t\,\operatorname{{\rm ad}}_{u}}(v)$ and $\Psi(b)=-udt$. The
corresponding maps $S^{1}\to\mathbb{Q}_{\infty}(S^{1}\vee S^{1})$ are thus
homotopic but not pointed homotopic.

##  3 Complete subgroups of complete Lie algebras

In this section $M$ will always denote a complete Lie algebra, ungraded and
with no differential. We first briefly describe the exponential and logarithm
maps in this setting and refer to [15, §2.4] for details. Given $UM$ the
universal enveloping algebra of $M$ consider the ideal $I$ of $UM$ generated
by $M$. By completing with respect to the filtration given by $I^{0}=UM$,
$I^{1}=I$ and $I^{n}=I^{n-1}I$, $n\geq 2$, we get

$\widehat{UM}=\varprojlim_{n\geq
0}UM/I^{n}{\quad\text{and}\quad}\widehat{I}=\varprojlim_{n\geq 1}I/I^{n},$

together with the well known bijections (see for instance [29, Chapter 4]),
inverses of each other,

$\textstyle{\widehat{I}\,\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\scriptscriptstyle\cong}$$\scriptstyle{\exp}$$\textstyle{{\,1+\widehat{I}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\log}$

given by

$\exp(x)=e^{x}=\sum_{n\geq
0}\frac{x^{n}}{n!}{\quad\text{and}\quad}\log(1+x)=\sum_{n\geq
1}(-1)^{n+1}\frac{x^{n}}{n}.$

Now, the diagonal $\Delta\colon UM\to UM$ clearly sends $I^{n}$ to
$J^{n}=\sum_{i+j=n}I^{i}\otimes I^{j}$ and thus it induces an algebra morphism
denoted in the same way

$\Delta\colon\widehat{UM}\to\widehat{UM}\widehat{\otimes}\widehat{UM}=\varprojlim_{n}(UM\otimes
UM)/J^{n}.$

The primitive and grouplike elements of $UM$ are respectively defined by

$P=\\{x\in\widehat{I},\,\,\Delta x=x\otimes 1+1\otimes
x\\}\,\,\text{and}\,\,G=\\{1+y\in
1+\widehat{I},\,\,\Delta(1+y)=(1+y)\otimes(1+y)\\}.$

Then, $\exp$ and $\log$ restrict to isomorphisms between $P$ and $G$. However,
see [15, Proposition 2.3], the inclusion $M\to P$ is in fact an isomorphism
and thus we have:

$\textstyle{M\,\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\scriptscriptstyle\cong}$$\scriptstyle{\exp}$$\textstyle{{\,G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\log}$

Now, the multiplication on $\widehat{UM}$ restricts to a product $\cdot$ on
$G$ for which it becomes a group. This in turn induces a group structure on
$M$ via the classical Baker-Campbell-Hausdorff product,

$x*y=\log(e^{x}\cdot e^{y}),\quad x,y\in M.$

######  Remark 3.1.

Consider now the particular case in which $M\subset{\rm Der}_{0}L$ is a
complete sub Lie algebra of $0$-derivations of a given cdgl $L$. In this
particular instance the injection $M\hookrightarrow\operatorname{\text{\rm
Hom}}(L,L)$ extends to an algebra morphism,
$\xi\colon\widehat{UM}\to\operatorname{\text{\rm Hom}}(L,L)$ where the product
on the endomorphisms of $L$ is given by composition. By restricting $\xi$ to
the grouplike elements we inject $G$ into the linear automorphisms of $L$.
Then, given $\theta\in M$, we abuse of notation and write
$e^{\theta}=\xi\circ\exp(\theta)$, that is,

$e^{\theta}=\sum_{n\geq 0}\frac{\theta^{n}}{n!}$

where the product is now composition. This is then a well defined linear
automorphism of $L$. It is easy to check that it commutes with the Lie bracket
in general, and with the differential whenever $\theta$ is a cycle.

Coming back to the general setting, if $E$ is a complete sub Lie algebra of
the complete Lie algebra $M$, then $\exp(E)$ is a subgroup of $G$. The purpose
of this section is to show that the converse also holds.

######  Theorem 3.2.

If $R$ is a complete subgroup of $(M,*)$, closed under multiplication by
scalars, then it is a complete sub Lie algebra of $L$.

Given $R$ a subgroup of $(M,*)$ denote by

$\Gamma^{1}=R\supset\dots\supset\Gamma^{i}\supset\Gamma^{i+1}\supset\dots$

its descending central series in which $\Gamma^{i+1}=(\Gamma^{i},\Gamma^{1})$
denotes the commutator subgroup. Accordingly, given $x,y\in R$ we denote,

$(x,y)=x\ast y\ast x^{-1}\ast y^{-1}=x\ast y\ast(-x)\ast(-y).$

A usual, the group $R$ is said to be complete if the natural map
$R\stackrel{{\scriptstyle\cong}}{{\to}}\varprojlim_{n}R/\Gamma^{n}$ is an
isomorphism.

######  Lemma 3.3.

If $(R,\ast)$ is nilpotent and closed under multiplication by scalars, then

$(x,y)-[x,y]\in\Gamma^{3}{\quad\text{for all}\quad}x,y\in R.$

###### Proof.

We first show that, given $n\geq 3$, there are elements $r_{i}\in\Gamma^{i}$,
for $3\leq i\leq n$, such that

$\bigl{(}(x,y)-[x,y]\bigr{)}\ast r_{3}\ast r_{4}\ast\dots\ast r_{n}\in
L^{n+1}.$

For $n=3$ take

$r_{3}=-\frac{1}{2}\Bigl{(}\bigl{(}x,(x,y)\bigr{)}\ast\bigl{(}y,(x,y)\bigr{)}\Bigr{)}\in\Gamma^{3}$

and a straightforward computation shows that in fact
$\bigl{(}(x,y)-[x,y]\bigr{)}*r_{3}\in L^{4}$. By induction, assume we have
constructed $r_{3},\dots,r_{n-1}$ as above and write

$\bigl{(}(x,y)-[x,y]\bigr{)}\ast r_{3}\ast r_{4}\ast\dots\ast
r_{n-1}=\sum_{k}\lambda_{k}c_{k}+\xi$

where $\xi\in L^{n+1}$, $\lambda_{k}\in\mathbb{Q}$ and for each $k$,

$c_{k}=\bigl{[}x_{1},[x_{2},[\dots,[x_{n-1},x_{n}]\bigr{]}\dots\bigr{]}$

where each $x_{i}$ is equal to $x$ or $y$. Then, consider

$\tilde{c}_{k}=\bigl{(}x_{1},(x_{2},(\dots,(x_{n-1},x_{n})\bigr{)}\dots\bigr{)}\in\Gamma^{n}$

and define

$r_{n}=-\prod_{k}\lambda_{k}\tilde{c}_{k}\in\Gamma^{n}.$

Clearly

$\bigl{(}(x,y)-[x,y]\bigr{)}\ast r_{3}\ast r_{4}\ast\dots\ast r_{n}\in
L^{n+1}$

Now, Since $R$ is nilpotent, $\Gamma^{q+1}=0$ for some $q$ and thus, $r_{n}=0$
for $n>q$. Hence,

$\bigl{(}(x,y)-[x,y]\bigr{)}\ast r_{3}\ast r_{4}\ast\dots\ast r_{q}\in
L^{n+1}{\quad\text{for all}\quad}n\geq 3.$

Thus, the above expression vanishes and

$(x,y)-[x,y]=(r_{3}\ast r_{4}\ast\dots\ast r_{q})^{-1}\in\Gamma^{3}$

∎

###### Proof of Theorem 3.2.

We assume first that $(R,*)$ is nilpotent of index $q$ so that so that
$\Gamma^{q+1}=0$. An iterated application of Lemma 3.3 implies that, for each
$n\geq 2$ and $x_{1},\dots,x_{n}\in R$,

$\bigl{(}x_{1},(x_{2},(\dots,(x_{n-1},x_{n})\bigr{)}\dots\bigr{)}-\bigl{[}x_{1},[x_{2},[\dots,[x_{n-1},x_{n}]\bigr{]}\dots\bigr{]}\in\Gamma^{n+1}.$
(9)

In particular, for $n=q$,

$\bigl{(}x_{1},(x_{2},(\dots,(x_{q-1},x_{q})\bigr{)}\dots\bigr{)}=\bigl{[}x_{1},[x_{2},[\dots,[x_{q-1},x_{q}]\bigr{]}\dots\bigr{]}.$
(10)

To conclude, we prove that for any $1\leq p\leq q$, any sum $\sum_{i}y_{i}$,
in which each $y_{i}$ is a bracket of length $p$ of elements of $R$, stays in
$R$. We proceed by descending induction on $p$. For $p=q$ and in view of (10),
each $y_{i}\in\Gamma^{q}$ which is abelian. Thus,

$\sum_{i}y_{i}=\prod_{i}y_{i}\in R.$

Assume the assertion holds if each $y_{i}\in L^{>p}$ and assume $y_{i}\in
L^{p}$ for each $i$. By formula (9) there are elements $z_{i}\in\Gamma^{p}$ so
that $y_{i}-z_{i}\in\Gamma^{p+1}$. Hence,

$a_{p}=\sum_{i}y_{i}-\prod_{i}z_{i}\in\Gamma^{p+1}.$

On the other hand this is a sum of elements, each of which is a bracket of
length at least $p+1$ of elements of $R$. Thus, by induction hypothesis,
$a_{p}\in R$. We then consider

$a_{p+1}=\sum_{i}y_{i}-\prod_{i}z_{i}*a_{p}^{-1}\in\Gamma^{p+2}$

which again is an element of $R$. In this way we construct elements
$a_{p},\dots,a_{q}\in R$ so that, for $a=a_{q}*\dots*a_{p}$,

$\sum_{i}y_{i}-\prod_{i}z_{i}*a^{-1}\in\Gamma^{q+1}=0$

Thus,

$\sum_{i}y_{i}=\prod_{i}z_{i}*a^{-1}\in R$

and the assertion holds for $R$ nilpotent.

Finally, if $R$ is complete, by the above it is the limit
$\varprojlim_{n}R/\Gamma^{n}$ of Lie subalgebras and the theorem follows. ∎

##  4 Evaluation fibrations

Given $X$ and $Y$ simplicial sets, the evaluation fibration is given by

$\operatorname{{\rm map}}^{*}(X,Y)\longrightarrow\operatorname{{\rm
map}}(X,Y)\stackrel{{\scriptstyle\operatorname{{\rm
ev}}}}{{\longrightarrow}}Y$

where $\operatorname{{\rm ev}}$ denotes the evaluation at the base point. We
will need a particular version of certain results, some of them already known
under a different perspective, describing cdgl models of evaluation
fibrations.

Given any cdgc $C$ and any dgl $L$, consider the dgl structure on
$\operatorname{\text{\rm Hom}}(C,L)$ with the usual differential $Df=d\circ
f-(-1)^{|f|}f\circ d$, and the convolution Lie bracket given by

$[f,g]=[\,,\,]\circ(f\otimes g)\circ\Delta.$

that is,

$[f,g](c)=\sum_{i}(-1)^{|g|\,|c_{i}|}[f(c_{i}),g(c_{i}^{\prime})],\quad\text{with}\quad\Delta
c=\sum_{i}c_{i}\otimes c_{i}^{\prime}\,.$

######  Remark 4.1.

If $L$ is a cdgl then $\operatorname{\text{\rm Hom}}(C,L)$ is also complete.
Indeed, let $\\{F^{n}\\}_{n\geq 1}$ be a filtration for which
$L\cong\varprojlim_{n}L/F^{n}$ and consider the filtration $\\{G^{n}\\}_{n\geq
1}$ of $\operatorname{\text{\rm Hom}}(C,L)$ given by
$G^{n}=\operatorname{\text{\rm Hom}}(C,F^{n})$. Then,

$\operatorname{\text{\rm Hom}}(C,L)\cong\varprojlim_{n}\operatorname{\text{\rm
Hom}}(C,L/F^{n})\cong\varprojlim_{n}\operatorname{\text{\rm Hom}}(C,L)/G^{n}.$

The same applies to note that $\operatorname{\text{\rm Hom}}(\overline{C},L)$
is also a sub cdgl of $\operatorname{\text{\rm Hom}}(C,L)$ and there is a cdgl
isomorphism,

$\operatorname{\text{\rm Hom}}(C,L)\cong\operatorname{\text{\rm
Hom}}(\overline{C},L)\,\widetilde{\times}\,L,$ (11)

where both $L$ and $\operatorname{\text{\rm Hom}}(\overline{C},L)$ are sub
cdgl’s and $[x,f]=\operatorname{{\rm ad}}_{x}\circ f$ for $x\in L$ and
$f\in\operatorname{\text{\rm Hom}}(\overline{C},L)$. In particular, this
twisted product is naturally a complete dgl.

Finally observe that the homotopy type of the cdgl $\operatorname{\text{\rm
Hom}}(C,L)$ is an invariant of the homotopy type of $C$. Indeed, if
$\varphi\colon C\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}C^{\prime}$
is a cdgc quasi-isomorphism, the cdgl quasi-isomorphism
$\varphi_{*}\colon\operatorname{\text{\rm
Hom}}(C^{\prime},L)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\operatorname{\text{\rm
Hom}}(C,L)$ trivially induces quasi-isomorphisms $\operatorname{\text{\rm
Hom}}(C^{\prime},L)/{G^{\prime}}^{n}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\operatorname{\text{\rm
Hom}}(C,L)/{G}^{n}$. Thus, by the generalized Goldman-Millson Theorem, see §1,
$\varphi_{*}$ is a weak equivalence of cdgl’s.

