# Commutative homotopical algebra embeds into non-commutative homotopical
algebra

Ricardo Campos  Ricardo Campos
Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS,
UPS, F-31062 Toulouse Cedex 9, France<EMAIL_ADDRESS>,
Dan Petersen  Dan Petersen
Matematiska Institutionen, Stockholms Universitet, 106 91 Stockholm, Sweden
<EMAIL_ADDRESS>,  Daniel Robert-Nicoud  Daniel Robert-Nicoud
<EMAIL_ADDRESS>and  Felix Wierstra  Felix Wierstra
Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science
Park 105-107, 1098 XG Amsterdam, Netherlands<EMAIL_ADDRESS>
###### Abstract.

Over a field of characteristic zero, we show that the forgetful functor from
the homotopy category of commutative dg algebras to the homotopy category of
dg associative algebras is faithful. In fact, the induced map of derived
mapping spaces gives an injection on all homotopy groups at any basepoint. We
prove similar results both for unital and non-unital algebras, and also Koszul
dually for the universal enveloping algebra functor from dg Lie algebras to dg
associative algebras. An important ingredient is a natural model for these
derived mapping spaces as Maurer–Cartan spaces of complete filtered dg Lie
algebras (or curved Lie algebras, in the unital case).

###### Key words and phrases:

Rational homotopy theory, universal enveloping algebras, deformation theory,
operads, Koszul duality

###### 2020 Mathematics Subject Classification:

Primary 18M70; secondary 13D10, 13D03, 16E40, 17B35, 55P62

##  1\. Commutativity and associativity

######  1.1.

The forgetful functor from commutative rings to associative algebras is a
canonical example of a _fully faithful functor_ : the notion of a homomorphism
$A\to A^{\prime}$ does not depend on whether $A$ and $A^{\prime}$ are
considered as commutative or just associative algebras.

######  1.2.

In the setting of homotopical algebra, the analogous statement is emphatically
false. Namely, when a forgetful functor $C\to D$ is fully faithful, this can
be informally understood as saying that the category $C$ consists of objects
of $D$ satisfying some _property_ , and the forgetful functor merely forgets
about this property. Examples could be the forgetful functor from finite sets
to all sets, or as just mentioned, the forgetful functor from commutative
rings to associative algebras. But in a homotopy-theoretic situation we should
not be asserting the _property_ that $xy=yx$; rather, we should supply a
_particular_ homotopy between $xy$ and $yx$ (and then higher homotopies,
etc.), and these homotopies are additional _structure_ on the algebra.

######  1.3.

To be more explicit, let us consider the two categories
$\mathrm{Ho}(\mathsf{CDGA})$ and $\mathrm{Ho}(\mathsf{DGA})$ of non-unital
commutative dg algebras, and non-unital dg algebras respectively, localized at
the class of quasi-isomorphisms. We always work over a field $\mathbf{K}$ of
characteristic zero in what follows. In this case both categories can be
modeled by the Hinich model structure [Hin97b], in which all objects are
fibrant. This means in particular that the set of morphisms $A\to A^{\prime}$
in the homotopy category can be computed as the set of homotopy classes of
morphisms $\mathrm{Q}A\to A^{\prime}$, where $\mathrm{Q}A$ denotes a cofibrant
replacement of $A$. Now on the one hand a cofibrant object of $\mathsf{CDGA}$
very rarely remains cofibrant in $\mathsf{DGA}$; on the other hand, the
notions of homotopy in the two categories are also different. The consequence
is that if $A,A^{\prime}$ are commutative dg algebras, then morphisms $A\to
A^{\prime}$ in $\mathrm{Ho}(\mathsf{DGA})$ are given by an a priori larger set
of maps modulo an a priori coarser notion of homotopy than morphisms in
$\mathrm{Ho}(\mathsf{CDGA})$. We denote similarly by
$\mathrm{Ho}(\mathsf{CDGA}^{u})$ and $\mathrm{Ho}(\mathsf{DGA}^{u})$ the
homotopy categories of unital commutative dg algebras and unital dg algebras,
respectively. Everything we have said in this paragraph remains true in the
unital case, too.

######  1.4.

In fact, the functor
$\mathrm{Ho}(\mathsf{DGA}^{u})\to\mathrm{Ho}(\mathsf{DGA}^{u})$ is _not_ full.
Take for example $A$ to be the polynomial ring $\mathbf{K}[x,y]$, considered
as a commutative dg algebra concentrated in degree $0$. It is cofibrant in the
category $\mathsf{CDGA}^{u}$, but not in $\mathsf{DGA}^{u}$. A cofibrant
replacement in $\mathsf{DGA}^{u}$ is given by non-commutative polynomial ring
$\mathbf{K}\langle x,y,z\rangle$ with $|z|=1$ and $dz=xy-yx$. Therefore, for
any $A^{\prime}$ in $\mathsf{CDGA}^{u}$, we have

$\hom_{\mathrm{Ho}(\mathsf{CDGA}^{u})}(A,A^{\prime})\cong
H_{0}(A^{\prime})\times H_{0}(A^{\prime}),$

but

$\hom_{\mathrm{Ho}(\mathsf{DGA}^{u})}(A,A^{\prime})\cong
H_{0}(A^{\prime})\times H_{0}(A^{\prime})\times H_{1}(A^{\prime}).$

The same example would work just as well if we had worked with non-unital
algebras. Our main result is the following.

######  1.5 Theorem.

Over a base field of characteristic zero, the forgetful functors

$\mathrm{Ho}(\mathsf{CDGA})\longrightarrow\mathrm{Ho}(\mathsf{DGA})\qquad\text{and}\qquad\mathrm{Ho}(\mathsf{CDGA}^{u})\longrightarrow\mathrm{Ho}(\mathsf{DGA}^{u})$

are faithful.

######  1.6.

The argument we will use for this result will in fact prove a stronger result
at the level of simplicial mapping spaces. We denote by
$\mathbf{R}\mathrm{map}_{\mathcal{C}}$ the derived mapping space of a model
category $\mathcal{C}$.

######  1.7 Theorem.

Over a base field of characteristic zero, the natural maps

$\mathbf{R}\mathrm{map}_{\mathsf{CDGA}}(A,A^{\prime})\longrightarrow\mathbf{R}\mathrm{map}_{\mathsf{DGA}}(A,A^{\prime})$

and

$\mathbf{R}\mathrm{map}_{\mathsf{CDGA}^{u}}(A,A^{\prime})\longrightarrow\mathbf{R}\mathrm{map}_{\mathsf{DGA}^{u}}(A,A^{\prime})$

induced by the forgetful functors give rise to an injection on all homotopy
groups at any basepoint, for any $A,A^{\prime}\in\mathsf{CDGA}$, resp.
$\mathsf{CDGA}^{u}$.

######  1.8.

For the homotopy groups $\pi_{n}$ with $n>0$ this recovers a theorem of Amrani
[Amr15], so only the result about $\pi_{0}$ (i.e. 1.5) is new. The assertion
about $\pi_{0}$ is significantly trickier to prove.

######  1.9.

Although the structure of the proof is very similar in both the unital and
non-unital case, the argument is technically simpler when dealing with not
necessarily unital algebras. For the rest of the introduction we focus on this
case.

######  1.10 Remark.

The set of homotopy classes of maps $A\to A^{\prime}$ in
$\mathrm{Ho}(\mathsf{DGA})$ can also be understood in terms of
$\mathsf{A}_{\infty}$-morphisms. This has the advantage that one does not have
to cofibrantly replace $A$. Namely,
$\hom_{\mathrm{Ho}(\mathsf{DGA})}(A,A^{\prime})$ can be identified with the
set of $\infty$-morphisms of $\mathsf{A}_{\infty}$-algebras from $A$ to
$A^{\prime}$, modulo $\mathsf{A}_{\infty}$-homotopy. Similarly
$\hom_{\mathrm{Ho}(\mathsf{CDGA})}(A,A^{\prime})$ is the set of
$\mathsf{C}_{\infty}$-morphisms from $A$ to $A^{\prime}$ modulo
$\mathsf{C}_{\infty}$-homotopy. Thus, 1.5 can be reformulated as saying that
if two $\mathsf{C}_{\infty}$-morphisms are $\mathsf{A}_{\infty}$-homotopic,
then they are also $\mathsf{C}_{\infty}$-homotopic. Such a result seems
potentially quite useful since the notion of $\mathsf{A}_{\infty}$-homotopy is
much simpler than the notion of $\mathsf{C}_{\infty}$-homotopy, the reason
being that the $\mathsf{A}_{\infty}$-operad is non-symmetric. Indeed, suppose
that $f,g:A\to A^{\prime}$ are $\mathsf{A}_{\infty}$-morphisms between
$\mathsf{A}_{\infty}$-algebras, and let

$\mathrm{B}f,\mathrm{B}g:\mathrm{B}A\to\mathrm{B}A^{\prime}$

be the corresponding maps of coassociative coalgebras. By a homotopy between
$f$ and $g$ we mean a $(\mathrm{B}f,\mathrm{B}g)$-coderivation
$H:\mathrm{B}A\to\mathrm{B}A^{\prime}$ such that $\partial
H=\mathrm{B}f-\mathrm{B}g$. We refer to [LH03, Section 1.3.4] for more
details. The notion of homotopy of $\infty$-morphisms over a symmetric operad
is more complicated and is discussed in detail in [DP16].

