# Rational Homotopy Theory

Alexander A. Voronov  School of Mathematics, University of Minnesota,
Minneapolis, MN 55455 and Kavli IPMU (WPI), UTIAS, The University of Tokyo,
Kashiwa, Chiba 277-8583, Japan

###### Abstract.

This is a survey of Rational Homotopy Theory, intended for a Mathematical
Physics readership.

Research is supported in part by World Premier International Research Center
Initiative (WPI Initiative), MEXT, Japan, and Travel grant #585720 from the
Simons Foundation.

###### Contents

  1. 1 Rational homotopy category of spaces
  2. 2 Differential graded commutative algebras
  3. 3 Minimal Sullivan models of DGCAs
  4. 4 Sullivan minimal models of spaces
    1. 4.1 Real algebraic models of manifolds
    2. 4.2 Rational algebraic models of spaces
    3. 4.3 Realizations
  5. 5 Rationalization
  6. 6 Model categories
  7. 7 The fundamental theorem of RHT
  8. 8 Quillen minimal models of spaces
  9. 9 Some physical applications
    1. 9.1 Hypothesis H
    2. 9.2 Mysterious Triality and cyclic loop spaces of $S^{4}$
    3. 9.3 Other applications to physics

## Introduction

Rational Homotopy Theory (RHT) is a story of success in Algebraic Topology. It
sacrifices the all-time favorite homotopy equivalence relation on topological
spaces for the weaker rational homotopy equivalence, ignoring all the torsion
in homotopy and homology groups, and provides a complete algebraic invariant
of the rational equivalence class of a good enough space $X$, the isomorphism
class of a differential graded (dg-) commutative algebra (DGCA) $M(X)$, called
the Sullivan minimal model of $X$: two such spaces are rationally equivalent
if and only if their Sullivan minimal models are isomorphic. Moreover, with
suitable model-category structures, the homotopy category of good enough DGCAs
becomes equivalent to the rational homotopy category of good enough spaces.
Another feature of the DGCA $M(X)$ is that it is rather easily computable,
making the invariant $M(X)$ an effective tool of topology. A parallel,
historically earlier algebraic model of a space $X$ is given by a differential
graded Lie algebra (DGLA) $Q(X)$, called the Quillen minimal model, and
provides practically the same features. Koszul duality between the commutative
and Lie operads, [GK94], brings the two models together.

Perhaps, one of the most spectacular mathematical applications of RHT is the
result of D. Sullivan and M. Vigué-Poirrier [VPS76, FOT08] which states a
sufficiently general simply connected Riemannian manifold has infinitely many
closed geodesics. RHT has made a strong appearance in Mathematical Physics via
H. Sati’s _Hypothesis H_ , which states, roughly speaking, that the Sullivan
minimal model of the 4-sphere $S^{4}$ captures the dynamics of fields in
M-theory in the low-energy (UV) limit, also known as 11d supergravity [Sat18].
Hypothesis H progresses to the dimensional reductions of M-theory with its
dynamics described by the iterated cyclic loop space
$\mathcal{L}_{c}^{k}S^{4}$ of the 4-sphere [FSS17, FSS20, FSS21, GS21, SV23b,
SV23a].

In this survey, we review the main ideas and results of RHT. RHT cannot
complain about the lack of surveys, see [Hes07, Ber12, Men15, Moe17, Wan19,
Hol21]. What distinguishes this survey from the variety of previous ones is
that it, first of all, keeps an eye on Mathematical Physics. A novel feature
of this survey is that more recent developments which engage
$L_{\infty}$-algebras in lieu of DGLAs that are free as graded Lie algebras
are put forward as Quillen minimal models. Another novelty, which may get a
rise out of seasoned rational homotopy theorists, is the use of the graded
symmetric algebra notation $S(V)$ in lieu of $\wedge V$, as well as swapping
the standard abbreviation CDGA for DGCA. The reason for the first deviation
from the standard convention is that in the category of graded vector spaces,
the graded exterior algebra is not free as a graded commutative algebra and
not isomorphic to the graded symmetric algebra. The reason for the second
deviation from the standards is that the placement of modifiers may be
commutative but not necessarily associative: D(GC)A $\neq$ (DG)CA $=$ C(DG)A.
Whatever abbreviation the practitioners use, they always mean D(GC)A in this
context.

Some novelties in exposition do not mean novelties in the content: this
article presents only known results. The only possible exception, which may
appear for the first time in written form, is Example 4.3 on the Chern-Simons
interpretation of the “phantom” 3-form $g_{3}$ on the sphere $S^{2}$. And even
that is not due to the author of the survey.

Disclaimer: This survey, being limited in length, leaves aside a number of
remarkable developments of RHT, such as, for example, those related to
$L_{\infty}$-algebras and their nerves [Hin97, Get09, SS12, BM13, Ber15, LM15,
BFMT20, RNV20]. Likewise, we do not pretend to cover all the applications of
RHT to physics. The choice of topics obviously reflects the bias of the
author.