For the remaining of this section we fix a connected cdgl $L$ and a connected
simplicial set $X$ of finite type whose minimal model is denoted by
$L^{\prime}$. Recall from §1 the notation $\\{{L^{\prime}}^{n}\\}_{\geq 1}$
for the filtration associated to $L^{\prime}$ and that ${\mathscr{C}}$ denotes
the chain coalgebra functor defined in (5). We begin with a reformulation of
[11, Proposition 12.25]:

######  Theorem 4.2.

The realization of the short exact cdgl sequence

$0\to\varinjlim_{n}\operatorname{\text{\rm
Hom}}(\overline{\mathscr{C}}(L^{\prime}/{L^{\prime}}^{n}),L)\longrightarrow\varinjlim_{n}\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L^{\prime}/{L^{\prime}}^{n}),L)\stackrel{{\scriptstyle\operatorname{{\rm
ev}}}}{{\longrightarrow}}L\to 0$

has the homotopy type of the fibration sequence

$\operatorname{{\rm map}}^{*}(X,\langle
L\rangle)\longrightarrow\operatorname{{\rm map}}(X,\langle
L\rangle)\stackrel{{\scriptstyle\operatorname{{\rm
ev}}}}{{\longrightarrow}}\langle L\rangle.$

In other words, there is a commutative diagram

$\textstyle{\operatorname{{\rm map}}^{*}(X,\langle
L\rangle)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{{\rm
map}}(X,\langle
L\rangle)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{{\rm
ev}}}$$\textstyle{\langle
L\rangle}$$\textstyle{\langle\varinjlim_{n}\operatorname{\text{\rm
Hom}}(\overline{\mathscr{C}}(L^{\prime}/{L^{\prime}}^{n}),L)\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\langle\varinjlim_{n}\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L^{\prime}/{L^{\prime}}^{n}),L)\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\langle
L\rangle,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$

where the vertical maps are homotopy equivalences.

Here, the inverse limit is taken in $\operatorname{{\bf cdgl}}$ and
$\varinjlim_{n}\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L^{\prime}/{L^{\prime}}^{n}),L)\stackrel{{\scriptstyle\operatorname{{\rm
ev}}}}{{\to}}L$ is induced by evaluating at $1\in\mathbb{Q}$ every morphism of
each $\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L^{\prime}/{L^{\prime}}^{n}),L)$.

###### Proof.

Recall from [11, Theorem 12.18] or [1, Theorem 6.6] that for any cdga model A
of any simplicial set $X$ and any cdgl $L$ there is a weak homotopy
equivalence

$\langle A\widehat{\otimes}L\rangle\simeq\operatorname{{\rm map}}(X,\langle
L\rangle)$

natural on any possible choice. Moreover, see [11, Proposition 12.25], there
is a commutative diagram of simplicial sets

$\textstyle{\operatorname{{\rm map}}^{*}(X,\langle
L\rangle)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{{\rm
map}}(X,\langle
L\rangle)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{{\rm
ev}}}$$\textstyle{\langle L\rangle}$$\textstyle{\langle
A^{+}\widehat{\otimes}L\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\langle
A\widehat{\otimes}L\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\langle
L\rangle,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$

where the vertical maps are homotopy equivalences and the bottom row is just
the realization of the cdgl short exact sequence $A^{+}\widehat{\otimes}L\to
A\widehat{\otimes}L\to L$.

On the other hand, if $A$ is a connected cdga of finite type, the cdgl
$A\widehat{\otimes}L$ (respec. $A^{+}\widehat{\otimes}L$) is naturally
isomorphic to $\operatorname{\text{\rm Hom}}(A^{\sharp},L)$ (respec.
$\operatorname{\text{\rm Hom}}(\overline{A^{\sharp}},L)$). Indeed, the
following chain of linear isomorphisms,

$A\widehat{\otimes}L=\varprojlim_{n}(A\otimes
L/F^{n})\cong\varprojlim_{n}\operatorname{\text{\rm
Hom}}(A^{\sharp},L/F^{n})\cong\operatorname{\text{\rm
Hom}}(A^{\sharp},\varprojlim_{n}L/F^{n})\cong\operatorname{\text{\rm
Hom}}(A^{\sharp},L),$

preserves differentials and Lie brackets.

Now, we choose

$A=\varinjlim_{n}{\mathscr{C}}^{*}(L^{\prime}/{L^{\prime}}^{n})$

which, by [11, Theorem 10.8], is a Sullivan model of $X$. Observe that since
each $L^{\prime}/{L^{\prime}}^{n}$ is of finite type, so is
${\mathscr{C}}^{*}(L^{\prime}/{L^{\prime}}^{n})$. Then, as complete tensor
products commute with inductive limits, the theorem follows from the
isomorphism

$A\widehat{\otimes}L\cong\varinjlim_{n}{\mathscr{C}}^{*}(L^{\prime}/{L^{\prime}}^{n})\widehat{\otimes}L\cong\varinjlim_{n}\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L^{\prime}/{L^{\prime}}^{n}),L)$

and its restriction to

$A^{+}\widehat{\otimes}L\cong\varinjlim_{n}{\mathscr{C}}^{+}(L^{\prime}/{L^{\prime}}^{n})\widehat{\otimes}L\cong\varinjlim_{n}\operatorname{\text{\rm
Hom}}(\overline{\mathscr{C}}(L^{\prime}/L^{\prime n}),L)$

where ${\mathscr{C}}^{+}$ denotes the augmentation ideal of
${\mathscr{C}}^{*}$.

∎

If $X$ is a nilpotent simplicial set of finite type the above result is
drastically simplified. For it, the following observation is important.

######  Remark 4.3.

Although outbound of the classical Quillen theory for simply connected spaces,
Neisendorfer defined in [24, §7] a “Lie model” of a nilpotent, finite type
complex $X$ as any dgl (not complete) quasi-isomorphic to
${\mathscr{L}}(A^{\sharp})$. Here, ${\mathscr{L}}$ is the functor in defined
in (5) and, as in the proof of Theorem 4.2, $A$ stands for a finite type
Sullivan model of $X$. On the other hand, if $L^{\prime}$ is the minimal
(cdgl) model of $X$, then $L^{\prime}$ and ${\mathscr{L}}(A^{\sharp})$ are
quasi-isomorphic, see [11, Theorem 10.2]. Moreover, as stated in §1, $\langle
L^{\prime}\rangle\simeq X_{\mathbb{Q}}$ which is also of the homotopy type of
the cdga realization of ${\mathscr{C}}^{*}{\mathscr{L}}(A^{\sharp})$. For a
compendium of the geometrical realization in the commutative side see for
instance [15, §1.6].

In what follows $\operatorname{\text{\rm
Hom}}(\overline{\mathscr{C}}(L^{\prime}),L)\widetilde{\times}L$ will always
denote the twisted product arising from the isomorphism (11) in Remark 4.1.
The following extends [5, Corollary 15] the following result we consider, as
in (11), the twisted product $\operatorname{\text{\rm
Hom}}(\overline{\mathscr{C}}(L^{\prime}),L)\widetilde{\times}L$.

######  Corollary 4.4.

Let $L^{\prime}$ be a Lie model of the nilpotent simplicial set $X$ of finite
type. Then, the realization of the short exact cdgl sequence

$0\to\operatorname{\text{\rm
Hom}}(\overline{\mathscr{C}}(L^{\prime}),L)\longrightarrow\operatorname{\text{\rm
Hom}}(\overline{\mathscr{C}}(L^{\prime}),L)\widetilde{\times}L{\longrightarrow}L\to
0$

has the homotopy type of the fibration sequence

$\operatorname{{\rm map}}^{*}(X,\langle
L\rangle)\longrightarrow\operatorname{{\rm map}}(X,\langle
L\rangle)\stackrel{{\scriptstyle\operatorname{{\rm
ev}}}}{{\longrightarrow}}\langle L\rangle.$

###### Proof.

As before, this fibration sequence has the homotopy type of the realization of
the cdgl sequence $A^{+}\widehat{\otimes}L\to A\widehat{\otimes}L\to L$ where
$A$ is the Sullivan minimal model of $X$. As $A$ is of finite type this
sequence becomes

$0\to\operatorname{\text{\rm
Hom}}(A_{+}^{\sharp},L)\longrightarrow\operatorname{\text{\rm
Hom}}(A^{\sharp}_{+},L)\widetilde{\times}L{\longrightarrow}L\to 0.$

Now, see the properties of the Quillen pair (5) in §1, since $A^{\sharp}$ is a
connected cdgc, it is quasi-isomorphic to
${\mathscr{C}}{\mathscr{L}}(A^{\sharp})$. Observe, that
${\mathscr{L}}(A^{\sharp})$ is the Neisendorfer model of $X$ which by Remark
4.3, is quasi-isomorphic to any given cdgl model of X. Hence, since
${\mathscr{C}}$ preserves quasi-isomorphisms,

$A^{\sharp}\simeq{\mathscr{C}}{\mathscr{L}}(A^{\sharp})\simeq{\mathscr{C}}(L^{\prime}).$

This implies, in view of the last observation in Remark 4.1, that the above
cdgl sequence is homotopy equivalent to

$0\to\operatorname{\text{\rm
Hom}}(\overline{\mathscr{C}}(L^{\prime}),L)\longrightarrow\operatorname{\text{\rm
Hom}}(\overline{\mathscr{C}}(L^{\prime}),L)\widetilde{\times}L{\longrightarrow}L\to
0$

and the corollary follows. ∎

We now consider the restriction of the evaluation fibration to a given path
component of $\operatorname{{\rm map}}(X,\langle L\rangle)$,

$\operatorname{{\rm map}}_{f}^{*}(X,\langle
L\rangle)\longrightarrow\operatorname{{\rm map}}_{f}(X,\langle
L\rangle)\stackrel{{\scriptstyle\operatorname{{\rm
ev}}}}{{\longrightarrow}}L,$

determined by a map $f\colon X\to\langle L\rangle$. Note that the fibre is the
non-connected complex of pointed homotopy classes of pointed maps freely
homotopic to $f$.

######  Remark 4.5.

Given a Lie model $L^{\prime}$ of a simplicial set $X$, the set
$\pi_{0}\operatorname{{\rm map}}(X,\langle L\rangle)$ of path components of
$\operatorname{{\rm map}}(X,\langle L\rangle)$ is the set of free homotopy
classes $\\{X,\langle L\rangle\\}$ which, by Corollary 2.5, is in bijective
correspondence with the set $\\{L^{\prime},L\\}/H_{0}(L)$. In the same way, by
Corollary 2.2, the set $\pi_{0}\operatorname{{\rm map}}^{*}(X,\langle
L\rangle)$ of pointed homotopy classes $\\{X,\langle L\rangle\\}^{*}$ is
identified with $\\{L^{\prime},L\\}$.

Now, if $X$ is nilpotent, by Corollary 4.4,

$\langle\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L^{\prime}),L)\rangle\simeq\operatorname{{\rm
map}}(X,\langle L\rangle){\quad\text{and}\quad}\langle\operatorname{\text{\rm
Hom}}(\overline{\mathscr{C}}(L^{\prime}),L)\rangle\simeq\operatorname{{\rm
map}}^{*}(X,\langle L\rangle).$

As a result,

$\pi_{0}\operatorname{{\rm map}}(X,\langle
L\rangle)\cong\\{L^{\prime},L\\}/H_{0}(L)\cong\widetilde{\operatorname{{\rm
MC}}}\bigl{(}\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L^{\prime}),L)\bigr{)}$

while, on the other hand,

$\pi_{0}\operatorname{{\rm map}}^{*}(X,\langle
L\rangle)\cong\\{L^{\prime},L\\}\cong\widetilde{\operatorname{{\rm
MC}}}\bigl{(}\operatorname{\text{\rm
Hom}}(\overline{\mathscr{C}}(L^{\prime}),L)\bigr{)}.$

These correspondences can be explicitly described:

Let $q\colon\overline{\mathscr{C}}(L^{\prime})\to L^{\prime}$ be the degree
$-1$ linear map defined by $q(sx)=x$ if $x\in L^{\prime}$ and $q(\land^{\geq
2}sL^{\prime})=0$. Denote in the same way the only possible extension of $q$
to ${\mathscr{C}}(L^{\prime})$ by $q(1)=0$. Now choose a map $f\colon
X\to\langle L\rangle$ which can be assumed to be pointed since $L$ is
connected. Then, the pointed homotopy class $[f]^{*}$ is identified to the
homotopy class of a cdgl morphism $\varphi\colon L^{\prime}\to L$ which
corresponds to the gauge class of the MC element

$\overline{\varphi}=\varphi\circ
q\colon\overline{\mathscr{C}}(L^{\prime})\longrightarrow L,$ (12)

On the other hand, the free homotopy class $[f]$ corresponds to the class of
$\varphi$ in $\\{L^{\prime},L\\}/H_{0}(L)$ which is identified to the gauge
class of the MC element