######  1.11.

This paper may be considered as a sequel, or coda, to our previous paper
[CPRNW19]. We say that a functor $F\colon C\to D$ is _injective on isomorphism
classes_ if $F(X)\cong F(Y)$ implies $X\cong Y$. In Theorem A of [CPRNW19] we
proved the result that
$\mathrm{Ho}(\mathsf{CDGA})\to\mathrm{Ho}(\mathsf{DGA})$ is injective on
isomorphism classes over a field of characteristic zero (and similarly in the
unital case). This result and 1.5 are morally similar in that both try to
probe the failure of the forgetful functor to be fully faithful (as clearly a
fully faithful functor is injective on isomorphism classes). Putting the two
papers together we see that the homotopy theory of commutative dg algebras
embeds faithfully into the non-commutative homotopy theory in a precise sense.
What is also striking is that the essential structure of the two arguments is
extremely similar.

######  1.12.

Let us try to bring out this similarity. The proof of [CPRNW19, Theorem A]
quickly reduces to the following statement: given two
$\mathsf{C}_{\infty}$-algebra structures on the same chain complex, and an
$\mathsf{A}_{\infty}$-isotopy between them, there must also exist a
$\mathsf{C}_{\infty}$-isotopy between them. Similarly 1.5 reduces to the
statement that given two $\mathsf{C}_{\infty}$-morphisms between
$\mathsf{C}_{\infty}$-algebras and an $\mathsf{A}_{\infty}$-homotopy between
them, there also exists a $\mathsf{C}_{\infty}$-homotopy between them. To
treat both situations on the same footing, we say simply that we are given two
$\mathsf{C}_{\infty}$-objects $\xi$ and $\xi^{\prime}$ related by an
$\mathsf{A}_{\infty}$-gauge, and we want to find a $\mathsf{C}_{\infty}$-gauge
from $\xi$ to $\xi^{\prime}$. More precisely, in both cases the problems are
best described in terms of deformation problems governed by dg Lie algebras.
In our previous paper we considered two dg Lie algebras whose Maurer–Cartan
elements correspond to $\mathsf{A}_{\infty}$-algebra structures (resp.
$\mathsf{C}_{\infty}$-algebra structures) on some fixed chain complex, such
that gauges correspond to $\mathsf{A}_{\infty}$-isotopies (resp.
$\mathsf{C}_{\infty}$-isotopies). Similarly, in the current paper we have two
dg Lie algebras that describe the set of $\mathsf{A}_{\infty}$-morphisms
(resp. $\mathsf{C}_{\infty}$-morphisms) between two fixed such algebras, with
gauges now corresponding to $\mathsf{A}_{\infty}$-homotopies (resp.
$\mathsf{C}_{\infty}$-homotopies). In both cases there is an inclusion of the
Lie algebra encoding $\mathsf{C}_{\infty}$-structures into the Lie algebra
encoding $\mathsf{A}_{\infty}$-structures.

######  1.13.

Once the problem is formulated in terms of $\mathsf{A}_{\infty}$\- and
$\mathsf{C}_{\infty}$-structures, it is natural to try to work one coefficient
at a time, which amounts to noting that the dg Lie algebras in question come
with natural filtrations (or gradings) by arity of the $\mathsf{A}_{\infty}$\-
and $\mathsf{C}_{\infty}$-operations. Let $\tau_{\leq n}\xi$ be the truncation
of $\xi$ to a $\mathsf{C}_{n}$-object, discarding all operations of arity
bigger than $n$. By an inductive obstruction theory argument one can in both
cases show that $\tau_{\leq n}\xi$ and $\tau_{\leq n}\xi^{\prime}$ are
$\mathsf{C}_{n}$-gauge equivalent for every $n$. Indeed, if we have an
equivalence between $\tau_{\leq n}\xi$ and $\tau_{\leq n}\xi^{\prime}$, then
there is a well defined obstruction class for finding a gauge equivalence
between $\tau_{\leq(n+1)}\xi$ and $\tau_{\leq(n+1)}\xi^{\prime}$. More
specifically, there are _two_ classes $\mathrm{o}_{n+1}^{\mathsf{A}}\in
H^{\mathsf{A}}$ and $\mathrm{o}_{n+1}^{\mathsf{C}}\in H^{\mathsf{C}}$, which
are obstructions for finding an $\mathsf{A}_{n+1}$-gauge and a
$\mathsf{C}_{n+1}$-gauge, respectively, both lying in certain cohomology
groups. Now the great news in both situations is that $H^{\mathsf{C}}$ is a
direct summand of $H^{\mathsf{A}}$, and $\mathrm{o}_{n+1}^{\mathsf{C}}$ is the
projection of $\mathrm{o}_{n+1}^{\mathsf{A}}$ onto $H^{\mathsf{C}}$. But
$\mathrm{o}_{n+1}^{\mathsf{A}}$ must vanish, since we had an
$\mathsf{A}_{\infty}$-gauge from $\xi$ to $\xi^{\prime}$ by assumption, and
then the obstruction $\mathrm{o}_{n+1}^{\mathsf{C}}$ vanishes, too. (In
[CPRNW19] the groups $H^{\mathsf{C}}$ and $H^{\mathsf{A}}$ are Harrison and
Hochschild cohomology groups, respectively, and Harrison cohomology is a
direct summand of Hochschild cohomology by a result of Barr [Bar68].) But at
this point one gets stuck: there is no natural way to deduce from the fact
that $\tau_{\leq n}\xi$ is $\mathsf{C}_{n}$-gauge equivalent to $\tau_{\leq
n}\xi^{\prime}$ for all $n$, that $\xi$ and $\xi^{\prime}$ are
$\mathsf{C}_{\infty}$-gauge equivalent. Although we can find a gauge at each
step, this does not imply that transfinite composition of all these gauges is
well defined or convergent.

######  1.14.

To make the argument work one needs to be more careful in the construction.
Instead of saying at each step that a certain obstruction class vanishes in
cohomology, so that _some_ gauge exists, we need to be able to choose a
_specific_ gauge, which is small enough in a suitable sense, so that the
sequence of gauges converges in the limit. To accomplish this, we pass from
cohomology to cochains and consider the natural complete filtrations on
everything in sight. The difficulties involved in making this strategy work
can be bundled up into a criterion which we proved in [CPRNW19].

######  1.15 Theorem ([CPRNW19, Theorem 1.7]).

Let $i\colon\mathfrak{h}\to\mathfrak{g}$ be a morphism of complete filtered dg
Lie algebras over a field of characteristic zero. Suppose that there exists
$r\colon\mathfrak{g}\to\mathfrak{h}$ such that $r\circ
i=\mathrm{id}_{\mathfrak{h}}$, and $r$ is a morphism of filtered
$\mathfrak{h}$-modules. Then the map
$\mathrm{MC}_{\bullet}(\mathfrak{h})\longrightarrow\mathrm{MC}_{\bullet}(\mathfrak{g})$
induces an injection on all homotopy groups for all basepoints. In particular,
the set of gauge equivalence classes of Maurer–Cartan elements of
$\mathfrak{h}$ injects into the gauge equivalence classes of Maurer–Cartan
elements of $\mathfrak{g}$.

######  1.16.

The assumption that $\mathfrak{h}\to\mathfrak{g}$ admits a retraction as
$\mathfrak{h}$-modules immediately implies that for any Maurer–Cartan element
$\alpha$ of $\mathfrak{h}$, there is a split injection in homology
$H_{\bullet}(\mathfrak{h}^{\alpha})\to H_{\bullet}(\mathfrak{g}^{\alpha})$,
where $(-)^{\alpha}$ denotes the effect of twisting the differential by the
Maurer–Cartan element $\alpha$. In terms of the heuristic outline of §1.13,
this corresponds to the fact that $H^{\mathsf{C}}$ is a direct summand of
$H^{\mathsf{A}}$, and that all obstruction classes vanish. Having the
retraction be filtration-preserving is what gives the extra leverage and
allows passage to the limit. We can now also justify the remark made in §1.8,
that the assertion about higher homotopy groups is easier than the assertion
about $\pi_{0}$ in 1.7. Indeed, for $n>0$ a theorem of Berglund [Ber15]
furnishes an identification
$\pi_{n}(\mathrm{MC}_{\bullet}(\mathfrak{h}),\alpha)\cong
H_{n-1}(\mathfrak{h}^{\alpha})$, so for the injection on higher homotopy
groups it is not necessary to suppose compatibility of the retraction with the
filtrations, and the convergence problems are irrelevant.

######  1.17.

In order to apply 1.15 we need suitable models for mapping spaces in
$\mathsf{CDGA}$ and $\mathsf{DGA}$ in terms of Maurer–Cartan spaces of
complete filtered dg Lie algebras. Before stating the result let us recall
some notation and preliminaries. If $A$ is a dg algebra, we denote by
$\mathrm{B}A$ its bar construction. If furthermore $A$ is commutative, we
write $\mathrm{B}_{\mathsf{Com}}A$ for its “commutative” bar construction,
i.e. the Koszul dual Lie coalgebra of $A$. Recall also that the space of
linear maps from a dg Lie coalgebra to a commutative dg algebra is itself a dg
Lie algebra. (This is the “linear dual” of the perhaps more familiar fact that
the tensor product of a Lie algebra and a commutative ring is a Lie algebra.)
Similarly the space of linear maps from a dg coalgebra to a dg algebra is
itself a dg algebra, and in particular a dg Lie algebra.