Conventions: When we use the term “space,” we will mean a topological space.
Moreover, we will assume all spaces to be _path-connected_ unless stated
otherwise. All manifolds will be smooth ($C^{\infty}$) by default. In graded
algebra, $\lvert a\rvert$ will denote the degree of an element $a$.

Acknowledgments: I am grateful to my collaborator Hisham Sati, who introduced
me to the beautiful interaction of rational homotopy with physics, Alexey
Bondal and Mikhail Kapranov, with whom I had numerous fruitful discussions at
Kavli IPMU, and Eli Schlossberg, who shared his notes on rational homotopy and
deformation theories with me.

##  1\. Rational homotopy category of spaces

A continuous map $f:X\to Y$ between topological spaces is called a _rational
$($homotopy$)$ equivalence_ if it induces an isomorphism on rational homology:
$f_{*}:H_{\bullet}(X;\mathbb{Q})\to H_{\bullet}(Y;\mathbb{Q})$. (Compare this
to a stronger notion of weak equivalence: to compare apples to apples and
oranges to oranges, let us say a continuous map $f:X\to Y$ of simply connected
spaces is called a _weak $($homotopy$)$ equivalence_ if it induces an
isomorphism $f_{*}:H_{\bullet}(X;\mathbb{Z})\to H_{\bullet}(Y;\mathbb{Z})$ on
integral homology.) We say that two spaces $X$ and $Y$ are _rationally
equivalent_ if they may be connected by a sequence of rational equivalences
$X\leftarrow X_{1}\to\dots\leftarrow X_{n}\rightarrow Y$. The equivalence
class of a space $X$ up to rational equivalence is called the _rational
homotopy type of_ $X$. If we add formal inverses of rational equivalences to
the category of spaces, i.e., we localize the category with respect to
rational equivalences, we obtain the _rational homotopy category of spaces_.

######  Example 1.1.

The unique map $\mathbb{R}{\mathbb{P}}^{2}\to\\{*\\}$ from the real projective
plane to a singleton is a rational equivalence, because the homology groups
$H_{n}(\mathbb{R}{\mathbb{P}}^{2};\mathbb{Z})$, $n>0$, are all torsion.

######  Example 1.2.

Let $S^{2n-1}$, $n\geq 1$, be an odd-dimensional sphere and
$K(\mathbb{Z},2n-1)$ be the Eilenberg-MacLane space. Let $S^{2n-1}\to
K(\mathbb{Z},2n-1)$ be a map realizing a generator of
$\pi_{2n-1}(K(\mathbb{Z},2n-1))=\mathbb{Z}$. This map is a rational
equivalence, because it induces an isomorphism on rational homotopy groups:

$\pi_{i}(S^{2n-1})\otimes\mathbb{Q}\cong\pi_{i}(K(\mathbb{Z},2n-1))\otimes\mathbb{Q}=\begin{cases}\mathbb{Q},&\text{if
$i=2n-1$,}\\\ 0,&\text{otherwise},\end{cases}$

cf. the remark at the end of the second paragraph of Section 6. Thus,
$S^{2n-1}$ is a _rational Eilenberg-MacLane space_. The same argument does not
work for even-dimensional spheres $S^{2n}$, because $S^{2n}$ has an extra
rational homotopy group in degree $4n-1$:

$\pi_{i}(S^{2n})\otimes\mathbb{Q}\cong\begin{cases}\mathbb{Q},&\text{if $i=2n$
or $4n-1$,}\\\ 0,&\text{otherwise}.\end{cases}$

##  2\. Differential graded commutative algebras

RHT assigns an algebraic model, namely a DGCA, see below, to a space. Such an
algebra is supposed to be large enough to carry all the information about the
rational homotopy type of the space. The theory works further to replace this
large DGCA with a much smaller model, called a minimal Sullivan model. The
prototypical example of an excessively large model is the de Rham algebra
$\Omega^{\bullet}(X)$ of a manifold $X$. In this case, the ground field has to
be extended to $\mathbb{R}$, and we are dealing with the _real_ , rather than
rational, _homotopy type_ of $X$.

A _differential graded commutative algebra $($DGCA$)$_ over a field $k$ is a
graded associative $k$-algebra $A=\bigoplus_{n\in\mathbb{Z}}A^{n}$ with a
multiplication

$\displaystyle A\otimes A$ $\displaystyle\to A,$ $\displaystyle a\otimes b$
$\displaystyle\mapsto ab,$

which respects the grading: $A^{m}\otimes A^{n}\to A^{m+n}$, is _graded
commutative_ :

$ba=(-1)^{\lvert a\rvert\lvert b\rvert}ab,$

has a unit $1\in A$, and has a differential $d:A\to A$, a $k$-linear operator
of degree 1 such that $d^{2}=0$ and it is a (graded) _derivation_ of $A$:

$d(ab)=(da)b+(-1)^{\lvert a\rvert}a(db).$

The derivation property, also known as the _Leibniz rule_ , implies that the
differential kills the constants: $d(1)=0$. A DGCA _homomorphism_ is a linear
map $A\to B$ which respects the DGCA structure: the multiplication, unit,
grading, and differential. In this survey, a DGCA will by default be
nonnegatively graded ($A=\bigoplus_{n\geq 0}A^{n}$) over a field $k$ of
characteristic 0.