$\overline{\varphi}=\varphi\circ
q\colon{\mathscr{C}}(L^{\prime})\longrightarrow L,$ (13)

With this notation, consider the component $\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L^{\prime}),L)^{\overline{\varphi}}\to L$ of
$\operatorname{\text{\rm Hom}}({\mathscr{C}}(L^{\prime}),L)\to L$ at the MC
element $\overline{\varphi}$. Recall from §1 that $\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L^{\prime}),L)^{\overline{\varphi}}$ is, by definition,
the connected cover of $(\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L^{\prime}),L),D_{\overline{\varphi}})$ where
$D_{\overline{\varphi}}$ is the differential $D$ in $\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L^{\prime}),L)$ perturbed by $\overline{\varphi}$. Then we
have:

######  Proposition 4.6.

The realization of the cdgl morphism

$\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L^{\prime}),L)^{\overline{\varphi}}\longrightarrow L$

has the homotopy type of the fibration

$\operatorname{{\rm map}}_{f}(X,\langle
L\rangle)\stackrel{{\scriptstyle\operatorname{{\rm
ev}}}}{{\longrightarrow}}L.$

###### Proof.

Recall that $\langle\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L^{\prime}),L)^{\overline{\varphi}}\rangle\simeq\langle\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L^{\prime}),L)\rangle^{\overline{\varphi}}$. To finish,
note that in view of Corollary 4.4 and Remark 4.5,
$\langle\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L^{\prime}),L)\rangle^{\overline{\varphi}}\to\langle
L\rangle$ is of the homotopy type of $\operatorname{{\rm map}}_{f}(X,\langle
L\rangle)\stackrel{{\scriptstyle\operatorname{{\rm
ev}}}}{{\longrightarrow}}L$. ∎

######  Remark 4.7.

(1) Observe that $\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L^{\prime}),L)^{\overline{\varphi}}\to L$ is not in
general a surjective morphism, i.e., it is not a cdgl fibration. Indeed, see
(11), in

$(\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L^{\prime}),L),D_{\overline{\varphi}})\cong(\operatorname{\text{\rm
Hom}}(\overline{\mathscr{C}}(L^{\prime}),L)\widetilde{\times}L,D_{\overline{\varphi}}),$

$D_{\overline{\varphi}}(x)=dx+[\overline{\varphi},x]=dx-(-1)^{|x|}\operatorname{{\rm
ad}}_{x}\circ\overline{\varphi}$ for any $x\in L$. Therefore, elements of
$L_{0}$ are not, in general, $D_{\overline{\varphi}}$-cycles and thus, they do
not belong to $\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L^{\prime}),L)^{\overline{\varphi}}$.

(2) Note also that, in view of Corollary 4.4 and Remark 4.5, the fibre of the
map $\langle\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L^{\prime}),L)\rangle^{\overline{\varphi}}\to\langle
L\rangle$ is the non-connected complex

${\amalg}_{[\psi]}\,\langle\operatorname{\text{\rm
Hom}}(\overline{\mathscr{C}}(L^{\prime}),L)\rangle^{\overline{\psi}}$

where: $[\psi]$ runs through homotopy classes of morphisms $\psi\colon
L^{\prime}\to L$ such that, when considered $\overline{\psi}$ in
$\operatorname{\text{\rm Hom}}({\mathscr{C}}(L),L)$, this is a
${\operatorname{{\rm MC}}}$ element gauge related to $\overline{\varphi}$. A
short computation shows that this amounts to the existence of $x\in L_{0}$
such that $e^{\operatorname{{\rm ad}}_{x}}\circ\varphi=\psi$. In other terms,
and with the notation in Definition 2.3,
$[\psi]=[x]{\scriptstyle\,\bullet\,}[\varphi]$ with $[x]\in H_{0}(L)$. Each of
these components is of the homotopy type of the corresponding component of
$\operatorname{{\rm map}}^{*}_{f}(X,\langle L\rangle)$.

######  Remark 4.8.

In all of the above, if $L$ is a Lie model of a connected simplicial set $Y$
of finite type, then the fibrations realized in Theorems 4.2, Corollary 4.4
and Proposition 4.6 are, respectively, the fibration

$\operatorname{{\rm
map}}^{*}(X,\mathbb{Q}_{\infty}Y)\longrightarrow\operatorname{{\rm
map}}(X,\mathbb{Q}_{\infty}Y)\stackrel{{\scriptstyle\operatorname{{\rm
ev}}}}{{\longrightarrow}}\mathbb{Q}_{\infty}Y$

and each of its components.

##  5 Derivations models of evaluation fibrations

The results in the past section can be expressed in terms of derivations. To
avoid excessive technicalities we assume $X$ nilpotent in what follows. Again
$L^{\prime}$ denotes the minimal model of $X$ and $L$ is a connected cdgl.
Given a cdgl morphism $\varphi\colon L^{\prime}\to L$, consider the dgl
morphism,

$\widetilde{\varphi}=\varphi\alpha\colon{\mathscr{L}}{\mathscr{C}}(L^{\prime})\longrightarrow
L,$

with $\alpha$ the adjunction map, see §1. Next, endow

${s^{-1}}{\rm
Der}_{\widetilde{\varphi}}({\mathscr{L}}{\mathscr{C}}(L^{\prime}),L)$

with the desuspended differential,

$D({s^{-1}}\theta)=-{s^{-1}}[d,\theta]=-{s^{-1}}(d\circ\theta-(-1)^{|\theta|}\theta\circ
d),$

and a Lie bracket,

$[s^{-1}\gamma,{s^{-1}}\eta]={s^{-1}}\theta,$

where $\theta$ is first defined on
${s^{-1}}\overline{{\mathscr{C}}}(L^{\prime})$ by

$-[\,\,,\,]\circ(\gamma{s^{-1}}\otimes\eta{s^{-1}})\circ\overline{\Delta}\circ
s.$

and then extended to ${\mathscr{L}}{\mathscr{C}}(L^{\prime})$ as a
$\widetilde{\varphi}$-derivation.

Consider the twisted product

$({s^{-1}}{\rm
Der}_{\widetilde{\varphi}}({\mathscr{L}}{\mathscr{C}}(L^{\prime}),L)\widetilde{\times}L,D)$

where ${s^{-1}}{\rm
Der}_{\widetilde{\varphi}}({\mathscr{L}}{\mathscr{C}}(L^{\prime}),L)$ is a sub
dgl and

$[x,{s^{-1}}\theta]=(-1)^{|x|}{s^{-1}}(\operatorname{{\rm
ad}}_{x}\circ\,\theta),\qquad Dx=dx-{s^{-1}}(\operatorname{{\rm
ad}}_{x}\circ\,\overline{\varphi}\circ s),\quad x\in L,$

Here, $\operatorname{{\rm ad}}_{x}\circ\,\theta$ and $\operatorname{{\rm
ad}}_{x}\circ\,{\overline{\varphi}}\circ s$ denote the
$\widetilde{\varphi}$-derivations which are these compositions on
${s^{-1}}\overline{\mathscr{C}}(L^{\prime})$, and $d$ is the differential on
$L$. Then, Corollary 4.4 translates to the following which extends [6, Theorem
3]:

######  Theorem 5.1.

For any cdgl morphism $\varphi\colon L^{\prime}\to L$,

$0\to{s^{-1}}{\rm
Der}_{\widetilde{\varphi}}({\mathscr{L}}{\mathscr{C}}(L^{\prime}),L)\longrightarrow{s^{-1}}{\rm
Der}_{\widetilde{\varphi}}({\mathscr{L}}{\mathscr{C}}(L^{\prime}),L)\,\widetilde{\times}\,L\longrightarrow
L\to 0$ (14)

is a cdgl short exact sequence whose realization has the homotopy type of the
fibration sequence

$\operatorname{{\rm map}}^{*}(X,\langle
L\rangle)\longrightarrow\operatorname{{\rm map}}(X,\langle
L\rangle)\stackrel{{\scriptstyle\operatorname{{\rm
ev}}}}{{\longrightarrow}}\langle L\rangle.$ (15)

###### Proof.

As perturbing a cdgl does not interfere with its realization, see (4),
Corollary 4.4 amounts to say that (15) is homotopy equivalent to the
realization of the cdgl sequence,

$0\to(\operatorname{\text{\rm
Hom}}(\overline{\mathscr{C}}(L^{\prime}),L),D_{\overline{\varphi}})\longrightarrow(\operatorname{\text{\rm
Hom}}(\overline{\mathscr{C}}(L^{\prime}),L)\widetilde{\times}L,D_{\overline{\varphi}})\stackrel{{\scriptstyle\operatorname{{\rm
ev}}}}{{\longrightarrow}}L\to 0.$

Recall that $D_{\overline{\varphi}}$ stands for the perturbed differential by
the MC elements $\overline{\varphi}$ as in (12) and (13) respectively. Next,
as in [6, Theorem 3] we show that

$\Gamma\colon{s^{-1}}{\rm
Der}_{\widetilde{\varphi}}({\mathscr{L}}{\mathscr{C}}(L^{\prime}),L)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}(\operatorname{\text{\rm
Hom}}(\overline{\mathscr{C}}(L^{\prime}),L),D_{\overline{\varphi}}),\quad\Gamma(s^{-1}\theta)(c)=(-1)^{|\theta|}\theta(s^{-1}c),$
(16)

is a dgl isomorphism, which then exhibit ${s^{-1}}{\rm
Der}_{\widetilde{\varphi}}({\mathscr{L}}{\mathscr{C}}(L^{\prime}),L)$ as a
complete dgl.

It is trivially a linear isomorphism and a straightforward computation shows
it commutes with the Lie brackets. It remains to prove that $\Gamma$ commutes
with differentials. For it, let $\theta\in{\rm
Der}_{\widetilde{\varphi}}({\mathscr{L}}{\mathscr{C}}(L^{\prime}),L)$ and
$c\in\overline{\mathscr{C}}(L^{\prime})$ with
$\overline{\Delta}(c)=\sum_{i}c_{i}\otimes c^{\prime}_{i}$.

On the one hand,

$\displaystyle\bigl{(}D_{\varphi}\Gamma(s^{-1}\theta)\bigr{)}(c)=$
$\displaystyle
d\bigl{(}\Gamma(s^{-1}\theta)(c)\bigr{)}-(-1)^{|\theta|+1}\Gamma(s^{-1}\theta)(dc)+[\overline{\varphi},\Gamma(s^{-1}\theta)](c)$
$\displaystyle=$
$\displaystyle(-1)^{|\theta|}d\theta(s^{-1}c)+\theta(s^{-1}dc)+\sum_{i}(-1)^{|c_{i}|(|\theta|+1)+|\theta|}\sum_{i}[\overline{\varphi}(c_{i}),\theta(s^{-1}c^{\prime}_{i})],$

On the other hand,

$\displaystyle\Gamma\bigl{(}D(s^{-1}\theta)\bigr{)}(c)=$
$\displaystyle(-1)^{|\theta|}(d\circ\theta-(-1)^{|\theta|}\theta\circ
d)({s^{-1}}c)$ $\displaystyle=$
$\displaystyle(-1)^{|\theta|}d\theta(s^{-1}c)-\theta(d_{1}{s^{-1}}c)-\theta(d_{2}{s^{-1}}c)$
$\displaystyle=$
$\displaystyle(-1)^{|\theta|}d\theta(s^{-1}c)+\theta({s^{-1}}dc)-\theta\bigl{(}\frac{1}{2}\sum_{i}(-1)^{|c_{i}|}[s^{-1}c_{i},s^{-1}c^{\prime}_{i}]\bigr{)}$
$\displaystyle=$
$\displaystyle(-1)^{|\theta|}d\theta(s^{-1}c)+\theta({s^{-1}}dc)+\sum_{i}(-1)^{|c_{i}|+|\theta|(|c_{i}|+1)}[\overline{\varphi}(c_{i}),\theta(s^{-1}c^{\prime}_{i})],$

and both expressions coincide. In the last equality we use that
${\mathscr{C}}(L^{\prime})$ is cocommutative, $\theta$ is a
$\widetilde{\varphi}$ derivation and
$\widetilde{\varphi}({s^{-1}}c)=-\overline{\varphi}(c)$ for any
$c\in{\mathscr{C}}(L^{\prime})$.

To finish, note that the isomorphism $\Gamma$ trivially extends to

$(\operatorname{\text{\rm
Hom}}(\overline{\mathscr{C}}(L^{\prime}),L)\widetilde{\times}L,D_{\overline{\varphi}})\cong({s^{-1}}{\rm
Der}_{\widetilde{\varphi}}({\mathscr{L}}{\mathscr{C}}(L^{\prime}),L)\,\widetilde{\times}\,L,D).$
(17)

Note that, in view of the isomorphisms (16) and (17), together with Remark
4.1, the objects in (14) are cdgl’s and the maps are cdgl morphisms. ∎

As a consequence we can easily express each component of (15) in terms of
derivations. Again, let $\operatorname{{\rm map}}_{f}(X,\langle L\rangle)$
denote the component of $\operatorname{{\rm map}}(X,\langle L\rangle)$
containing the map $f\colon X\to\langle L\rangle$, which corresponds, via
Remark 4.5, to an element in $\\{L,L\\}/H_{0}(L)$ represented by the homotopy
class of a cdgl morphism $\varphi\colon L^{\prime}\to L$.