######  1.18 Proposition.

Let $A$ and $A^{\prime}$ be dg algebras over a field $\mathbf{K}$ of
characteristic zero. Then:

  * $\diamond$

$\mathbf{R}\mathrm{map}_{\mathsf{DGA}}(A,A^{\prime})\simeq\mathrm{MC}_{\bullet}(\hom_{\mathbf{K}}(\mathrm{B}A,A^{\prime}))$.

  * $\diamond$

If $A$ and $A^{\prime}$ are commutative, then
$\mathbf{R}\mathrm{map}_{\mathsf{CDGA}}(A,A^{\prime})\simeq\mathrm{MC}_{\bullet}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Com}}A,A^{\prime}))$.

Here, both $\hom_{\mathbf{K}}(\mathrm{B}A,A^{\prime})$ and
$\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Com}}A,A^{\prime})$ are considered as
complete filtered dg Lie algebras, with the filtrations induced by the
coradical filtrations of $\mathrm{B}A$ and $\mathrm{B}_{\mathsf{Com}}A$
respectively. In the commutative case, the projection
$\mathrm{B}A\to\mathrm{B}_{\mathsf{Com}}A$ induces the natural map
$\mathbf{R}\mathrm{map}_{\mathsf{DGA}}(A,A^{\prime})\longrightarrow\mathbf{R}\mathrm{map}_{\mathsf{CDGA}}(A,A^{\prime}).$

######  1.19 Remark.

Versions of 1.18 are already in the literature, see in particular Berglund
[Ber15, Proposition 6.1]. But note the lack of boundedness or finite-
dimensionality assumptions here.

######  1.20 Remark.

We expect the first equivalence
$\mathbf{R}\mathrm{map}_{\mathsf{DGA}}(A,A^{\prime})\simeq\mathrm{MC}_{\bullet}(\hom_{\mathbf{K}}(\mathrm{B}A,A^{\prime}))$
to remain valid over fields of arbitrary characteristic, cf. [dKW21].

######  1.21.

The proof of 1.7 is now nearly complete: by 1.15 and 1.18, all that is left to
do is to construct a certain left inverse to the map

$\hom_{\mathbf{K}}\left(\mathrm{B}_{\mathsf{Com}}A,A^{\prime}\right)\longrightarrow\hom_{\mathbf{K}}\left(\mathrm{B}A,A^{\prime}\right).$

We will obtain it as follows. We can identify $\mathrm{B}A$ with the
_universal enveloping conilpotent coalgebra_ of $\mathrm{B}_{\mathsf{Com}}A$.
By a dual version of the Poincaré–Birkhoff–Witt theorem, we can identify
$\mathrm{B}A\cong\mathrm{Sym}^{c}(\mathrm{B}_{\mathsf{Com}}A)$ as
$\mathrm{B}_{\mathsf{Com}}A$-comodules. From this, one can in particular
construct a splitting of the projection
$\mathrm{B}A\to\mathrm{B}_{\mathsf{Com}}A$.

######  1.22.

Again we see that the arguments of this paper follow closely those of
[CPRNW19], where it also turned out that the retraction needed to apply 1.15
was obtained from the Poincaré–Birkhoff–Witt theorem.

######  1.23.

As indicated in §1.9, there are additional complications in the unital case:
namely, when the algebras have units, we need to replace dg Lie algebras with
_curved_ Lie algebras throughout, as we explain in Section 5.

######  1.24.

In our previous paper [CPRNW19] we did not only study the forgetful functor
$\mathsf{CDGA}\to\mathsf{DGA}$. A major portion of the paper concerned instead
the _Koszul dual_ of this functor. Since associative algebras are Koszul self-
dual, and commutative algebras are Koszul dual to Lie algebras, this Koszul
dual is the universal enveloping algebra functor
$\mathsf{DGLA}\to\mathsf{DGA}$. In [CPRNW19] the results we obtained about the
universal enveloping algebra functor were slightly weaker than those for the
forgetful functor. Indeed, we could prove that
$\mathrm{Ho}(\mathsf{CDGA})\to\mathrm{Ho}(\mathsf{DGA})$ is injective on
isomorphism classes, but we could only prove that the universal enveloping
algebra functor $\mathrm{Ho}(\mathsf{DGLA})\to\mathrm{Ho}(\mathsf{DGA})$ is
injective on isomorphism classes up to _derived completion_ of the dg Lie
algebras involved. By contrast, in this paper we are able to prove a “Koszul
dual” of 1.5 with no additional completeness hypotheses.

######  1.25 Theorem.

Let $\mathfrak{g},\mathfrak{h}$ be two dg Lie algebras over a field of
characteristic zero. For any choice of basepoint, the natural map

$\mathbf{R}\mathrm{map}_{\mathsf{DGLA}}(\mathfrak{g},\mathfrak{h})\longrightarrow\mathbf{R}\mathrm{map}_{\mathsf{DGA}}(U\mathfrak{g},U\mathfrak{h})$

induces an injection on all homotopy groups.

######  1.26.

This theorem is less striking than its commutative and associative counterpart
1.5 in that there is no longer a large difference in behavior between the
classical (non-dg) and the homotopical situation. Indeed, consider the
classical universal enveloping algebra functor from Lie algebras to
associative algebras over a field of characteristic zero. This functor is
clearly not full (consider e.g. $\mathfrak{g}=\mathfrak{h}$ a free Lie
algebra). However, the universal enveloping algebra functor _does_ turn out to
be faithful, but it is not automatic: it is a consequence of the
Poincaré–Birkhoff–Witt theorem. This is also exactly what happens in the
homotopical situation.

######  1.27.

Some experts will perhaps not find anything surprising beyond this point: what
remains of the paper is mostly to fill in details in the proof of 1.7 sketched
in §1.21. We first explain the dual form of the PBW theorem mentioned in §1.21
in Section 2, and verify that the retraction obtained from dual PBW satisfies
the conditions of 1.15 in Section 3. After that we give the proof of 1.18 in
Section 4. In Section 5 we then treat the unital case, proving in particular a
unital version of 1.18 involving curved Lie algebras. Finally, in Section 6 we
consider the Koszul dual situation which concerns the universal enveloping
algebra functor from dg Lie algebras to dg algebras.

######  1.28.

We mostly follow the same conventions as in [CPRNW19]. We always work over a
field $\mathbf{K}$ of characteristic $0$. All the filtrations that we consider
are concentrated in positive filtration degree.

######  1.29.

As in [CPRNW19] we systematically use the language of algebraic operads, and
the results are obtained by studying the interplay between the operads
$\mathsf{Lie}$, $\mathsf{Ass}$ and $\mathsf{Com}$. In fact, the only property
of these operads that we end up using (besides their Koszulness) is that the
natural morphism $\mathsf{Lie}\to\mathsf{Ass}$ admits a left inverse in the
category of infinitesimal bimodules over the operad $\mathsf{Lie}$. The
methods of this paper extend to show the following more general theorem, whose
proof we leave to the interested reader:

######  1.30 Theorem.

Let $f\colon\mathsf{P}\to\mathsf{Q}$ be a morphism of binary Koszul operads in
characteristic zero with $\mathsf{P}(n)$ and $\mathsf{Q}(n)$ finite
dimensional for all $n$. Let $\mathsf{Q}^{!}\to\mathsf{P}^{!}$ be the induced
morphism between the Koszul dual operads. Suppose that there exists a morphism
of infinitesimal $\mathsf{P}$-bimodules $s\colon\mathsf{Q}\to\mathsf{P}$ such
that $s\circ f=\mathrm{id}_{\mathsf{P}}$. Then both the forgetful functor from
$\mathsf{Q}^{!}$-algebras to $\mathsf{P}^{!}$-algebras and the derived
operadic pushforward functor $\mathbf{L}f_{!}$ from $\mathsf{P}$-algebras to
$\mathsf{Q}$-algebras have the property that they induce injections on all
homotopy groups of derived mapping spaces for all basepoints.

######  1.31.

Acknowledgments. We would like to thank Victor Roca i Lucio for discussions
concerning his thesis and Joost Nuiten for discussions about curved Lie
algebras. The first author was supported by the grant ANR-20-CE40-0016
HighAGT, the second author was supported by the grant ERC-2017-STG 759082 and
a Wallenberg Academy Fellowship, and the fourth author was supported by the
Dutch Research Organization (NWO) grant number VI.Veni.202.046.

##  2\. On the Poincaré–Birkhoff–Witt theorem

######  2.1.

In this section we will explain some generalities around the
Poincaré–Birkhoff–Witt theorem. Our perspective will be that the statement of
the Poincaré–Birkhoff–Witt theorem is that there is an isomorphism of
infinitesimal $\mathsf{Lie}$-bimodules

($\dagger$) $\mathsf{Ass}\cong\mathsf{Com}\circ\mathsf{Lie}.$

We may also write the right hand side as $\mathrm{Sym}(\mathsf{Lie})$, which
makes the infinitesimal $\mathsf{Lie}$-bimodule structure more transparent
(cf. [CPRNW19, Remark 2.9]). In [CPRNW19, Section 2] we explained how to
deduce ($\dagger$ ‣ 2.1) from (an abstract form of) the usual PBW theorem,
i.e. that for a Lie algebra $\mathfrak{g}$ there is an isomorphism of
$\mathfrak{g}$-modules $U\mathfrak{g}\cong\mathrm{Sym}(\mathfrak{g})$. We will
now explain how to deduce the standard form of the PBW theorem from ($\dagger$
‣ 2.1), and then explain that the exact same reasoning also implies the
following dual form of the PBW theorem.