######  Corollary 5.2.

The realization of the cdgl morphism

$({s^{-1}}{\rm
Der}_{\widetilde{\varphi}}({\mathscr{L}}{\mathscr{C}}(L^{\prime}),L)\,\widetilde{\times}\,L,D)^{0}\longrightarrow
L$

has the homotopy type of the fibration

$\operatorname{{\rm map}}_{f}(X,\langle
L\rangle)\stackrel{{\scriptstyle\operatorname{{\rm
ev}}}}{{\longrightarrow}}\langle L\rangle.$

Recall from §1 that given $M$ any cdgl, $M^{0}$ stands for its connected
cover, or equivalently, its component at the MC element $0$.

###### Proof.

This is an immediate application of Proposition 4.6 and the isomorphism (17).
∎

We finish by describing, also in terms of derivations, the homotopy long exact
sequence of the evaluation fibration. For it, fix a pointed map $f\colon
X\to\langle L\rangle$ as the base point of both $\operatorname{{\rm
map}}_{f}^{*}(X\langle L\rangle)$ and $\operatorname{{\rm map}}_{f}(X,\langle
L\rangle)$

On the other hand, let $\varphi\colon L^{\prime}\to L$ the cdgl morphism whose
homotopy class in $\\{L^{\prime},L\\}$ identifies the pointed homotopy class
$[f]^{*}$. Recall again that, by Remark 4.5 and in the orbit set
$\\{L^{\prime},L\\}/H_{0}(L)$, this morphism correspond to free homotopy class
$[f]$. Consider the twisted chain complex

$({\rm Der}_{\varphi}(L^{\prime},L)\widetilde{\times}sL,D)$

which has ${\rm Der}_{\varphi}(L^{\prime},L)$ as subcomplex and

$Dsx=-sdx+ad_{x}\circ\varphi,\quad x\in L.$

Then, we prove the following which, in particular, recovers [11, Theorem
12.35]:

######  Theorem 5.3.

For the chosen basepoints, the homotopy long exact sequence of the fibration

$\operatorname{{\rm map}}^{*}_{f}(X,\langle
L\rangle)\longrightarrow\operatorname{{\rm map}}_{f}(X,\langle
L\rangle)\stackrel{{\scriptstyle\operatorname{{\rm
ev}}}}{{\longrightarrow}}\langle L\rangle$ (18)

is isomorphic to the homology long exact sequence of

$0\to{\rm Der}_{\varphi}(L^{\prime},L)^{0}\longrightarrow({\rm
Der}_{\varphi}(L^{\prime},L)\,\widetilde{\times}\,sL)^{0}\longrightarrow sL\to
0.$

In particular,

$\pi_{*}(\operatorname{{\rm map}}_{f}^{*}(X,\langle L\rangle),[f]^{*})\cong
H_{*}\bigl{(}{\rm
Der}_{\varphi}(L^{\prime},L)\bigr{)}\,\,\text{and}\,\,\pi_{*}\operatorname{{\rm
map}}_{f}(X,\langle L\rangle)\cong H_{*}({\rm
Der}_{\varphi}(L^{\prime},L)\widetilde{\times}sL).$

###### Proof.

On the one hand, since the adjunction map
$\alpha\colon{\mathscr{L}}{\mathscr{C}}(L^{\prime})\stackrel{{\scriptstyle\simeq}}{{\to}}L^{\prime}$
is a quasi isomorphism, the map of chain complexes,

$\alpha^{*}\colon{\rm
Der}_{\varphi}(L^{\prime},L)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}{\rm
Der}_{\widetilde{\varphi}}({\mathscr{L}}{\mathscr{C}}(L^{\prime}),L),$ (19)

is also a quasi-isomorphism, see for instance [6, Lemma 3.4]. Extend it to
another quasi-isomorphism

$\alpha^{*}\times\operatorname{{\rm id}}_{sL}\colon({\rm
Der}_{\varphi}(L^{\prime},L)\widetilde{\times}sL,D)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}({\rm
Der}_{\widetilde{\varphi}}({\mathscr{L}}{\mathscr{C}}(L^{\prime}),L)\widetilde{\times}sL,D^{\prime})$
(20)

by setting

$D^{\prime}sx=-sdx+\theta,\quad x\in L,$

where $\theta$ is the $\widetilde{\varphi}$-derivation which extends the map
$\operatorname{{\rm ad}}_{x}\circ\overline{\varphi}\circ s\colon
s^{-1}{\mathscr{C}}(L^{\prime})\to L$. In particular, $\theta$ coincides with
$\operatorname{{\rm ad}}_{x}\circ\varphi$ on $L^{\prime}\cong
s^{-1}sL^{\prime}$. This shows that (20) is indeed a chain map and thus, a
quasi-isomorphism. From (19) and (20) we obtain a commutative diagram of chain
complexes where the top sequence is the suspension of (14):

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Der}_{\widetilde{\varphi}}({\mathscr{L}}{\mathscr{C}}(L^{\prime}),L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({\rm
Der}_{\widetilde{\varphi}}({\mathscr{L}}{\mathscr{C}}(L^{\prime}),L)\widetilde{\times}sL,D^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{sL\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Der}_{\varphi}(L^{\prime},L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{({\rm
Der}_{\varphi}(L^{\prime},L)\widetilde{\times}sL,D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{sL\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

Thus, by Corollary 5.2, the homology long exact sequence of the component at
$0$ of the top row is precisely the homotopy long exact sequence of (18). For
it note that, by Remark 4.7(2), the component of $\operatorname{{\rm
map}}_{f}^{*}(X,\langle L\rangle)$ containing $[f]^{*}$ has the homotopy type
of $\langle\operatorname{\text{\rm
Hom}}(\overline{\mathscr{C}}(L^{\prime}),L)^{\overline{\varphi}}\rangle$ and
$(\operatorname{\text{\rm
Hom}}(\overline{\mathscr{C}}(L^{\prime}),L),D_{\overline{\varphi}})\cong
s^{-1}{\rm
Der}_{\widetilde{\varphi}}({\mathscr{L}}{\mathscr{C}}(L^{\prime}),L)$.

∎

##  6 Nilpotent monoids of homotopy automorphisms and classifying fibrations

Whenever $X$ is a simply connected finite with minimal model $L$, it is well
known (see [27, 31], [2] in the fiberwise context or [3] for the relative
case) that the fibration,

$X_{\mathbb{Q}}\longrightarrow B\operatorname{{\rm
aut}}^{*}_{1}(X_{\mathbb{Q}})\longrightarrow B\operatorname{{\rm
aut}}_{1}(X_{\mathbb{Q}})$

is modeled by a dgl fibration sequence of the form

$L\stackrel{{\scriptstyle\operatorname{{\rm
ad}}}}{{\longrightarrow}}\widetilde{\rm Der}L\longrightarrow\widetilde{\rm
Der}L\widetilde{\times}sL.$

Under the same hypothesis, the above fibration is the simply connected cover
of the universal classifying fibration

$X_{\mathbb{Q}}\longrightarrow B\operatorname{{\rm
aut}}^{*}(X_{\mathbb{Q}})\longrightarrow B\operatorname{{\rm
aut}}(X_{\mathbb{Q}})$

which cannot be modeled in general, even if again $X$ is simply connected.
Indeed, as the following result shows, $B\operatorname{{\rm
aut}}^{*}(X_{\mathbb{Q}})$ and $B\operatorname{{\rm aut}}(X_{\mathbb{Q}})$ do
not lie, in general, in the image of the realization functor. In particular,
they are not nilpotent spaces.

######  Proposition 6.1.

Both, $B\operatorname{{\rm aut}}^{*}(S^{n}_{\mathbb{Q}})$ and
$B\operatorname{{\rm aut}}(S^{n}_{\mathbb{Q}})$, do not have the homotopy type
of the realization of any connected cdgl.

###### Proof.

Suppose $B\operatorname{{\rm aut}}(S^{n}_{\mathbb{Q}})\simeq\langle L\rangle$
for a given connected cdgl $L$. Then there is a group isomorphism

$\pi_{0}\operatorname{{\rm
aut}}(S^{n}_{\mathbb{Q}})\cong\pi_{1}B\operatorname{{\rm
aut}}(S^{n}_{\mathbb{Q}})\cong H_{0}(L)$

where in the latter, the group structure is given by the BCH product. However,
by Remark 4.5, $\pi_{0}\operatorname{{\rm
aut}}(S^{n}_{\mathbb{Q}})=[\mathbb{L}(x),\mathbb{L}(x)]$. This is just the
automorphisms of a one dimensional vector space which is identified with the
multiplicative group $\mathbb{Q}^{*}=\mathbb{Q}-\\{0\\}$.

Now, assume there is an isomorphism
$\psi\colon\mathbb{Q}^{*}\stackrel{{\scriptstyle\cong}}{{\to}}H_{0}(L)$ and
let $a=\psi(2)$. Recall that, for the BCH product, $\mu a*\nu
a=(\lambda+\mu)a$ for $\mu,\nu\in\mathbb{Q}$. Thus, given
$\lambda\in\mathbb{Q}^{*}$ with $\psi(\lambda)=\frac{1}{2}a$ we have
$\psi(\lambda^{2})=\frac{1}{2}a*\frac{1}{2}a=a=\psi(2)$. Then $\lambda^{2}=2$
which is a contradiction. The same argument also works for the pointed case.

∎

Despite this result we will be able to realize distinguished classifying
fibrations, also by means of cdgl’s of derivations.

Let $X$ be a connected finite complex and let $\pi_{0}\operatorname{{\rm
aut}}(X)$ be the group of homotopy classes of self homotopy equivalences of
$X$, often denoted by ${\cal E}(X)$. In the same way,
$\pi_{0}\operatorname{{\rm aut}}^{*}(X)$ is the group ${\cal E}^{*}(X)$ of
based homotopy classes of pointed self homotopy equivalences of $X$. Denote by

$\zeta\colon{\mathcal{E}}^{*}(X)\longrightarrow{\mathcal{E}}(X)$ (21)

the natural surjection which induces the bijection
${\mathcal{E}}^{*}(X)/\pi_{1}(X)\cong{\mathcal{E}}(X)$.

######  Definition 6.2.

For a given subgroup $G\subset{\mathcal{E}}(X)$ we consider the sub monoids
$\operatorname{{\rm aut}}_{G}(X)\subset\operatorname{{\rm aut}}(X)$ and
$\operatorname{{\rm aut}}_{G}^{*}(X)\subset\operatorname{{\rm aut}}^{*}(X)$
defined by

$\operatorname{{\rm aut}}_{G}(X)=\\{f\in\operatorname{{\rm aut}}(X),\,\,[f]\in
G\\},\quad\operatorname{{\rm aut}}_{G}^{*}(X)=\\{f\in\operatorname{{\rm
aut}}^{*}(X),\,\,[f]\in G\\}$

Note that $\pi_{0}\operatorname{{\rm aut}}_{G}(X)=G$ while
$\pi_{0}\operatorname{{\rm aut}}_{G}^{*}(X)$ is the subgroup
$G^{*}\subset{\mathcal{E}}^{*}(X)$ consisting of pointed homotopy classes of
pointed homotopy automorphisms in $\operatorname{{\rm aut}}_{G}^{*}(X)$. In
other terms,

$G^{*}=\zeta^{-1}(G)=\\{[f]^{*}\in{\mathcal{E}}^{*}(X),\,\,\text{with $[f]\in
G$}\\}.$

Thus, note that $G^{*}$ is preserved by the action of $\pi_{1}(X)$ and

$G^{*}/\pi_{1}(X)\cong G.$

Then, the evaluation fibration $\operatorname{{\rm
map}}^{*}(X,X)\longrightarrow\operatorname{{\rm map}}(X,X)\longrightarrow X$
restricts to a fibration $\operatorname{{\rm
aut}}_{G}^{*}(X)\to\operatorname{{\rm aut}}_{G}(X)\to X$ which is extended on
the right to provide a fibration sequence

$X\longrightarrow B\operatorname{{\rm aut}}^{*}_{G}(X)\longrightarrow
B\operatorname{{\rm aut}}_{G}(X).$

This fibration sequence, see [16], is universal with respect to fibrations
with fiber $X$ and whose “structure group” is a subgroup of $G$. In other
terms, the set

$\operatorname{{\cal F}\text{\rm ib}}_{G}(X,B)$ (22)

of equivalence classes of fibration sequences $X\to E\to B$ over $B$ for which
the image of the natural map $\pi_{1}(B)\to{\mathcal{E}}(X)$ lies in $G$, is
in bijective correspondence with the set of (free) homotopy classes
$\\{B,B\operatorname{{\rm aut}}_{G}(X)\\}$.

The counterpart of the above in the pointed setting is the following:

######  Definition 6.3.

Given $\pi\subset{\mathcal{E}}^{*}(X)$ a subgroup of pointed homotopy classes
of pointed equivalences, define

$\operatorname{{\rm aut}}_{\pi}^{*}(X)=\\{f\in\operatorname{{\rm
aut}}^{*}(X),\,\,[f]^{*}\in\pi\\}.$

######  Remark 6.4.

Note that in general, given $G\subset{\mathcal{E}}(X)$ and
$\pi\subset{\mathcal{E}}^{*}(X)$, $\operatorname{{\rm aut}}_{G}^{*}(X)$ and
$\operatorname{{\rm aut}}_{\pi}^{*}(X)$ are different submonoids of
$\operatorname{{\rm aut}}^{*}(X)$. In particular, for the trivial subgroup
$\\{1\\}\subset{\mathcal{E}}(X)$, $\operatorname{{\rm aut}}_{1}^{*}(X)$ is the
non connected monoid of pointed maps freely homotopic to $\operatorname{{\rm
id}}_{X}$. However, for $\\{1\\}\subset{\mathcal{E}}^{*}(X)$,
$\operatorname{{\rm aut}}_{1}^{*}(X)$ is the connected monoid of pointed maps
based homotopic to $\operatorname{{\rm id}}_{X}$.