######  2.2 Proposition.

Let $L$ be a conilpotent dg Lie coalgebra, and denote by $U^{c}L$ its
universal enveloping conilpotent coalgebra. There is an isomorphism of
$L$-comodules $U^{c}(L)\cong\mathrm{Sym}^{c}(L)$.

######  2.3.

The material in this section is known to experts, but we do not know a
reference which states it explicitly, and the perspective is perhaps somewhat
novel. The original dual PBW theorem is due to Michaelis [Mic80, Mic85], who
considers more generally any _locally finite_ Lie coalgebra, not necessarily
conilpotent. The fact that the isomorphism ($\dagger$ ‣ 2.1) can be promoted
to an isomorphism of infinitesimal bimodules seems to have been first noticed
by Griffin [Gri14]. Other closely related works are Dotsenko–Tamaroff [DT21]
and Fresse [Fre09, Section 10.2].

######  2.4.

Let $\mathsf{P}\to\mathsf{Q}$ be a morphism of operads. The restriction
functor from $\mathsf{Q}$-algebras to $\mathsf{P}$-algebras has a left
adjoint, which we denote $\mathscr{U}_{\mathsf{Q}}$. In the case of
$\mathsf{Lie}\to\mathsf{Ass}$, the functor $\mathscr{U}_{\mathsf{Ass}}$ is the
usual universal enveloping algebra functor.

######  2.5.

In general, we can identify algebras over an operad $\mathsf{P}$ with left
$\mathsf{P}$-modules concentrated in arity $0$. The functor
$\mathscr{U}_{\mathsf{Q}}$ makes sense for left $\mathsf{P}$-modules as well,
and in a sense it is more naturally considered as such. If $V$ is a left
$\mathsf{P}$-module, then $\mathscr{U}_{\mathsf{Q}}V$ is the relative
composition product

($\ast$)
$\mathscr{U}_{\mathsf{Q}}V\coloneqq\mathsf{Q}\circ_{\mathsf{P}}V=\mathrm{coeq}(\mathsf{Q}\circ\mathsf{P}\circ
V\rightrightarrows\mathsf{Q}\circ V),$

i.e. the coequalizer of two parallel arrows, one of which is given by the left
$\mathsf{P}$-module structure on $V$ and the other by the right
$\mathsf{P}$-module structure on $\mathsf{Q}$ induced by the morphism
$\mathsf{P}\to\mathsf{Q}$.

######  2.6.

We may now reverse the perspective by fixing $V$ and taking ($\ast$ ‣ 2.5) to
define a functor of $\mathsf{Q}$. We immediately notice that ($\ast$ ‣ 2.5)
makes no reference to the morphism $\mathsf{P}\to\mathsf{Q}$; it only involves
the right $\mathsf{P}$-module structure on $\mathsf{Q}$. For simplicity, from
now on we will now fix $V=A$ to be a $\mathsf{P}$-algebra. Then, in differing
levels of generality,111We remark that the fact that $\mathsf{P}$-bimodules
are neither more nor less general than infinitesimal $\mathsf{P}$-bimodules is
reflected in the fact that $\mathsf{P}$-algebras are neither more nor less
general than $A$-modules. we may consider ($\ast$ ‣ 2.5) as defining either:

  * $\diamond$

a functor $\mathsf{Q}\longmapsto\mathscr{U}_{\mathsf{Q}}A$ from right
$\mathsf{P}$-modules to chain complexes,

  * $\diamond$

a functor $\mathsf{Q}\longmapsto\mathscr{U}_{\mathsf{Q}}A$ from
$\mathsf{P}$-bimodules to $\mathsf{P}$-algebras, or

  * $\diamond$

a functor $\mathsf{Q}\longmapsto\mathscr{U}_{\mathsf{Q}}A$ from infinitesimal
$\mathsf{P}$-bimodules to $A$-modules.

The last bullet point is the least obvious here. An infinitesimal left
$\mathsf{P}$-module structure on $\mathsf{Q}$ consists of a map

$\mathsf{P}\circ(\mathsf{P};\mathsf{Q})\to\mathsf{Q}$

satisfying a suitable associativity condition, with notation as in [LV12,
Section 6.1]. Applying the relative composition product
$(-)\circ_{\mathsf{P}}A$ to both sides, this yields a map

$\mathsf{P}\circ(A;\mathscr{U}_{\mathsf{Q}}A)\longrightarrow\mathscr{U}_{\mathsf{Q}}A$

defining an $A$-module structure on $\mathscr{U}_{\mathsf{Q}}A$.

######  2.7.

Specializing this to $\mathsf{Lie}\to\mathsf{Ass}$ and using ($\dagger$ ‣
2.1), we have that for any dg Lie algebra $\mathfrak{g}$

$U\mathfrak{g}=\mathscr{U}_{\mathsf{Ass}}\mathfrak{g}\cong\mathscr{U}_{\mathsf{Com}\circ\mathsf{Lie}}\mathfrak{g}=\mathsf{Com}\circ\mathsf{Lie}\circ_{\mathsf{Lie}}\mathfrak{g}\cong\mathsf{Com}\circ\mathfrak{g}=\mathrm{Sym}(\mathfrak{g})$

as $\mathfrak{g}$-modules, recovering the classical PBW theorem.

######  2.8.

We can now dualize the whole discussion. All cooperads, coalgebras, and
comodules are implicitly assumed to be conilpotent in what follows. Suppose
that $\mathsf{E}\to\mathsf{D}$ is a morphism of cooperads. In this situation,
there is a natural restriction functor from $\mathsf{E}$-coalgebras to
$\mathsf{D}$-coalgebras. Dually to the case of operads, it has a _right_
adjoint, which we will denote $\mathscr{U}^{c}_{\mathsf{E}}$. A
$\mathsf{D}$-coalgebra is a left $\mathsf{D}$-comodule concentrated in arity
$0$, and we may define $\mathscr{U}^{c}_{\mathsf{E}}M$ more generally for any
left $\mathsf{D}$-comodule $M$ by the formula

$\mathscr{U}^{c}_{\mathsf{E}}M\coloneqq\mathsf{E}\circ^{\mathsf{D}}M,$

where the right-hand side is the equalizer of the diagram

$\mathsf{E}\circ M\rightrightarrows\mathsf{E}\circ\mathsf{D}\circ M$

dual to ($\ast$ ‣ 2.5). Again we fix a $\mathsf{D}$-coalgebra $C$, and
consider the functor $\mathsf{E}\mapsto\mathscr{U}^{c}_{\mathsf{E}}C$. This
may be considered as a functor from right $\mathsf{D}$-comodules to chain
complexes, from $\mathsf{D}$-bicomodules to $\mathsf{D}$-coalgebras, or from
infinitesimal $\mathsf{D}$-bicomodules to $C$-comodules.

######  2.9.

In the special case of the morphism $\mathsf{coAss}\to\mathsf{coLie}$, the
functor $\mathscr{U}^{c}_{\mathsf{coAss}}=U^{c}$ is the universal enveloping
conilpotent coalgebra. Taking the linear dual in ($\dagger$ ‣ 2.1) shows that
there is an isomorphism of infinitesimal $\mathsf{coLie}$-comodules

($\dagger\dagger$) $\mathsf{coAss}\cong\mathsf{coCom}\circ\mathsf{coLie}.$

Dualizing the argument of §2.7 in the evident way proves 2.2.

##  3\. Construction of a retraction

######  3.1.

We write $\hom_{\mathbf{K}}$ for the internal hom of chain complexes over
$\mathbf{K}$. If $C$ is a dg coalgebra and $A$ is a dg algebra, then
$\hom_{\mathbf{K}}(C,A)$ has a natural dg algebra structure defined by the
_convolution product_. If we write the coproduct of $C$ using Sweedler
notation

$\Delta(x)=\sum x_{(1)}\otimes x_{(2)},$

then the convolution product of $f,g\in\hom_{\mathbf{K}}(C,A)$ is defined as

$(f\ast g)(x)\coloneqq\sum f(x_{(1)})g(x_{(2)})$

for $x\in C$. This is readily verified to be associative. By the same
formulas, if $M$ is a right $C$-comodule and $N$ is a right $A$-module, then
$\hom_{\mathbf{K}}(M,N)$ is functorially a right $\hom_{K}(C,A)$-module.

######  3.2.

In a similar way, if $L$ is a dg Lie coalgebra and $A$ is a commutative dg
algebra, then $\hom_{\mathbf{K}}(L,A)$ is a dg Lie algebra. Again using
Sweedler’s notation for the Lie cobracket of $L$, the bracket of
$f,g\in\hom_{\mathbf{K}}(L,A)$ is given by

$[f,g](x)\coloneqq\sum f(x_{(1)})g(x_{(2)})$

for $x\in L$. If $M$ is an $L$-comodule and $N$ an $A$-module, then
$\hom_{\mathbf{K}}(M,N)$ is functorially a $\hom_{\mathbf{K}}(L,A)$-module.

######  3.3.

If $\mathsf{D}$ is a reduced cooperad, then any $\mathsf{D}$-coalgebra comes
equipped with a canonical filtration as follows. The cooperad $\mathsf{D}$ has
a natural increasing filtration by arity

$0=\tau_{\leq 0}\mathsf{D}\xhookrightarrow{\quad}\tau_{\leq
1}\mathsf{D}\xhookrightarrow{\quad}\tau_{\leq
2}\mathsf{D}\xhookrightarrow{\quad}\cdots$

where $\tau_{\leq k}\mathsf{D}$ denotes the truncation of $\mathsf{D}$ at
arity $k$, i.e. the cooperad obtained by setting all operations of arity above
$k$ to zero. Each of the truncations is a $\mathsf{D}$-cobimodule in a natural
way. If $C$ is a $\mathsf{D}$-coalgebra, then
$C\cong\mathsf{D}\circ^{\mathsf{D}}C$ inherits a filtration given by
$\tau_{\leq k}\mathsf{D}\circ^{\mathsf{D}}C$ in filtration degree $k-1$,
called the _coradical filtration_ of $C$. The $\mathsf{D}$-coalgebra $C$ is
said to be _conilpotent_ if this filtration is exhaustive. With this
description of the coradical filtration, it becomes obvious that if
$\mathsf{E}\to\mathsf{D}$ is a morphism of reduced cooperads, then the natural
map

$\mathscr{U}^{c}_{\mathsf{E}}C=\mathsf{E}\circ^{\mathsf{D}}C\longrightarrow\mathsf{D}\circ^{\mathsf{D}}C\cong
C$

is compatible with the respective coradical filtrations, where
$\mathscr{U}^{c}_{\mathsf{E}}C$ is endowed with the filtration induced by the
arity filtration of $\mathsf{E}$. More generally, if
$\mathsf{E}_{1}\to\mathsf{E}_{2}$ is a morphism of right
$\mathsf{D}$-comodules, $\mathsf{D}$-bicomodules, or infinitesimal
$\mathsf{D}$-bicomodules, then the induced morphism

$\mathscr{U}^{c}_{\mathsf{E}_{1}}C\longrightarrow\mathscr{U}^{c}_{\mathsf{E}_{2}}C$

is compatible with the filtrations induced by the arity filtrations of
$\mathsf{E}_{1}$ and $\mathsf{E}_{2}$.

######  3.4 Proposition.

Let $L$ be a conilpotent dg Lie coalgebra, let $U^{c}L$ be its universal
enveloping conilpotent coalgebra, and let $A$ a commutative dg algebra. We
consider the map of complete filtered dg Lie algebras

$\hom_{\mathbf{K}}(L,A)\longrightarrow\hom_{\mathbf{K}}(U^{c}L,A)$

induced by the projection $U^{c}L\to L$, where both sides are filtered by the
respective coradical filtrations. This map admits a filtration-preserving left
inverse which is a morphism of $\hom_{\mathbf{K}}(L,A)$-modules.

###### Proof.

The dual PBW theorem in the form ($\dagger\dagger$ ‣ 2.9) shows that the
projection

$\mathsf{coAss}\longrightarrow\mathsf{coLie}$

admits a right inverse

$\mathsf{coLie}\to\mathsf{coAss}$

as a morphism of infinitesimal $\mathsf{coLie}$-bicomodules. Such a right
inverse induces a map

$L=\mathsf{coLie}\circ^{\mathsf{coLie}}L\longrightarrow\mathsf{coAss}\circ^{\mathsf{coLie}}L=U^{c}L,$

which is right inverse to the projection $U^{c}L\to L$. The discussion of §2.8
shows that this is a morphism of $L$-comodules. Applying
$\hom_{\mathbf{K}}(-,A)$ gives a map

$\hom_{\mathbf{K}}(U^{c}L,A)\longrightarrow\hom_{\mathbf{K}}(L,A),$

which is left inverse to
$\hom_{\mathbf{K}}(L,A)\longrightarrow\hom_{\mathbf{K}}(U^{c}L,A)$, and it is
straightforward to see from §3.2 that it is a morphism of
$\hom_{\mathbf{K}}(L,A)$-modules. The discussion of §3.3 shows that this
retraction is compatible with the filtrations, where the convolution algebras
inherit filtrations from the coradical filtrations of the coalgebras. ∎

##  4\. Models for mapping spaces of algebras

######  4.1.

Hinich [Hin97b] has constructed a simplicial model structure on unbounded dg
algebras over an operad $\mathsf{P}$ over a field $\mathbf{K}$ of
characteristic zero. This model structure is right transferred from the model
structure on unbounded chain complexes along the forgetful functor. The
resulting weak equivalences are the morphisms of $\mathsf{P}$-algebras that
are quasi-isomorphisms of chain complexes, and the fibrations are the
morphisms that are degree-wise surjections. Any triangulated algebra [LV12,
Section B.6.7] is cofibrant. In particular, if $\mathsf{P}$ is a Koszul
operad, then a canonical cofibrant replacement functor is given by the bar-
cobar resolution. The simplicial enrichment is given by

$\mathrm{map}_{\mathsf{P}}(A,A^{\prime})\coloneqq\hom_{\mathsf{P}\text{-}\mathsf{alg}}(A,A^{\prime}\otimes\Omega_{\bullet})\in\mathsf{sSet},$

where $\Omega_{\bullet}$ is Sullivan’s algebra of polynomial differential
forms on the simplices.

######  4.2.

For a complete filtered dg Lie algebra $\mathfrak{g}$, Hinich [Hin97a] has
also introduced the _Maurer–Cartan simplicial set_

$\mathrm{MC}_{\bullet}(\mathfrak{g})\coloneqq\mathrm{MC}(\mathfrak{g}\leavevmode\nobreak\
\widehat{\otimes}\leavevmode\nobreak\ \Omega_{\bullet}),$

where $\mathfrak{g}\leavevmode\nobreak\ \widehat{\otimes}\leavevmode\nobreak\
\Omega_{\bullet}\coloneqq\varprojlim(\mathfrak{g}/F^{n}\mathfrak{g}\otimes\Omega_{\bullet})$.
It is a non-trivial result that this simplicial set is in fact a Kan complex.
Note in particular that
$\mathrm{MC}_{0}(\mathfrak{g})=\mathrm{MC}(\mathfrak{g})$. Since
$\mathfrak{g}$ has a complete filtration, we may also exponentiate
$\mathfrak{g}_{0}$ to get a _gauge group_ acting on
$\mathrm{MC}(\mathfrak{g})$, and two Maurer–Cartan elements of $\mathfrak{g}$
lie in the same connected component of $\mathrm{MC}_{\bullet}(\mathfrak{g})$
if and only if they are gauge equivalent.

######  4.3.

We can now prove 1.18 from the introduction.

######  4.4 Proposition (1.18).

Let $A$ and $A^{\prime}$ be dg algebras over a field $\mathbf{K}$ of
characteristic zero. Then:

  * $\diamond$

$\mathbf{R}\mathrm{map}_{\mathsf{DGA}}(A,A^{\prime})\simeq\mathrm{MC}_{\bullet}(\hom_{\mathbf{K}}(\mathrm{B}A,A^{\prime}))$.

  * $\diamond$

If $A$ and $A^{\prime}$ are commutative, then
$\mathbf{R}\mathrm{map}_{\mathsf{CDGA}}(A,A^{\prime})\simeq\mathrm{MC}_{\bullet}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Com}}A,A^{\prime}))$.

Here, both $\hom_{\mathbf{K}}(\mathrm{B}A,A^{\prime})$ and
$\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Com}}A,A^{\prime})$ are considered as
complete filtered dg Lie algebras, with the filtrations induced by the
coradical filtrations of $\mathrm{B}A$ and $\mathrm{B}_{\mathsf{Com}}A$
respectively. In the commutative case, the projection
$\mathrm{B}A\to\mathrm{B}_{\mathsf{Com}}A$ induces the natural map
$\mathbf{R}\mathrm{map}_{\mathsf{DGA}}(A,A^{\prime})\longrightarrow\mathbf{R}\mathrm{map}_{\mathsf{CDGA}}(A,A^{\prime})$.