Recall that a fibration of pointed spaces is pointed if it has a section.
Then, see for instance [16, Theorem 2.2], there exists a “universal” pointed
fibration,

$X\longrightarrow Z\longrightarrow B\operatorname{{\rm aut}}_{\pi}^{*}(X)$
(23)

such that: the set

$\operatorname{{\cal F}\text{\rm ib}}_{\pi}^{*}(X,B)$

of equivalence classes of pointed fibrations $F\to E\to B$ over $B$, with
fiber homotopically equivalent to $X$, and for which the image of the natural
map $\pi_{1}(B)\to{\mathcal{E}}^{*}(F)$ lies in $\pi$, is in bijective
correspondence with the set of free homotopy classes $\\{B,B\operatorname{{\rm
aut}}_{\pi}^{*}(X)\\}$.

From this point on, we fix a finite nilpotent complex $X$.

Let first $G$ be a subgroup of ${\cal E}(X)$ which acts nilpotently on
$H_{*}(X)$. Then, see [13, Theorem D], both $B\operatorname{{\rm aut}}_{G}(X)$
and $B\operatorname{{\rm aut}}_{G}^{*}(X)$ are nilpotent spaces and thus
$G\cong\pi_{1}B\operatorname{{\rm aut}}_{G}(X)$ and
$G^{*}\cong\pi_{1}B\operatorname{{\rm aut}}_{G}^{*}(X)$ are nilpotent groups.

The rationalization of $G$ or $G^{*}$ can be traced as follows: let

$\Gamma^{0}=H_{*}(X)\supset\dots\supset\Gamma^{i-1}\supset\Gamma^{i}\supset\dots\supset\Gamma^{q}=0$

be the central series of the $G$-action. That is, $\Gamma^{i}$ is generated by
$f_{*}(\alpha)-\alpha$, with $[f]\in G$ and $\alpha\in\Gamma^{i-1}$. Consider
$K\subset{\mathcal{E}}(X)$ the subgroup of self homotopy equivalences which
“stabilize” the above series. That is,

$K=\\{[f]\in{\mathcal{E}}(X),\,\,\text{$f_{*}$ induce the identity on
$\Gamma^{i}/\Gamma^{i+1}$, for all $i$}\,\\}.$

Then, also by [13, Theorem D], $K$ is nilpotent and the following is a
straightforward homological reformulation of [21, Theorem 3.3]:

######  Theorem 6.5.

The rationalization $K_{\mathbb{Q}}$ of $K$ is the subgroup of ${\cal
E}(X_{\mathbb{Q}})$ which stabilize the following series,

$\Gamma^{0}_{\mathbb{Q}}=H_{*}(X_{\mathbb{Q}})=H_{*}(X)_{\mathbb{Q}}\supset\dots\supset\Gamma^{i-1}_{\mathbb{Q}}\supset\Gamma^{i}_{\mathbb{Q}}\supset\dots\supset\Gamma^{q}_{\mathbb{Q}}=0.$
(24)

In other terms, a class $[f]$ of ${\cal E}(X_{\mathbb{Q}})$ is in
$K_{\mathbb{Q}}$ if it acts trivially on each
$\Gamma^{i-1}_{\mathbb{Q}}/\Gamma^{i}_{\mathbb{Q}}$ for all $i\geq 0$.
Moreover, the map $K\to K_{\mathbb{Q}}$, $[f]\mapsto[f_{\mathbb{Q}}]$, is the
rationalization. $\square$

As $G$ is a subgroup of $K$, it follows that $G_{\mathbb{Q}}$ is a subgroup of
$K_{\mathbb{Q}}$ and the map $G\to G_{\mathbb{Q}}$,
$[f]\mapsto[f_{\mathbb{Q}}]$, is also the rationalization. The same procedure
describes $G^{*}_{\mathbb{Q}}$ and $K^{*}_{\mathbb{Q}}$.

This defines maps of monoids,

$\operatorname{{\rm aut}}_{G}(X)\longrightarrow\operatorname{{\rm
aut}}_{G_{\mathbb{Q}}}(X_{\mathbb{Q}}),\qquad\operatorname{{\rm
aut}}_{G}^{*}(X)\longrightarrow\operatorname{{\rm
aut}}_{G_{\mathbb{Q}}}^{*}(X_{\mathbb{Q}}),$

both sending a map to its rationalization. Remark that, componentwise, these
maps are also the rationalizations as, for each map $f\colon X\to X$, pointed
in the second case,

$\operatorname{{\rm
map}}_{f_{\mathbb{Q}}}(X_{\mathbb{Q}},X_{\mathbb{Q}})\simeq\operatorname{{\rm
map}}_{f}(X,X)_{\mathbb{Q}},\quad\operatorname{{\rm
map}}_{f_{\mathbb{Q}}}^{*}(X_{\mathbb{Q}},X_{\mathbb{Q}})\simeq\operatorname{{\rm
map}}_{f}^{*}(X,X)_{\mathbb{Q}}.$

From all of the above, straightforward arguments yield:

######  Proposition 6.6.

Let $X$ be a finite complex and $G\subset{\mathcal{E}}(X)$ be a subgroup which
acts nilpotently on $H_{*}(X)$. Then, the rationalization of the classifying
fibration sequence

$X\longrightarrow B\operatorname{{\rm aut}}^{*}_{G}(X)\longrightarrow
B\operatorname{{\rm aut}}_{G}(X)$

has the homotopy type of the rational classifying fibration sequence

$X_{\mathbb{Q}}\longrightarrow B\operatorname{{\rm
aut}}^{*}_{G_{\mathbb{Q}}}(X_{\mathbb{Q}})\longrightarrow B\operatorname{{\rm
aut}}_{G_{\mathbb{Q}}}(X_{\mathbb{Q}}).$

$\square$

Now, let $L=({\widehat{\mathbb{L}}}(V),d)$ be the minimal Lie model of $X$. In
view of Corollary 2.2,
$\\{X_{\mathbb{Q}},X_{\mathbb{Q}}\\}^{*}\cong\\{L,L\\}$. In particular, every
pointed homotopy equivalence
$X_{\mathbb{Q}}\stackrel{{\scriptstyle\simeq}}{{\to}}X_{\mathbb{Q}}$ in
${\mathcal{E}}^{*}(X_{\mathbb{Q}})$ is identified with a homotopy class of a
quasi-isomorphism $L\stackrel{{\scriptstyle\cong}}{{\to}}L$ which is
necessarily an automorphism as $L$ is connected [11, Theorem 3.19].

######  Definition 6.7.

Denote by

${\cal G}\subset\operatorname{{\rm aut}}(L)/\sim,$

the subgroup of homotopy classes of automorphism of $L$ which, in view of the
above, is isomorphic to
$G_{\mathbb{Q}}^{*}\subset{\mathcal{E}}^{*}(X_{\mathbb{Q}})$.

As $G_{\mathbb{Q}}^{*}/\pi_{1}(X_{\mathbb{Q}})\cong G_{\mathbb{Q}}$, by
Corollary 2.5 we deduce that ${\cal G}$ is invariant by the $H_{0}(L)$ action
and

${\cal G}/H_{0}(L)\cong G_{\mathbb{Q}}.$

Denote also,

$\operatorname{{\rm aut}}_{{\cal G}}(L)=\\{\varphi\in\operatorname{{\rm
aut}}(L),\,[\varphi]\in{\cal G}\\},$

which is clearly a subgroup of $\operatorname{{\rm aut}}(L)$, and define ${\rm
Der}^{\mathcal{G}}L\subset{\rm Der}L$ by

${\rm Der}^{\mathcal{G}}_{\geq 1}L={\rm Der}_{\geq 1}L\quad\text{and}\quad{\rm
Der}^{\mathcal{G}}_{0}L=\\{\text{$\theta\in{\rm Der}_{0}L$ such that
$e^{\theta}\in\operatorname{{\rm aut}}_{{\cal G}}(L)$}\\}.$

Accordingly, for $K\subset{\mathcal{E}}(X)$ containing $G$ as in Theorem 6.5
we obtain ${\cal K}\subset\operatorname{{\rm aut}}(L)/\sim$ isomorphic to
$K_{\mathbb{Q}}^{*}$, $\operatorname{{\rm aut}}_{\cal
K}(L)=\\{\varphi\in\operatorname{{\rm aut}}(L),\,\,[\varphi]\in{\cal K}\\}$,
and ${\rm Der}^{\mathcal{K}}L\subset{\rm Der}L$, where

${\rm Der}^{\mathcal{K}}_{\geq 1}L={\rm Der}_{\geq 1}L\quad\text{and}\quad{\rm
Der}^{\mathcal{K}}_{0}L=\\{\text{$\theta\in{\rm Der}_{0}L$ such that
$e^{\theta}\in\operatorname{{\rm aut}}_{{\cal K}}(L)$}\\}.$

As observed above, ${\cal G}$ and $\operatorname{{\rm aut}}_{\cal G}(L)$ are
subgroups of ${\cal K}$ and $\operatorname{{\rm aut}}_{\cal K}(L)$
respectively. Therefore, ${\rm Der}^{\mathcal{G}}_{0}L$ is a subgroup of ${\rm
Der}^{\mathcal{K}}_{0}L$ for the Baker-Campbell-Hausdorff product.

Then, the precise statement of Theorem 0.1 is:

######  Theorem 6.8.

The maps

$L\stackrel{{\scriptstyle\operatorname{{\rm ad}}}}{{\longrightarrow}}{\rm
Der}^{\mathcal{G}}L\longrightarrow{\rm
Der}^{\mathcal{G}}L\widetilde{\times}sL$

constitute a cdgl fibration sequence whose realization is homotopy equivalent
to the classifying fibration

$X_{\mathbb{Q}}\longrightarrow B\operatorname{{\rm
aut}}^{*}_{G_{\mathbb{Q}}}(X_{\mathbb{Q}})\longrightarrow B\operatorname{{\rm
aut}}_{G_{\mathbb{Q}}}(X_{\mathbb{Q}})$

Here, $\operatorname{{\rm ad}}$ denotes the adjoint operator and $({\rm
Der}^{\mathcal{G}}L\widetilde{\times}sL,D)$ is the twisted product where $sL$
is an abelian Lie algebra and

$Dsx=-sdx+\operatorname{{\rm
ad}}_{x},\quad[\theta,sx]=(-1)^{|\theta|}s\theta(x),\quad x\in
L,\quad\theta\in{\rm Der}^{\mathcal{G}}(L).$ (25)

Observe that combining this with Corollary 4.4 we obtain

######  Corollary 6.9.

The realization of the cdgl sequence

$\operatorname{\text{\rm
Hom}}(\overline{\mathscr{C}}(L),L)\to\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L),L)\to L\stackrel{{\scriptstyle\operatorname{{\rm
ad}}}}{{\to}}{\rm Der}^{\mathcal{G}}L\to{\rm
Der}^{\mathcal{G}}L\widetilde{\times}sL$ (26)

is of the homotopy type of

$\operatorname{{\rm
map}}^{*}(X_{\mathbb{Q}},X_{\mathbb{Q}})\to\operatorname{{\rm
map}}(X_{\mathbb{Q}},X_{\mathbb{Q}})\stackrel{{\scriptstyle\operatorname{{\rm
ev}}}}{{\to}}X_{\mathbb{Q}}\to B\operatorname{{\rm
aut}}_{G_{\mathbb{Q}}}^{*}(X_{\mathbb{Q}})\to B\operatorname{{\rm
aut}}_{G_{\mathbb{Q}}}(X_{\mathbb{Q}})$

In particular, restricting to the corresponding components in the above cdgl
produces a Lie model of the fibration sequence,

$\operatorname{{\rm
aut}}_{G_{\mathbb{Q}}}^{*}(X_{\mathbb{Q}})\to\operatorname{{\rm
aut}}_{G_{\mathbb{Q}}}(X_{\mathbb{Q}})\stackrel{{\scriptstyle\operatorname{{\rm
ev}}}}{{\to}}X_{\mathbb{Q}}\to B\operatorname{{\rm
aut}}_{G_{\mathbb{Q}}}^{*}(X_{\mathbb{Q}})\to B\operatorname{{\rm
aut}}_{G_{\mathbb{Q}}}(X_{\mathbb{Q}})$

$\square$

We now focus on a subgroup $\pi\subset{\mathcal{E}}^{*}(X)$ of pointed
homotopy classes of pointed equivalences which acts nilpotently on
$\pi_{*}(X)$. Then, by [13, Theorem C], $B\operatorname{{\rm
aut}}_{\pi}^{*}(X)$ is a nilpotent complex, and so is $\pi$. As in the
homological case, the rationalization $\pi_{\mathbb{Q}}$ of $\pi$ is similarly
described as a subgroup of ${\mathcal{E}}^{*}(X_{\mathbb{Q}})$, this time with
the aid of [21, Theorem 3.3]. The analog of Definition 6.7 is:

######  Definition 6.10.

Define

$\Pi\subset\operatorname{{\rm aut}}(L)/\sim,$

as the subgroup of homotopy classes of automorphism of $L$ which, in view of
Corollary 2.2, is isomorphic to
$\pi_{\mathbb{Q}}\subset{\mathcal{E}}^{*}(X_{\mathbb{Q}})$. Consider also the
subgroup of $\operatorname{{\rm aut}}(L)$ given by

$\operatorname{{\rm aut}}_{\Pi}(L)=\\{\varphi\in\operatorname{{\rm
aut}}(L),\,[\varphi]\in\Pi\\},$

and define ${\rm Der}^{\Pi}L\subset{\rm Der}L$ by,

${\rm Der}^{\Pi}_{\geq 1}L={\rm Der}_{\geq 1}L\quad\text{and}\quad{\rm
Der}^{\Pi}_{0}L=\\{\text{$\theta\in{\rm Der}_{0}L$ such that
$e^{\theta}\in\operatorname{{\rm aut}}_{\Pi}(L)$}\\}.$

Finally, we will also assume that $\pi_{\mathbb{Q}}$ is preserved by the
action of $\pi_{1}(X_{\mathbb{Q}})$, or equivalently, $\Pi$ is preserved by
the action of $H_{0}(L)$. Then, Theorem 0.2 reads:

######  Theorem 6.11.

The maps

$L{\longrightarrow}L\widetilde{\times}{\rm Der}^{\Pi}L\longrightarrow{\rm
Der}^{\Pi}L$ (27)

constitute a cdgl fibration sequence whose realization is homotopy equivalent
to the classifying fibration

$X_{\mathbb{Q}}\longrightarrow Z_{\mathbb{Q}}\longrightarrow
B\operatorname{{\rm aut}}_{\pi_{\mathbb{Q}}}^{*}(X_{\mathbb{Q}}).$

Here $L\widetilde{\times}{\rm Der}^{\Pi}L$ is the twisted product in which
both terms are sub dgl’s and $[\theta,x]=\theta(x)$ for any $\theta\in{\rm
Der}^{\Pi}L$ and $x\in L$.

######  Remark 6.12.

Note that, with the notation and nomenclature of the classical reference [23,
§7], this theorem proves in particular that the geometric bar construction
$B(*,\operatorname{{\rm aut}}_{\pi}^{*}(X),X)$ has the rational homotopy type
of the realization of $L\widetilde{\times}{\rm Der}^{\Pi}L$. Indeed in the
universal classifying fibration (23), $Z$ is precisely $B(*,\operatorname{{\rm
aut}}_{\pi}^{*}(X),X)$.

We point out that Theorems 6.8 and 6.11 are accurate on their hypothesis. Let
$K$ be a subgroup of ${\mathcal{E}}^{*}(X_{\mathbb{Q}})$ or
${\mathcal{E}}(X_{\mathbb{Q}})$ acting nilpotently on
$\pi_{*}(X_{\mathbb{Q}})$ or $H_{*}(X_{\mathbb{Q}})$, and which does not
necessarily arise as the rationalization of a nilpotent action on the homotopy
or homology groups of $X$. Then, as shown in the following example, and even
for $K$ abelian and $X$ simply connected, $B\operatorname{{\rm
aut}}_{K}^{*}(X_{\mathbb{Q}})$ and $B\operatorname{{\rm
aut}}_{K}(X_{\mathbb{Q}})$ do not lie in general in the image of the
realization functor. In particular they fail to be nilpotent.

######  Example 6.13.

Let $X=S^{n}\vee S^{n}$, with $n\geq 2$, whose minimal model is
$M=(\mathbb{L}(x,y),0)$ with $|x|=|y|=n-1$. Since $X_{\mathbb{Q}}$ is simply
connected, free and pointed classes coincide. Consider the subgroup
$K\subset{\mathcal{E}}^{*}(X_{\mathbb{Q}})={\mathcal{E}}(X)$ generated by the
homotopy class of the automorphism $\varphi\colon
M\stackrel{{\scriptstyle\cong}}{{\to}}M$ given by $\varphi(x)=x+y$ and
$\varphi(y)=y$. Obviously, $K$ acts nilpotently on both
$\pi_{*}(X_{\mathbb{Q}})$ and $H_{*}(X_{\mathbb{Q}})$ as its central series
vanishes at the second stage. Now, if $L$ is a cdgl for which $\langle
L\rangle\simeq B\operatorname{{\rm aut}}_{K}^{*}(X_{\mathbb{Q}})$ it follows
that $H_{0}(L)$, endowed with the BCH group structure, must coincide with $K$
which is in turn isomorphic to $\mathbb{Z}$. However, a similar argument to
the one in Proposition 6.1 shows that this is not possible. The same applies
to $B\operatorname{{\rm aut}}_{K}(X_{\mathbb{Q}})$.

######  Remark 6.14.

Although all spaces involved in Theorems 6.8 and 6.11 are connected and
nilpotent, it is important to stress that it is absolutely essential to work
in $\operatorname{{\bf cdgl}}$ and not in the model category defined in [24]
whose objects are connected (not complete) dgl’s. Among others, this is the
main obstruction: fibrations in this model structure are surjective morphisms
but not every fibration of nilpotent complexes can be modeled by a surjective
dgl morphism. For instance, the simple example given by the universal cover of
the circle $\mathbb{R}\to S^{1}$ does not have a surjective model. However, it
can be modeled by the cdgl fibration (surjective only in non-negative
degrees),

$\varphi\colon({\widehat{\mathbb{L}}}(x,y),d)\longrightarrow(\mathbb{L}(x),0),\,\,\varphi(x)=x,\,\,\varphi(y)=0{\quad\text{where}\quad}|x|=0,\quad
dx=y.$

Note that the fibre of this morphism is given by,

$\ker\varphi=({\widehat{\mathbb{L}}}(a_{n})_{n\geq 0},d),\qquad
a_{n}=\operatorname{{\rm ad}}_{x}^{n}(y).$

Since each $a_{n}$ is a Maurer-Cartan element the realization of this cdgl is,
as it should, a discrete countable space. The same argument applies to
$X_{\mathbb{Q}}\to B\operatorname{{\rm
aut}}^{*}_{G_{\mathbb{Q}}}(X_{\mathbb{Q}})$ and $X_{\mathbb{Q}}\to Z$ of
Theorems 6.8 and 6.11 respectively, whose homotopy fibers $\operatorname{{\rm
aut}}_{G_{\mathbb{Q}}}(X_{\mathbb{Q}})$ and $\operatorname{{\rm
aut}}_{\pi_{\mathbb{Q}}}^{*}(X_{\mathbb{Q}})$ are non-connected.

The next sections are devoted to the proofs of Theorems 6.8 and 6.11.

##  7 Complete Lie algebras of derivations and twisted products

Given $C$ a cdgc and $L$ a (non necessarily complete) dgl, consider the
twisted product

$\operatorname{\text{\rm Hom}}(C,L)\longrightarrow\operatorname{\text{\rm
Hom}}(C,L)\,\widetilde{\times}\,{\rm Der}L\longrightarrow{\rm Der}L$ (28)

in which both terms are sub dgl’s and

$[\theta,f]=\theta\circ f,\quad\theta\in{\rm Der}L,\quad
f\in\operatorname{\text{\rm Hom}}({\mathscr{C}}(L),L).$

This dgl sequence, and its “dual” ${\rm
Coder}(C)\widetilde{\times}\operatorname{\text{\rm Hom}}(C,L)$, have already
proven to be useful in the modeling of certain simply connected spaces of
homotopy automorphisms, see [2, §3] and [3, §4]. However, on the one hand,
${\rm Der}L$ may fail to be complete even if $L$ is. On the other hand, even
if ${\rm Der}L$ is complete, $\operatorname{\text{\rm
Hom}}(C,L)\widetilde{\times}{\rm Der}L$ may fail to be so.

Also, for any given dgl $L$, consider the twisted products

${\rm Der}L\widetilde{\times}sL{\quad\text{and}\quad}L\widetilde{\times}{\rm
Der}L$ (29)

defined as in (25) and (27) respectively. Again, these may not be complete
dgl’s even if $L$ and ${\rm Der}L$ are. The following illustrates this
situation.

######  Example 7.1.

(1) Consider an odd dimensional sphere $S^{n}$ whose model is the abelian Lie
algebra $L=(\mathbb{L}(x),0)$ with $x$ of degree $n-1$. In this case
${\mathscr{C}}(L)=(\land sx,0)$. Then, $\operatorname{{\rm map}}(S^{n},S^{n})$
is modeled by $\operatorname{\text{\rm Hom}}({\mathscr{C}}(L),L)$ which is an
abelian Lie algebra generated by $x$, corresponding to $1\mapsto x$, and an
element $z$ of degree $-1$, corresponding to $sx\mapsto x$. On the other hand
${\rm Der}L$ is an abelian lie algebra generated by $\operatorname{{\rm
id}}_{L}$, a degree $0$ derivation which we denote by $\theta$. Hence, in this
case, (28) takes the form

$\operatorname{{Span}}\\{x,z\\}\longrightarrow\operatorname{{Span}}\\{x,z,\theta\\}\longrightarrow\operatorname{{Span}}\\{\theta\\}$

where, in the middle, $[\theta,x]=x$ and $[\theta,z]=z$. Observe then that the
bracket does not respect the usual filtration in the non twisted product. Note
also that, even though $\operatorname{\text{\rm Hom}}({\mathscr{C}}(L),L)$ and
${\rm Der}L$ are obviously complete as they are abelian,
$\operatorname{\text{\rm Hom}}({\mathscr{C}}(L),L)\,\widetilde{\times}\,{\rm
Der}L$ is not (with respect to the usual bracket filtration) as its completion
is easily seen to be just $\operatorname{{Span}}\\{\theta\\}$. That is, the
completion of (28) becomes

$\operatorname{\text{\rm Hom}}({\mathscr{C}}(L),L)\stackrel{{\scriptstyle
0}}{{\longrightarrow}}{\rm Der}L\stackrel{{\scriptstyle\operatorname{{\rm
id}}}}{{\longrightarrow}}{\rm Der}L.$

This also illustrates that the completion functor is not left exact in
general.

(2) For the same $L$, observe that the twisted dgl ${\rm
Der}L\widetilde{\times}sL$ becomes the dgl
$\operatorname{{Span}}\\{\theta,sx\\}$, with zero differential and
$[\theta,sx]=sx$, whose completion is ${\rm
Der}L=\operatorname{{Span}}\\{\theta\\}$. An analogous argument shows that
$L\widetilde{\times}{\rm Der}L$ fails also to be complete with respect to the
usual filtration.

The content of this section let us overcome these obstacles, at least with the
restrictions given in Theorems 6.8 and 6.11.

From now on we fix $L=({\widehat{\mathbb{L}}}(V),d)$ a connected, minimal,
free cdgl and a finite filtration of $V$ by graded subspaces:

$V=V^{0}\supset\dots\supset V^{i}\supset V^{i+1}\supset\dots\supset V^{q}=0,$
(30)

We first refine the usual filtration in $L$ by considering, for $n\geq 1$ and
$p\geq 0$,

${\widehat{\mathbb{L}}}^{n,p}(V)=\operatorname{{Span}}\\{\bigl{[}a_{1},[a_{2},[\dots,[a_{n-1},a_{n}]\bigr{]}\dots\bigr{]}\in{\widehat{\mathbb{L}}}^{n}(V),\,a_{i}\in
V^{\alpha_{i}}\,\text{and}\,\,{\textstyle\sum_{i=1}^{n}}\alpha_{i}=p\\}.$

Observe that ${\widehat{\mathbb{L}}}^{n,p}=0$ for $p\geq nq$. Then, for $n\geq
1$ and $0\leq p\leq nq-1$, define

$F^{n,p}={\widehat{\mathbb{L}}}^{n,p}(V)\oplus{\widehat{\mathbb{L}}}^{\geq
n+1}(V)$

and note that,

$\displaystyle{\widehat{\mathbb{L}}}(V)$ $\displaystyle=F^{1,0}\supset
F^{1,1}\supset\dots\supset F^{1,q-1}$ $\displaystyle\supset F^{2,0}\supset
F^{2,1}\supset\dots\supset F^{2,2q-1}$
$\displaystyle...............................................$
$\displaystyle\supset F^{n,0}\supset F^{n,1}\supset\dots\supset F^{n,nq-1}$
$\displaystyle...............................................$

Remark that $F^{n,p}$ ranks

$m=q+\dots+(n-1)q+p+1=\frac{(n-1)nq}{2}+p+1$ (31)

in the order given by this chain of inclusions.

######  Definition 7.2.

For $n,p$ and $m$ as above define

$F^{m}=F^{n,p}.$

######  Proposition 7.3.

The sequence $\\{F^{m}\\}_{m\geq 1}$ is a filtration of $L$. Furthermore, $L$
is complete with respect to this filtration.

###### Proof.

Observe that $[F^{n,p},F^{r,s}]\subset F^{n+r,p+s}$. Now, $F^{n,p}$ and
$F^{r,s}$ rank respectively $\frac{(n-1)nq}{2}+p+1$ and
$\frac{(r-1)rq}{2}+s+1$ in the filtration order. The sum of these integers is
smaller than $\frac{(n+r-1)(n+r)q}{2}+p+s+1$ which is the position of
$F^{n+r,p+s}$ in that order. This shows that the Lie bracket is filtration
preserving.