###### Proof.

Both assertions have essentially the same proof, so we will present only the
demonstration of the first one. The simplicial set
$\mathbf{R}\mathrm{map}_{\mathsf{DGA}}(A,A^{\prime})$ depends on a choice of
cofibrant replacement functor. We choose the bar-cobar resolution
$\Omega\mathrm{B}$. In simplicial degree $n$, the left hand side is given by

$\mathbf{R}\mathrm{map}_{\mathsf{DGA}}(A,A^{\prime})_{n}=\hom_{\mathsf{DGA}}\left(\Omega\mathrm{B}A,A^{\prime}\otimes\Omega_{n}\right),$

which by [LV12, Theorem 11.3.1] is the set of Maurer–Cartan elements of the dg
Lie algebra

$\hom_{\mathbf{K}}\left(\mathrm{B}A,A^{\prime}\otimes\Omega_{n}\right).$

Thus, we need to compare the two simplicial sets

$\mathrm{MC}\left(\hom_{\mathbf{K}}\left(\mathrm{B}A,A^{\prime}\right)\leavevmode\nobreak\
\widehat{\otimes}\leavevmode\nobreak\
\Omega_{\bullet}\right)\qquad\text{and}\qquad\mathrm{MC}\left(\hom_{\mathbf{K}}\left(\mathrm{B}A,A^{\prime}\otimes\Omega_{\bullet}\right)\right).$

The natural inclusion
$\hom_{\mathbf{K}}\left(\mathrm{B}A,A^{\prime}\right)\otimes\Omega_{\bullet}\xhookrightarrow{\quad}\hom_{\mathbf{K}}\left(\mathrm{B}A,A^{\prime}\otimes\Omega_{\bullet}\right)$
is a morphism of filtered modules when both sides are filtered by the
coradical filtration of $\mathrm{B}A$. Since the filtration on the right hand
side is complete, this map factors canonically through the completion, giving
a map

$\hom_{\mathbf{K}}\left(\mathrm{B}A,A^{\prime}\right)\leavevmode\nobreak\
\widehat{\otimes}\leavevmode\nobreak\
\Omega_{\bullet}\xhookrightarrow{\quad}\hom_{\mathbf{K}}\left(\mathrm{B}A,A^{\prime}\otimes\Omega_{\bullet}\right)$

which in general is not an isomorphism. Nevertheless, it induces a homotopy
equivalence of Maurer–Cartan spaces by [RNW19a, Theorem 4.1]. ∎

######  4.5 Remark.

We remark that [RNW19a, Theorem 4.1] does not explicitly consider the
completed tensor product, but the completed tensor product is implicitly used
in the arguments. We will discuss the proof of [RNW19a, Theorem 4.1] in more
detail in §§5.12–5.14.

######  4.6 Remark.

The model for mapping spaces constructed in 4.4 is evidently functorial in
both $A$ and $A^{\prime}$, defining a bifunctor from
$\mathsf{DGA}^{\mathrm{op}}\times\mathsf{DGA}$ (resp.
$\mathsf{CDGA}^{\mathrm{op}}\times\mathsf{CDGA}$) to Kan complexes. In fact
more is true: it is also functorial with respect to
$\mathsf{A}_{\infty}$-morphisms, resp. $\mathsf{C}_{\infty}$-morphisms, in
both arguments. Indeed, it is shown in [RNW19b] that the convolution algebra
construction $\hom_{\mathbf{K}}(-,-)$ is functorial with respect to
$\infty$-morphisms in either of its two arguments, but not simultaneously. But
an $\infty$-morphism of algebras induces a strict morphism between bar
constructions, so $\hom_{\mathbf{K}}(\mathrm{B}A,A^{\prime})$ is in fact
simultaneously functorial with respect to $\infty$-morphisms in both $A$ and
$A^{\prime}$. It straightforward to check that the quasi-isomorphisms of 1.18
are natural in both slots.

######  4.7 Remark.

The same model

$\mathbf{R}\mathrm{map}_{\mathsf{P}}(A,A^{\prime})\simeq\mathrm{MC}_{\bullet}\left(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{P}}A,A^{\prime})\right)$

works for algebras over any Koszul binary operad, by the same argument. It
also works outside of the binary case, with the difference that the
convolution algebra is an $\mathsf{L}_{\infty}$-algebra in that case, instead
of a dg Lie algebra [Wie19, Section 7].

######  4.8.

We can now prove the non-unital case of 1.7 from the introduction.

######  4.9 Theorem (1.7).

Over a base field of characteristic zero, the natural map

$\mathbf{R}\mathrm{map}_{\mathsf{CDGA}}(A,A^{\prime})\longrightarrow\mathbf{R}\mathrm{map}_{\mathsf{DGA}}(A,A^{\prime})$

induced by the forgetful functor induces an injection on all homotopy groups
for any $A,A^{\prime}\in\mathsf{CDGA}$ and any basepoint.

###### Proof.

By 4.4 we can model
$\mathbf{R}\mathrm{map}_{\mathsf{CDGA}}(A,A^{\prime})\to\mathbf{R}\mathrm{map}_{\mathsf{DGA}}(A,A^{\prime})$
by applying $\mathrm{MC}_{\bullet}$ to the morphism of complete filtered dg
Lie algebras
$\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Com}}A,A^{\prime})\to\hom_{\mathbf{K}}(\mathrm{B}A,A^{\prime})$.
By 1.15 we are done if we can construct a filtered retraction from
$\hom_{\mathbf{K}}(\mathrm{B}A,A^{\prime})$ onto
$\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Com}}A,A^{\prime})$ as filtered
$\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Com}}A,A^{\prime})$-modules. This is
exactly what is given by 3.4, since $\mathrm{B}A$ is canonically isomorphic to
$U^{c}\mathrm{B}_{\mathsf{Com}}A$ (this is well known, and can be proven by
dualizing the argument of [CPRNW19, Lemma 4.7]). ∎

##  5\. The unital case

######  5.1.

In this section, we show the unital version of Theorem 1.5. Recall that we
denote by $\mathsf{DGA}^{u}$ (resp. $\mathsf{CDGA}^{u}$) the categories of
unital associative (resp. commutative) dg algebras over our fixed base field
$\mathbf{K}$.

######  5.2 Theorem.

Over a base field of characteristic zero, the forgetful functor

$\mathrm{Ho}(\mathsf{CDGA}^{u})\longrightarrow\mathrm{Ho}(\mathsf{DGA}^{u})$

is faithful. More generally, the map
$\mathbf{R}\mathrm{map}_{\mathsf{CDGA}^{u}}(A,A^{\prime})\to\mathbf{R}\mathrm{map}_{\mathsf{DGA}^{u}}(A,A^{\prime})$
induced by the forgetful functor induces an injection on all homotopy groups,
for any $A,A^{\prime}\in\mathsf{CDGA}^{u}$.

######  5.3.

The main difference in the unital case is the following. Our strategy for
proving 1.5 was to model the two mapping spaces as Maurer–Cartan simplicial
sets of two dg Lie algebras, and then to apply our criterion 1.15. However, in
the categories of unital (commutative) dg algebras, it is simply not possible
that mapping spaces can be modeled by dg Lie algebras in this way. Indeed, dg
Lie algebras intrinsically model _pointed_ spaces, since the element $0$
provides a canonical basepoint of the Maurer–Cartan space. The space of maps
between two non-unital rings is canonically pointed by the zero map, but the
space of maps between two unital rings has no natural basepoint. In fact the
space of maps may even be empty: for example, there does not exist any
homomorphism $\mathbf{C}\to\mathbf{R}$, derived or otherwise. To overcome this
we need to systematically replace dg Lie algebras with _curved Lie algebras_.

######  5.4.

Indeed, from the point of view of rational homotopy theory, the choice of an
augmentation of a commutative dg algebra corresponds to fixing a basepoint of
a space. Thus, one may think that unital algebras without augmentations should
correspond to unbased spaces. It is a remarkable insight of Positselski
[Pos11] that Koszul duality can be extended to unital algebras by introducing
a _curvature_ term on the other side of the duality: the Koszul dual of a
unital commutative dg algebra is a _curved_ Lie coalgebra. Similarly counital
coalgebras are Koszul dual to curved algebras. It is then natural that
Maurer–Cartan spaces of curved Lie algebras can be fruitfully used to model
unbased spaces. Further operadic and homotopical studies of curved Lie (or
$\mathsf{L}_{\infty}$-) algebras have been pursued by [HM12, Bra12, Mau17,
LG19, LGL18, CCN21, RiL22].

######  5.5.

The proof of 5.2 will closely follow the proof of 1.5, except that we need to
replace 1.15 and 1.18 with the following curved counterparts.

######  5.6 Proposition.

Let $i\colon\mathfrak{h}\to\mathfrak{g}$ be a morphism of complete filtered
curved Lie algebras over a field of characteristic zero. Suppose that there
exists $r\colon\mathfrak{g}\to\mathfrak{h}$ such that $r\circ
i=\mathrm{id}_{\mathfrak{h}}$, and $r$ is a morphism of filtered
$\mathfrak{h}$-modules. Then the map
$\mathrm{MC}_{\bullet}(\mathfrak{h})\longrightarrow\mathrm{MC}_{\bullet}(\mathfrak{g})$
induces an injection on all homotopy groups for all basepoints.

######  5.7 Proposition.

Let $A$ and $A^{\prime}$ be unital dg algebras over a field $\mathbf{K}$ of
characteristic zero. Then:

  * $\diamond$

$\mathbf{R}\mathrm{map}_{\mathsf{DGA}^{u}}(A,A^{\prime})\simeq\mathrm{MC}_{\bullet}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{uAss}}A,A^{\prime}))$.

  * $\diamond$

If $A$ and $A^{\prime}$ are commutative, then
$\mathbf{R}\mathrm{map}_{\mathsf{CDGA}^{u}}(A,A^{\prime})\simeq\mathrm{MC}_{\bullet}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{uCom}}A,A^{\prime}))$.

Here, both $\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{uAss}}A,A^{\prime})$ and
$\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{uCom}}A,A^{\prime})$ are considered as
complete filtered curved Lie algebras, with the filtrations induced by the
coradical filtrations of $\mathrm{B}_{\mathsf{uAss}}A$ and
$\mathrm{B}_{\mathsf{uCom}}A$ respectively. In the commutative case, the
projection $\mathrm{B}_{\mathsf{uAss}}A\to\mathrm{B}_{\mathsf{uCom}}A$ induces
the natural map
$\mathbf{R}\mathrm{map}_{\mathsf{DGA}^{u}}(A,A^{\prime})\longrightarrow\mathbf{R}\mathrm{map}_{\mathsf{CDGA}^{u}}(A,A^{\prime}).$

######  5.8.

Once 5.6 and 5.7 are in place, the proof proceeds just as in the uncurved
case, by observing that the Poincaré–Birkhoff–Witt theorem implies the
existence of a retraction
$\mathrm{B}_{\mathsf{uCom}}A\to\mathrm{B}_{\mathsf{uAss}}A$. In the rest of
this section we explain why 5.6 and 5.7 are true.

######  5.9.

The proof of 5.6 formally reduces to its uncurved analogue, 1.15, by means of
_twisting_. Recall that if $\mathfrak{g}$ is a complete filtered dg Lie
algebra and $\alpha\in\mathrm{MC}(\mathfrak{g})$, then we may twist the
differential of $\mathfrak{g}$ by $\alpha$ to obtain a new complete filtered
dg Lie algebra $\mathfrak{g}^{\alpha}$. If we think of complete filtered dg
Lie algebras as models for pointed spaces, then this construction can be
understood geometrically as changing the basepoint: given a pointed space
$(X,x_{0})$ and a point $x_{1}\in X$, we obtain a new pointed space
$(X,x_{1})$. If $\mathfrak{g}$ is instead a complete filtered _curved_ Lie
algebra, then we may similarly twist $\mathfrak{g}$ by an _arbitrary_ element
$\alpha\in\mathfrak{g}_{-1}$, obtaining a new complete filtered curved Lie
algebra $\mathfrak{g}^{\alpha}$. See for instance [Bra12, Section 7.1]. In the
special case that $\alpha$ is a Maurer–Cartan element, the twisted algebra
$\mathfrak{g}^{\alpha}$ has vanishing curvature term; that is,
$\mathfrak{g}^{\alpha}$ is a dg Lie algebra when
$\alpha\in\mathrm{MC}(\mathfrak{g})$. Recalling that complete filtered curved
Lie algebras model unbased spaces, the geometric interpretation is instead
that if $X$ is an unbased space and $x_{1}\in X$ is a point, then $(X,x_{1})$
is a pointed space.

######  5.10.

It is now clear how 5.6 follows from 1.15. Indeed, suppose first in the
situation of 5.6 that $\mathrm{MC}(\mathfrak{h})=\varnothing$. In this case
the statement is vacuously true. Otherwise, pick a Maurer–Cartan element
$\alpha\in\mathfrak{h}$, and apply 1.15 to the morphism of dg Lie algebras
$\mathfrak{h}^{\alpha}\to\mathfrak{g}^{i(\alpha)}$.

######  5.11.

The proof of 5.7 follows closely the proof of 4.4. We need three ingredients
in order to carry out the argument:

  * $\diamond$

The bar-cobar construction
$\Omega_{\mathsf{uCom}}\mathrm{B}_{\mathsf{uCom}}A\to A$ is a cofibrant
replacement functor in the category $\mathsf{CDGA}^{u}$.

  * $\diamond$

If $A$ is a unital cdga and $L$ is a curved conilpotent Lie coalgebra, then
Maurer–Cartan elements of the curved Lie algebra $\hom_{\mathbf{K}}(L,A)$
naturally correspond to unital cdga morphisms $\Omega_{\mathsf{uCom}}L\to A$.

  * $\diamond$

The natural inclusion
$\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{uCom}}A,A^{\prime})\leavevmode\nobreak\
\widehat{\otimes}\leavevmode\nobreak\
\Omega_{\bullet}\hookrightarrow\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{uCom}}A,A^{\prime}\otimes\Omega_{\bullet})$
induces a homotopy equivalence on simplicial Maurer–Cartan spaces.

(And, of course, we also need the evident analogues of the above statements in
the associative case.) The fact that
$\Omega_{\mathsf{uCom}}\mathrm{B}_{\mathsf{uCom}}A\to A$ is a quasi-
isomorphism is part of curved Koszul duality [HM12, Proposition 5.2.8]).
Moreover, the algebra $\Omega_{\mathsf{uCom}}\mathrm{B}_{\mathsf{uCom}}A$ is
quasi-free, and its space of generators (i.e. $\mathrm{B}_{\mathsf{uCom}}A$)
is equipped with a “good” filtration, namely the filtration induced by the
weight grading of $\mathsf{uCom}^{\text{\raisebox{0.0pt}{\textexclamdown}}}$.
Hence the bar-cobar resolution is cofibrant, which shows the first bullet
point. The second bullet point is [LG19, Proposition 134]. Thus, all that
remains to explain is the last bullet point. The argument is a minor
modification of the one used to prove [RNW19a, Theorem 4.1]. We now recall
this argument and then explain how it should be modified.

######  5.12.

Let us first review a construction due to Getzler [Get09]. Getzler considers
the _Dupont contraction_ , which is an explicit contraction

$\Omega_{\bullet}$$C_{\bullet}$$p_{\bullet}$$i_{\bullet}$$s_{\bullet}$

in the category of simplicial chain complexes, where $C_{n}\subset\Omega_{n}$
is a subcomplex isomorphic to the cellular cochains on $\Delta_{n}$. For a
complete filtered dg Lie algebra $\mathfrak{g}$, we denote by
$\mathrm{MC}_{\bullet}(\mathfrak{g})^{s_{\bullet}=0}$ the subspace of
$\mathrm{MC}_{\bullet}(\mathfrak{g})$ consisting of Maurer–Cartan elements
which satisfy the “gauge condition” $s_{\bullet}(\alpha)=0$.222Getzler denotes
this simplicial set $\gamma_{\bullet}(\mathfrak{g})$. Getzler shows for
nilpotent $\mathfrak{g}$ that
$\mathrm{MC}_{\bullet}(\mathfrak{g})^{s_{\bullet}=0}$ is a Kan complex and
that
$\mathrm{MC}_{\bullet}(\mathfrak{g})^{s_{\bullet}=0}\hookrightarrow\mathrm{MC}_{\bullet}(\mathfrak{g})$
is a homotopy equivalence. The result extends to the complete filtered case as
an easy consequence of Getzler’s arguments, see [Ber15, Proposition 5.4].

######  5.13.

Returning to the inclusion of simplicial dg Lie algebras

$\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Com}}A,A^{\prime})\leavevmode\nobreak\
\widehat{\otimes}\leavevmode\nobreak\
\Omega_{\bullet}\xhookrightarrow{\quad}\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Com}}A,A^{\prime}\otimes\Omega_{\bullet}),$

the point is now that the Dupont contraction operates on both sides of the
inclusion. Thus we obtain a commuting diagram of simplicial sets

$\mathrm{MC}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Com}}A,A^{\prime})\leavevmode\nobreak\
\widehat{\otimes}\leavevmode\nobreak\
\Omega_{\bullet})^{s_{\bullet}=0}$$\mathrm{MC}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Com}}A,A^{\prime})\leavevmode\nobreak\
\widehat{\otimes}\leavevmode\nobreak\
\Omega_{\bullet})$$\mathrm{MC}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Com}}A,A^{\prime}\otimes\Omega_{\bullet}))^{s_{\bullet}=0}$$\mathrm{MC}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Com}}A,A^{\prime}\otimes\Omega_{\bullet})).$

What Getzler showed is precisely that the top horizontal arrow is an
equivalence. Repeating his argument with obvious modifications shows that the
bottom horizontal arrow is an equivalence, too. The right vertical arrow is
the one we want to prove is an equivalence. The left vertical arrow will turn
out to be an _isomorphism_ of simplicial sets, finishing the argument.

######  5.14.

To see that

$\mathrm{MC}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Com}}A,A^{\prime})\leavevmode\nobreak\
\widehat{\otimes}\leavevmode\nobreak\
\Omega_{n})^{s_{n}=0}\longrightarrow\mathrm{MC}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Com}}A,A^{\prime}\otimes\Omega_{n}))^{s_{n}=0}$

is a bijection for all $n$, we use a reinterpretation of Getzler’s gauge
condition due to Bandiera [Ban17]. Suppose that we are given a complete
filtered $\mathsf{L}_{\infty}$\- algebra $\mathfrak{G}$, and a contraction

$\mathfrak{G}$$\mathfrak{H}$$p$$i$$s$

of filtered chain complexes. To this data, the procedure of Homotopy Transfer
[LV12, Section 10.3] associates a canonical structure of complete filtered
$\mathsf{L}_{\infty}$-algebra on $\mathfrak{H}$, and Bandiera shows that
$\mathrm{MC}(\mathfrak{G})^{s=0}=\mathrm{MC}(\mathfrak{H})$. In our situation,
this means that the simplicial dg Lie algebra structure on
$\mathfrak{g}\leavevmode\nobreak\ \widehat{\otimes}\leavevmode\nobreak\
\Omega_{\bullet}$ induces a simplicial $\mathsf{L}_{\infty}$-structure on
$\mathfrak{g}\otimes C_{\bullet}=\mathfrak{g}\leavevmode\nobreak\
\widehat{\otimes}\leavevmode\nobreak\ C_{\bullet}$,333We have
$\mathfrak{g}\otimes C_{\bullet}=\mathfrak{g}\leavevmode\nobreak\
\widehat{\otimes}\leavevmode\nobreak\ C_{\bullet}$ since $C_{n}$ is a finite-
dimensional chain complex for all $n$. and that

$\mathrm{MC}_{\bullet}(\mathfrak{g})^{s_{\bullet}=0}=\mathrm{MC}(\mathfrak{g}\otimes
C_{\bullet}).$

But this finishes the result: we are reduced to showing that

$\mathrm{MC}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Com}}A,A^{\prime})\otimes
C_{n})\to\mathrm{MC}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Com}}A,A^{\prime}\otimes
C_{n}))$

is a bijection. But
$\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Com}}A,A^{\prime})\otimes
C_{n}\to\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Com}}A,A^{\prime}\otimes C_{n})$
is an isomorphism, since $C_{n}$ is a finite-dimensional chain complex.