On the other hand, since the differential of $L$ is decomposable each $F^{m}$
is a differential ideal. Moreover, we have

$dF^{m}\subset F^{m+1}$

We now check that $L$ is complete respect to this filtration. Since
$\cap_{n,p}F^{n,p}=0$ the natural map

${\widehat{\mathbb{L}}}(V)\longrightarrow\varprojlim_{m}{\widehat{\mathbb{L}}}(V)/F^{m}$

is injective. Finally, write a given element in
$\varprojlim_{n,p}{\widehat{\mathbb{L}}}(V)/F^{n,p}$ as a series,

$\sum_{n,p}x_{n,p}{\quad\text{with}\quad}x_{n,p}\in F^{n,p},$

and note that this is a well defined element in ${\widehat{\mathbb{L}}}(V)$
since, for each $m\geq 1$, $\sum_{n,p}x_{n,p}$ contains only a finite sum in
${\widehat{\mathbb{L}}}^{m}(V)$. This implies the surjectivity of the map
above. ∎

Note that considering in $L$ the usual filtration
$\\{{\widehat{\mathbb{L}}}^{\geq m}(V)\\}_{m\geq 1}$ for which it is complete,
the identity $(L,\\{{\widehat{\mathbb{L}}}^{\geq m}(V)\\}_{m\geq
1})\to(L,\\{F^{m}\\}_{m\geq 1})$ is a cdgl morphism.

######  Definition 7.4.

Denote by ${\mathcal{D}er}L\subset{\rm Der}L$ the connected sub dgl in which,

$\displaystyle{\mathcal{D}er}_{\geq 1}L={\rm Der}_{\geq 1}L,$
$\displaystyle{\mathcal{D}er}_{0}L=\\{\theta\in{\rm Der}_{0}L\cap\ker
D,\,\,\theta(V^{i})\subset V^{i+1}\oplus{\widehat{\mathbb{L}}}^{\geq
2}(V)\,\text{for all $i$}\,\\}.$

Note that this is a well defined Lie algebra and, since the differential in
$L$ is decomposable, it preserves the differential.

Using the new filtration on $L$ we first construct a filtration of
${\mathcal{D}er}_{0}L$ for which it becomes a cdgl. For it, note that

${\mathcal{D}er}_{0}L=\\{\theta\in{\rm Der}_{0}L\cap\ker
D,\,\,\theta(F^{m})\subset F^{m+1}\,\,\text{for all $m$}\,\\}$

Hence, we filter ${\mathcal{D}er}_{0}L$ by,

$\operatorname{{\mathcal{F}}}^{n}=\\{\theta\in{\mathcal{D}er}_{0}L,\,\,\theta(F^{m})\subset
F^{m+n},\,\,\text{for all $m$}\,\\},\quad n\geq 1,$

and a straightforward computation proves:

######  Lemma 7.5.

${\mathcal{D}er}_{0}L$ is complete with respect to
$\\{\operatorname{{\mathcal{F}}}^{n}\\}_{n\geq 1}$. $\square$.

To finish, we “extend” this filtration on ${\mathcal{D}er}_{0}L$ to
${\mathcal{D}er}L$ by means of the following procedure:

Consider first the subspaces of ${\rm Der}_{\geq 1}$,

$M^{n}=\\{\theta\in{\rm Der}_{\geq 1}L,\,\theta(L)\subset L^{\geq
n+1}\\},\qquad n\geq 1,$

and choose a refinement of this sequence by constant subspaces

$\displaystyle M^{1}=$ $\displaystyle{\cal J}^{1,0}={\cal J}^{1,1}=\dots={\cal
J}^{1,q-1}\supset$ $\displaystyle M^{2}=$ $\displaystyle{\cal J}^{2,0}={\cal
J}^{2,1}=\dots={\cal J}^{2,2q-1}\supset$
$\displaystyle..................................................$
$\displaystyle M^{n}=$ $\displaystyle{\cal J}^{n,0}={\cal J}^{n,1}=\dots={\cal
J}^{n,nq-1}\supset$
$\displaystyle..................................................$

As in (31), ${\cal J}^{n,p}$ ranks $m=\frac{(n-1)nq}{2}+p+1$ in the order
given by this chain, and define

${\cal J}^{m}={\cal J}^{n,p}.$

The only purpose of extracting this refinement is simply to assure that,

$[\operatorname{{\mathcal{F}}}^{n},{\cal J}^{m}]\subset{\cal J}^{m+n},\quad
n,m\geq 1.$

Finally, define:

$E^{n}=\operatorname{{\mathcal{F}}}^{n}_{0}\oplus(\oplus_{m=1}^{n-1}{\cal
J}^{n-m}_{m})\oplus{\rm Der}_{\geq n}L,\qquad n\geq 1,$ (32)

######  Proposition 7.6.

${\mathcal{D}er}L$ is a cdgl with respect to the filtration
$\\{E^{n}\\}_{n\geq 1}$.

###### Proof.

Note that, $E^{1}=\operatorname{{\mathcal{F}}}^{1}_{0}\oplus{\rm Der}_{\geq
1}L={\mathcal{D}er}L$. Also, since the differential in $L$ is decomposable,
$d{\cal J}^{m}_{p}\subset{\cal J}^{m+1}_{p-1}$ for $p\geq 2$, and $d{\cal
J}^{m}_{1}\subset\operatorname{{\mathcal{F}}}^{k+1}_{0}$, for all $m\geq 1$.
Finally $[{\cal J}^{n},{\cal J}^{m}]\subset{\cal J}^{n+m}$ while, as noted
above, $[\operatorname{{\mathcal{F}}}^{n},{\cal J}^{m}]\subset{\cal J}^{m+n}$
for $n,m\geq 1$. With this data one easily checks that $\\{E^{n}\\}_{\geq 1}$
is a filtration of ${\mathcal{D}er}L$.

Now, observe that for $p\geq 1$ and $n\geq 0$,

$E^{n}_{p}=\begin{cases}{\rm Der}_{n}L&\text{if $n\leq p,$},\\\ {\cal
J}^{n-p}_{p},&\text{if $n>p$}.\end{cases}$

Hence, for $p\geq 1$,

$\varprojlim_{n}({\mathcal{D}er}L/E^{n})_{p}=\varprojlim_{n}{\rm
Der}_{p}L/{\cal J}^{n}_{p}={\rm Der}_{p}L.$

On the other hand, for $p=0$, and in view of Lemma 7.5,

$\varprojlim_{n}({\mathcal{D}er}L/E^{n})_{0}=\varprojlim_{n}{\mathcal{D}er}_{0}/\operatorname{{\mathcal{F}}}^{n}={\mathcal{D}er}_{0}L,$

∎

Next, given
$f\colon({\widehat{\mathbb{L}}}(V),d)\stackrel{{\scriptstyle\cong}}{{\to}}({\widehat{\mathbb{L}}}(V),d)$
an element of $\operatorname{{\rm aut}}L$, denote by

$f_{*}\colon V\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}V$

the induced automorphism on the indecomposables. Then, we prove:

######  Proposition 7.7.

$\exp\bigl{(}{\mathcal{D}er}_{0}L\bigr{)}=\\{f\in\operatorname{{\rm
aut}}(L),\,\,(f_{*}-\operatorname{{\rm id}}_{V})(V^{i})\subset
V^{i+1}\,\,\text{for all $i$}\,\\}.$

###### Proof.

Let $\theta\in{\mathcal{D}er}_{0}L$ and call $f=e^{\theta}$. Then
$f-\operatorname{{\rm id}}_{L}=\sum_{n\geq 1}\frac{\theta^{n}}{n!}$ which
clearly satisfies $(f-\operatorname{{\rm id}}_{L})(V^{i})\subset
V^{i+1}\oplus{\widehat{\mathbb{L}}}^{\geq 2}(V)$ for all $i$. Hence
$(f_{*}-\operatorname{{\rm id}}_{V})(V^{i})\subset V^{i+1}$ also for all $i$.

Conversely, let $f\in\operatorname{{\rm aut}}(L)$ be such that $f(v)-v\in
V^{i+1}\oplus{\widehat{\mathbb{L}}}^{\geq 2}(V)$ for all $v\in V^{i}$ and all
$i$. Define a linear morphism,

$\theta\colon V\longrightarrow L,\qquad\theta(v)=\sum_{n\geq
1}(-1)^{n}\frac{(f-\operatorname{{\rm id}}_{L})^{n}(v)}{n}$

Recursively, one sees that the component of $(f-\operatorname{{\rm
id}}_{L})^{n}(v)$ in ${\widehat{\mathbb{L}}}^{m,r}(V)$ is zero for $n$ large
enough. Thus, $\theta$ is well defined and can be extended to a derivation
$\theta\in{\mathcal{D}er}_{0}(L)$. Finally, since the formula that defines
$\theta$ is that of $\log f$, computing $e^{\theta}$ recovers $f$. ∎

Next, recall from Remark 4.1 that, for any cdgc $C$ and any cdgl $L$,
$\operatorname{\text{\rm Hom}}(C,L)$ is complete with respect to the natural
filtration given by that of $L$ . However, we will need a new filtration which
is “compatible” with the one we just constructed for ${\mathcal{D}er}(L)$.

######  Definition 7.8.

Let $C$ be a cdgc and $L=({\widehat{\mathbb{L}}}(V),d)$ be a minimal cdgl
filtered by $\\{F^{m}\\}_{m\geq 1}$ of Definition 7.2. For $m\geq 1$, define

${I}^{m}=\oplus_{k\geq 0}\operatorname{\text{\rm Hom}}(C,F_{k}^{m-k})$

where $F^{n}=L$ for $n\leq 0$.

######  Proposition 7.9.

The sequence $\\{{I}^{m}\\}_{m\geq 1}$ is a filtration of
$\operatorname{\text{\rm Hom}}(C,L)$ for which it becomes a cdgl.

###### Proof.

Clearly $I^{1}=\operatorname{\text{\rm Hom}}(C,L)$, $I^{m+1}\subset I^{m}$ and
$D{I}^{m}\subset{I}^{m}$ for all $m$. On the other hand, given
$\varphi\in\operatorname{\text{\rm Hom}}(C,F^{n-i}_{i})\subset I^{n}$,
$\psi\in\operatorname{\text{\rm Hom}}(C,F^{m-j}_{j})\subset I^{m}$ and $c\in
C$ with $\Delta(c)=\sum_{i}c_{i}\otimes c_{i}^{\prime}$,

$[\varphi,\psi](c)=\pm\sum_{i}[\varphi(c_{i}),\psi(c_{i}^{\prime})]\in
F^{n+m-i-j}_{i+j}.$

This implies that $[I^{n},I^{m}]\subset I^{n+m}$ and therefore
$\\{{I}^{m}\\}_{m\geq 1}$ is a filtration. A degree-wise analogous argument to
that in Remark 4.1 shows that $\operatorname{\text{\rm Hom}}(C,L)$ is complete
with respect to this new filtration. ∎

Next, in the restriction of the twisted product in (28) to

$\operatorname{\text{\rm Hom}}(C,L)\,\widetilde{\times}\,{\mathcal{D}er}L,$

consider $\\{J^{n}\\}_{n\geq 1}=\\{I^{n}\times E^{n}\\}_{n\geq 1}$ with
$\\{I^{n}\\}_{n\geq 1}$ of Definition 7.8 and $\\{E^{n}\\}_{n\geq 1}$ as in
(32). Then, a direct inspection proves:

######  Proposition 7.10.

The sequence $\\{{J}^{n}\\}_{n\geq 1}$ is a filtration of
$\operatorname{\text{\rm Hom}}(C,L)\,\widetilde{\times}\,{\mathcal{D}er}L$ for
which it becomes a cdgl. $\square$

Furthermore, from Propositions 7.6, 7.9 and 7.10, and with respect to the
corresponding filtrations, we deduce

######  Corollary 7.11.

The maps

$\operatorname{\text{\rm Hom}}(C,L)\longrightarrow\operatorname{\text{\rm
Hom}}(C,L)\,\widetilde{\times}\,{\mathcal{D}er}L\longrightarrow{\mathcal{D}er}L$

form a short exact sequence of cdgl morphism and thus, it is a cdgl fibration.
$\square$

Finally, in the twisted products

${\mathcal{D}er}L\widetilde{\times}sL{\quad\text{and}\quad}L\widetilde{\times}{\mathcal{D}er}L,$

obtained as restrictions of those in (29), we consider the respective
sequences

$\\{E^{n}\times sF^{n}\\}_{n\geq 1}{\quad\text{and}\quad}\\{F^{n}\times
E^{n}\\}_{n\geq 1}.$

By procedures analogous to those used in this section one obtains:

######  Proposition 7.12.

These sequences are filtrations for which
${\mathcal{D}er}L\widetilde{\times}sL$ and
$L\widetilde{\times}{\mathcal{D}er}L,$ are cdgl’s. $\square$

##  8 Modeling free and pointed classifying fibrations

In this section we complete the proofs of Theorems 6.8 and 6.11. We therefore
adopt (and fix) the notation in §6 and start with the following:

######  Lemma 8.1.

The maps

$L\stackrel{{\scriptstyle\operatorname{{\rm ad}}}}{{\longrightarrow}}{\rm
Der}^{\mathcal{G}}L\longrightarrow{\rm
Der}^{\mathcal{G}}L\widetilde{\times}sL,$

constitute a well defined cdgl sequence.