######  5.15 Remark.

One can also show directly that $\mathrm{MC}(\mathfrak{g}\otimes
C_{\bullet})\to\mathrm{MC}(\mathfrak{g}\leavevmode\nobreak\
\widehat{\otimes}\leavevmode\nobreak\ \Omega_{\bullet})$ is a homotopy
equivalence, without passing through Getzler’s gauge condition, although
ultimately the arguments are quite similar [RN19, Theorem 3.3].

######  5.16.

Let us now return to the unital case, and to the inclusion of simplicial
curved Lie algebras

$\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{uCom}}A,A^{\prime})\leavevmode\nobreak\
\widehat{\otimes}\leavevmode\nobreak\
\Omega_{\bullet}\xhookrightarrow{\quad}\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{uCom}}A,A^{\prime}\otimes\Omega_{\bullet}).$

Again, we get a commuting diagram

$\mathrm{MC}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{uCom}}A,A^{\prime})\leavevmode\nobreak\
\widehat{\otimes}\leavevmode\nobreak\
\Omega_{\bullet})^{s_{\bullet}=0}$$\mathrm{MC}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{uCom}}A,A^{\prime})\leavevmode\nobreak\
\widehat{\otimes}\leavevmode\nobreak\
\Omega_{\bullet})$$\mathrm{MC}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{uCom}}A,A^{\prime}\otimes\Omega_{\bullet}))^{s_{\bullet}=0}$$\mathrm{MC}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{uCom}}A,A^{\prime}\otimes\Omega_{\bullet})).$

The horizontal arrows are still equivalences: indeed, this reduces to the
uncurved case by “twisting” as in §§5.9–5.10. The left vertical arrow is still
an isomorphism: the interpretation of the gauge condition in terms of Homotopy
Transfer is proven in the curved setting in [Get18]. To be precise, Getzler
shows that in the curved setting it also holds
$\mathrm{MC}(\mathfrak{G})^{s=0}=\mathrm{MC}(\mathfrak{H})$, where
$\mathfrak{H}$ inherits from homotopy transfer a curved
$\mathsf{L}_{\infty}$-algebra structure. As before, the conclusion follows
from

$\displaystyle\mathrm{MC}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{uCom}}A,A^{\prime})\leavevmode\nobreak\
\widehat{\otimes}\leavevmode\nobreak\ \Omega_{\bullet})^{s_{\bullet}=0}$
$\displaystyle{}=\mathrm{MC}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{uCom}}A,A^{\prime})\otimes
C_{\bullet})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}$
$\displaystyle{}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathrm{MC}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{uCom}}A,A^{\prime}\otimes
C_{\bullet}))=\mathrm{MC}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{uCom}}A,A^{\prime}\otimes\Omega_{\bullet}))^{s_{\bullet}=0}.$

##  6\. Enveloping algebras

######  6.1.

In this section we prove 1.25. We write $\mathrm{B}_{\mathsf{Lie}}$ for the
bar construction from dg Lie algebras to dg cocommutative coalgebras and
$\Omega_{\mathsf{Lie}}$ for its left adjoint cobar construction.

######  6.2 Lemma.

Let $\mathfrak{g}$ and $\mathfrak{h}$ be dg Lie algebras over a field of
characteristic zero. There is a commutative diagram

$\mathrm{MC}_{\bullet}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Lie}}\mathfrak{g},\mathfrak{h}))$$\mathrm{MC}_{\bullet}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Lie}}\mathfrak{g},U\mathfrak{h})).$$\mathbf{R}\mathrm{map}_{\mathsf{DGLA}}(\mathfrak{g},\mathfrak{h})$$\mathbf{R}\mathrm{map}_{\mathsf{DGA}}(U\mathfrak{g},U\mathfrak{h}).$$\sim$$\sim$

in which the vertical arrows are homotopy equivalences, the upper horizontal
arrow is induced by the natural map $\mathfrak{h}\to U\mathfrak{h}$, and the
lower horizontal arrow is induced from the universal enveloping algebra
functor.

###### Proof.

We remark that the left vertical arrow is constructed by the exact same
arguments as in 4.4. As in the proof of 4.4, the two simplicial mapping spaces
depend on the choice of a cofibrant replacement of $\mathfrak{g}$ and
$U\mathfrak{g}$, respectively. For $\mathfrak{g}$ we choose the bar-cobar
resolution
$\Omega_{\mathsf{Lie}}\mathrm{B}_{\mathsf{Lie}}\mathfrak{g}\stackrel{{\scriptstyle\sim}}{{\to}}\mathfrak{g}$.
But for $U\mathfrak{g}$ we choose instead to take the resolution

$\Omega\mathrm{B}_{\mathsf{Lie}}\mathfrak{g}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}U\mathfrak{g}.$

This map is constructed as follows: the universal enveloping algebra functor
preserves quasi-isomorphisms by the PBW theorem, so from the bar-cobar
resolution of $\mathfrak{g}$ we get a quasi-isomorphism
$U\Omega_{\mathsf{Lie}}\mathrm{B}_{\mathsf{Lie}}\mathfrak{g}\stackrel{{\scriptstyle\sim}}{{\to}}U\mathfrak{g}$.
But if $C$ is a cocommutative coalgebra, then $\Omega C\cong
U\Omega_{\mathsf{Lie}}C$ [CPRNW19, Lemma 4.7], so
$U\Omega_{\mathsf{Lie}}\mathrm{B}_{\mathsf{Lie}}\mathfrak{g}\cong\Omega\mathrm{B}_{\mathsf{Lie}}\mathfrak{g}$.
The algebra $\Omega\mathrm{B}_{\mathsf{Lie}}\mathfrak{g}$ is triangulated,
hence cofibrant, by the same argument that shows that any bar-cobar resolution
is cofibrant.

With these choices of cofibrant replacement, the simplicial enrichment
constructed by Hinich (§4.1) gives

$\mathbf{R}\mathrm{map}_{\mathsf{DGLA}}(\mathfrak{g},\mathfrak{h})_{n}=\mathrm{Hom}_{\mathsf{DGLA}}(\Omega_{\mathsf{Lie}}\mathrm{B}_{\mathsf{Lie}}\mathfrak{g},\mathfrak{h}\otimes\Omega_{n})=\mathrm{MC}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Lie}}\mathfrak{g},\mathfrak{h}\otimes\Omega_{n}))$

and

$\mathbf{R}\mathrm{map}_{\mathsf{DGA}}(U\mathfrak{g},U\mathfrak{h})_{n}=\mathrm{Hom}_{\mathsf{DGA}}(\Omega\mathrm{B}_{\mathsf{Lie}}\mathfrak{g},U\mathfrak{h}\otimes\Omega_{n})=\mathrm{MC}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Lie}}\mathfrak{g},U\mathfrak{h}\otimes\Omega_{n}))$

with the map between them given by the universal enveloping algebra functor.
But now we have in each simplicial degree a commuting diagram

$\mathrm{MC}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Lie}}\mathfrak{g},\mathfrak{h})\leavevmode\nobreak\
\widehat{\otimes}\leavevmode\nobreak\
\Omega_{n})$$\mathrm{MC}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Lie}}\mathfrak{g},U\mathfrak{h})\leavevmode\nobreak\
\widehat{\otimes}\leavevmode\nobreak\
\Omega_{n})$$\mathrm{MC}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Lie}}\mathfrak{g},\mathfrak{h}\otimes\Omega_{n}))$$\mathrm{MC}(\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Lie}}\mathfrak{g},U\mathfrak{h}\otimes\Omega_{n}))$

with the horizontal arrows induced by $\mathfrak{h}\to U\mathfrak{h}$, which
produces the commuting square of the lemma. The vertical arrows realize to
homotopy equivalences by [RNW19a, Theorem 4.1]. ∎

###### Proof of 1.25.

Applying 6.2 and 1.15, it suffices to show that the map

$\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Lie}}\mathfrak{g},\mathfrak{h})\to\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Lie}}\mathfrak{g},U\mathfrak{h})$

induced by the embedding $\mathfrak{h}\to U\mathfrak{h}$ admits a filtration-
preserving left inverse as a map of
$\hom_{\mathbf{K}}(\mathrm{B}_{\mathsf{Lie}}\mathfrak{g},\mathfrak{h})$-modules.
But $\mathfrak{h}\to U\mathfrak{h}$ admits a left inverse as a morphism of
$\mathfrak{h}$-modules, by the usual PBW theorem, and this gives the
retraction we want by the functoriality explained in §3.2. It clearly respects
the filtrations, since the relevant filtrations are induced by the coradical
filtration on $\mathrm{B}_{\mathsf{Lie}}\mathfrak{g}$. ∎

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