###### Proof.

First, recall that ${\cal G}$ is identified with the group
$G_{\mathbb{Q}}^{*}$ which acts nilpotently on $H_{*}(X_{\mathbb{Q}})$ as
$G_{\mathbb{Q}}$ does. Associated to this $G^{*}_{\mathbb{Q}}$-action, we find
the series (24) of Theorem 6.5 which, via the isomorphism
$s^{-1}\widetilde{H}_{*}(X_{\mathbb{Q}})\cong V$, is then of the form

$V=V^{0}\supset\dots\supset V^{i}\supset V^{i+1}\supset\dots\supset V^{q}=0.$

Hence, for ${\cal K}$ and $\operatorname{{\rm aut}}_{\cal K}L$ as in
Definition 6.7, it follows that

$\operatorname{{\rm aut}}_{\cal K}(L)=\\{f\in\operatorname{{\rm
aut}}(L),\,\,(f_{*}-\operatorname{{\rm id}}_{V})(V^{i})\subset
V^{i+1}\,\,\text{for all $i$}\,\\}.$

We then apply Proposition 7.7 to conclude that ${\rm Der}^{\mathcal{K}}L$ is
precisely ${\mathcal{D}er}L$ of Definition 7.4 for the above filtration of
$V$. Hence, by Proposition 7.6, ${\rm Der}^{\mathcal{K}}L$ is a well defined
cdgl. Now, recall from Definition 6.7 that ${\rm Der}^{\mathcal{G}}_{0}L$ is a
subgroup of ${\rm Der}^{\mathcal{K}}_{0}L$ for the BCH product which,
furthermore, is $\mathbb{Q}$-local as
$e^{\lambda\theta}=(e^{\theta})^{\lambda}\in{\cal G}$ for any
$\lambda\in\mathbb{Q}$ and $\theta\in{\rm Der}^{\mathcal{G}}_{0}L$. Then,
apply Theorem 3.2 to deduce that ${\rm Der}^{\mathcal{G}}_{0}L$ is a sub Lie
algebra of ${\rm Der}^{\mathcal{K}}_{0}L$. We see now that $D({\rm
Der}_{1}L)\subset{\rm Der}^{\mathcal{G}}_{0}L$ by observing that,

$e^{D\theta}\sim\operatorname{{\rm id}}L,\qquad\text{for any $\theta\in{\rm
Der}_{1}L$}.$

On the one hand, since the differential in $L=({\widehat{\mathbb{L}}}(V),d)$
is decomposable $(D\theta)(V)\subset{\widehat{\mathbb{L}}}^{\geq 2}(V)$ and
thus $e^{D\theta}$ is always well defined. On the other hand, consider the
derivation $\eta$ of degree $1$ in $L\widehat{\otimes}\land(t,dt)$ defined by

$\eta(v)=t\theta(v){\quad\text{for}\quad}v\in
V{\quad\text{and}\quad}\eta\bigl{(}L\otimes\land^{+}(t,dt)\bigr{)}=0$

Then, $D\eta$ is a cycle in ${\rm
Der}_{0}\bigl{(}L\widehat{\otimes}\land(t,dt)\bigr{)}$ and thus, $e^{D\eta}$
is an automorphism of $L\widehat{\otimes}\land(t,dt)$. Therefore, the map

$\Phi\colon L\to L\widehat{\otimes}\land(t,dt),\qquad\Phi(x)=e^{D\eta}(x),$

is a homotopy between $\varepsilon_{0}\circ\Psi=\operatorname{{\rm id}}_{L}$
and $\varepsilon_{1}\circ\Psi=e^{D\theta}$.

All of the above shows that ${\rm Der}^{\mathcal{G}}L$, and thus ${\rm
Der}^{\mathcal{G}}L\widetilde{\times}sL$, are sub dgl’s of ${\rm
Der}^{\mathcal{K}}L$ and ${\rm Der}^{\mathcal{K}}L\widetilde{\times}sL$
respectively, which are complete by Propositions 7.6 and 7.12. Hence, ${\rm
Der}^{\mathcal{G}}L$ and ${\rm Der}^{\mathcal{G}}L\widetilde{\times}sL$ are
also complete for the induced filtrations and all the objects in the statement
are cdgl’s.

We finish by checking that the image of the adjoint operator lies in ${\rm
Der}^{\mathcal{G}}L$. By definition, this amounts to say that, for each $x\in
L_{0}$, the homotopy class of $e^{\operatorname{{\rm ad}}_{x}}$ lives in
${\cal G}$. But this is trivial as, with the notation in Definition 2.3,
$[e^{\operatorname{{\rm ad}}_{x}}]=[e^{\operatorname{{\rm
ad}}_{x}}]{\scriptstyle\,\bullet\,}[\operatorname{{\rm id}}_{L}]\in{\cal G}$.
∎

Next, consider the restriction of (28) to

$0\to\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L),L)\longrightarrow\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L),L)\,\widetilde{\times}\,{\rm
Der}^{\mathcal{G}}L{\longrightarrow}{\rm Der}^{\mathcal{G}}L\to 0.$ (33)

Recall that both terms in the middle are sub dgl’s and $[\theta,f]=\theta\circ
f$ with $\theta\in{\rm Der}^{\mathcal{G}}L$ and $f\in\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L),L)$.

Fix the “universal” MC element $q\colon{\mathscr{C}}(L)\to L$ of
$\operatorname{\text{\rm Hom}}({\mathscr{C}}(L),L)$ defined in Remark 4.5.
With the notation in (13), $q=\overline{\operatorname{{\rm id}}}_{L}$ as it
represents the homotopy class of the identity in $\\{L,L\\}/H_{0}(L)$ which,
in turn, is identified with $\operatorname{{\rm id}}_{X_{\mathbb{Q}}}$.

Note that (33) is a cdgl fibration sequence, so is its realization,

$\langle\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L),L)\rangle\longrightarrow\langle\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L),L)\,\widetilde{\times}\,{\rm
Der}^{\mathcal{G}}L\rangle{\longrightarrow}\langle{\rm
Der}^{\mathcal{G}}L\rangle.$

We restrict this fibration, in which the base $\langle{\rm
Der}^{\mathcal{G}}L\rangle$ is connected, to the path component of the total
space $\langle\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L),L)\,\widetilde{\times}\,{\rm
Der}^{\mathcal{G}}L\rangle$ containing the $0$-simplex $q$. We then obtain
another fibration

$F\longrightarrow\langle\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L),L)\,\widetilde{\times}\,{\rm
Der}^{\mathcal{G}}L\rangle^{q}\longrightarrow\langle{\rm
Der}^{\mathcal{G}}L\rangle.$ (34)

for which we identify its fiber and total space in the next results.

######  Lemma 8.2.

$F={\amalg}_{[\varphi]\in{\cal G}/H_{0}(L)}\,\,\langle\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L),L)\rangle^{\overline{\varphi}}$.

###### Proof.

Observe that $F$ does not have to be connected as it is formed by all the path
components of $\langle\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L),L)\rangle$ lying in $\langle\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L),L)\,\widetilde{\times}\,{\rm
Der}^{\mathcal{G}}L\rangle^{q}$. That is, and with notation in Remark 4.5,

$F={\amalg}_{\varphi}\,\,\langle\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L),L)\rangle^{\overline{\varphi}},$

where $\varphi\colon L\to L$ runs through the homotopy classes of cdgl
morphisms such thatthe MC element $\overline{\varphi}=\varphi q$ of
$\operatorname{\text{\rm Hom}}({\mathscr{C}}(L),L)$ is gauge related to $q$
when considered as MC elements of $\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L),L)\,\widetilde{\times}\,{\rm Der}^{\mathcal{G}}L$. That
is, $\overline{\varphi}$ and $q$ represent the same element in
$\widetilde{\operatorname{{\rm MC}}}(\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L),L)\,\widetilde{\times}\,{\rm Der}^{\mathcal{G}}L)$.

Now, again in view of Remark 4.5, $\overline{\varphi}$ and $q$ are already
gauge related in $\operatorname{\text{\rm Hom}}({\mathscr{C}}(L),L)$ if and
only if $\varphi\sim\operatorname{{\rm id}}_{L}$. If this is not the case, but
still $\overline{\varphi}$ is gauge related to $q$ in $\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L),L)\,\widetilde{\times}\,{\rm Der}^{\mathcal{G}}L$,
there must be a derivation,

$\theta\in{\rm Der}^{\mathcal{G}}_{0}L{\quad\text{such
that}\quad}\theta\,{\mathscr{G}}\,q=\overline{\varphi}.$

That is,

$e^{\operatorname{{\rm
ad}}_{\theta}}(\overline{\varphi})-\frac{e^{\operatorname{{\rm
ad}}_{\theta}}-1}{\operatorname{{\rm ad}}_{\theta}}(D\theta)=q.$

Since $\theta$ is a cycle and $\overline{\varphi}=\varphi q$, this becomes,

$e^{\theta}=\varphi,{\quad\text{that is,}\quad}\varphi\in\operatorname{{\rm
aut}}_{\cal G}L.$

Conversely, given an automorphism $\varphi\in\operatorname{{\rm aut}}_{\cal
G}L$ consider $\theta=\log\varphi\in{\mathcal{D}er}_{0}L$ and one easily
checks, as above, that $\theta\,{\mathscr{G}}\,q=\overline{\varphi}$. ∎

######  Lemma 8.3.

$(\operatorname{\text{\rm Hom}}({\mathscr{C}}(L),L)\,\widetilde{\times}\,{\rm
Der}^{\mathcal{G}}L)^{q}\simeq L$.

###### Proof.

Consider the map,

$\gamma\colon L\longrightarrow(\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L),L)\,\widetilde{\times}\,{\rm
Der}^{\mathcal{G}}L,D_{q}),\quad\gamma(x)=-x+\operatorname{{\rm ad}}_{x},\quad
x\in L.$

Note that in view of (11), $\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L),L)\cong\operatorname{\text{\rm
Hom}}(\overline{\mathscr{C}}(L),L)\widetilde{\times}L$. Also, as shown in
Lemma 8.1, $\operatorname{{\rm ad}}_{x}\in{\rm Der}^{\mathcal{G}}L$ for any
$x\in L$. Thus, $\gamma$ is well defined.

On the other hand, recall from (11) and (33) that, in

$\operatorname{\text{\rm Hom}}({\mathscr{C}}(L),L)\,\widetilde{\times}\,{\rm
Der}^{\mathcal{G}}L\cong\operatorname{\text{\rm
Hom}}(\overline{\mathscr{C}}(L),L)\widetilde{\times}L\widetilde{\times}{\rm
Der}^{\mathcal{G}}L,$

the differential $D_{q}$ is defined as:

$\displaystyle D_{q}f=$ $\displaystyle Df+[q,f],\quad
f\in\operatorname{\text{\rm Hom}}({\mathscr{C}}(L),L),$ $\displaystyle
D_{q}x=$ $\displaystyle dx+[q,x]=dx-(-1)^{|x|}\operatorname{{\rm ad}}_{x}\circ
q,\quad x\in L,$ $\displaystyle D_{q}\theta=$ $\displaystyle
D\theta+[q,\theta]=D\theta-(-1)^{|\theta|}\theta\circ q,\quad\theta\in{\rm
Der}^{\mathcal{G}}.$

Thus, a simple computation shows that $\gamma$ commutes with differentials. We
then show that the component of $\gamma$ at $0$,

$\gamma\colon
L\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}(\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L),L)\,\widetilde{\times}\,{\rm
Der}^{\mathcal{G}}L,D_{q})^{0}=(\operatorname{\text{\rm
Hom}}({\mathscr{C}}(L),L)\,\widetilde{\times}\,{\rm Der}^{\mathcal{G}}L)^{q}$

is a quasi-isomorphism.

We recall again that $q=\overline{\operatorname{{\rm id}}}_{L}$ with the
notation in (13). In this particular case, the cdgl isomorphism in (16) and
the quasi-isomorphism of chain complexes in (19) become respectively,

$\Gamma\colon{s^{-1}}{\rm
Der}_{\alpha}({\mathscr{L}}{\mathscr{C}}(L),L)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}(\operatorname{\text{\rm
Hom}}(\overline{\mathscr{C}}(L),L),D_{q}){\quad\text{and}\quad}\alpha^{*}\colon{\rm
Der}L\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}{\rm
Der}_{\alpha}({\mathscr{L}}{\mathscr{C}}(L^{\prime}),L).$

The composition $\Gamma\circ s^{-1}\alpha^{*}$ is then a quasi-isomorphism of
chain complexes

$s^{-1}{\rm
Der}L\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}(\operatorname{\text{\rm
Hom}}(\overline{\mathscr{C}}(L),L),D_{q})$

which trivially extends, by the identity on $L\widetilde{\times}{\rm
Der}^{\mathcal{G}}$, to a quasi-isomorphism

$(s^{-1}{\rm Der}L\widetilde{\times}L\widetilde{\times}{\rm
Der}^{\mathcal{G}}L,\widetilde{D})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}(\operatorname{\text{\rm