A DGCA $A$ is called _$($ homologically$)$ connected_ if $A^{0}=k$
(respectively, $H^{0}(A)=k$). It is called _$($ homologically$)$ simply
connected_ if $A^{0}=k$ and $A^{1}=0$, (respectively, $H^{0}(A)=k$ and
$H^{1}(A)=0$).

A DGCA $(S(V),d)$ is called _semifree_ , if it is based on the free graded
commutative algebra $S(V)=\bigoplus_{n\geq 0}S^{n}(V)$ on a positively graded
vector space $V=\bigoplus_{n>0}V^{n}$.

##  3\. Minimal Sullivan models of DGCAs

RHT provides a minimal Sullivan model for each good enough DGCA. A _Sullivan
algebra_ is a semifree DGCA $(S(V),d)$ such that the differential $d$
satisfies the following nilpotency condition, also known as the _Sullivan
condition_ : there exists a filtration of the generating space $V$ by graded
subspaces:

$0=V(0)\subset V(1)\subset\dots$

such that $V=\bigcup_{n\geq 0}V(n)$ and $d(V(n))\subset S(V(n-1))$ for all
$n\geq 1$. A Sullivan algebra is automatically connected.

A semifree DGCA $(S(V),d)$ is called _minimal_ if $d(V)\subset S^{\geq 2}(V)$.

Every simply connected minimal DGCA $S(V)$ is necessarily Sullivan: the
filtration associated with the grading does the job. Indeed, we have $V^{1}=0$
and for

$V(n)\colonequals\bigoplus_{2\leq i\leq n}V^{i},\qquad n\geq 0,$

the Sullivan nilpotency condition follows from $d(V^{n})\subset S^{\geq
2}(V)^{n+1}\subset S(V(n-1))$ for all $n\geq 1$.

A DGCA _quasi-isomorphism_ is a DGCA homomorphism $A\to B$ which induces and
isomorphism on the cohomology of the differentials
$H^{\bullet}(A,d_{A})\xrightarrow{\sim}H^{\bullet}(B,d_{B})$. In this case we
write $A\xrightarrow{\operatorname{q-is}}B$ and say that $A$ (or
$A\xrightarrow{\operatorname{q-is}}B$, to be more precise) is a _model_ of
$B$. Sullivan and minimal Sullivan models play an important role in RHT for a
variety of reasons, mostly related to the fact that they are cofibrant DGCAs
in a suitable model-category structure, see Section 6.

######  Theorem 1 ([FHT01, Theorems 14.11-14.12]).

  1. $(1)$

Every quasi-isomorphism between minimal Sullivan algebras is an isomorphism.

  2. $(2)$

Every homologically connected DGCA has a minimal Sullivan model, unique up to
isomorphism.

We conclude this section with a structure theorem for Sullivan algebras.

######  Theorem 2 ([FHT01, Theorem 14.9]).

Every connected Sullivan algebra is isomorphic to the tensor product of a
minimal Sullivan algebra and a semifree contractible algebra $S(U\oplus dU)$,
where $U$ is a positively graded vector space.

##  4\. Sullivan minimal models of spaces

Every manifold $X$ has a DGCA associated to it, the de Rham algebra
$(\Omega^{\bullet}(X),d_{\operatorname{dR}})$. This is quite a familiar a DGCA
over the reals $\mathbb{R}$ whose cohomology is the de Rham cohomology
$H^{\bullet}(X;\mathbb{R})$. Sullivan [Sul77] has generalized this
construction to work for an arbitrary topological space $X$ and produce a DGCA
$A_{\operatorname{PL}}(X)$ over the rationals $\mathbb{Q}$ whose cohomology is
isomorphic to the rational cohomology of $X$:

$H^{\bullet}(A_{\operatorname{PL}}(X))\cong H^{\bullet}(X;\mathbb{Q}).$

The DGCA $A_{\operatorname{PL}}(X)$, called the polynomial de Rham algebra or
the algebra of polynomial differential forms on $X$, is defined in Section
4.2. By definition, a Sullivan minimal model $M(X)$ of a path-connected space
$X$ is a minimal Sullivan model of $A_{\operatorname{PL}}(X)$. Since $M(X)$ is
defined uniquely up to isomorphism, it is usually called _the_ Sullivan
minimal model of $X$. One has

(1) $H^{\bullet}(M(X))\cong H^{\bullet}(X;\mathbb{Q})$

and

(2)
$M(X)^{>0}/(M(X)^{>0})^{2}\cong\operatorname{Hom}_{\mathbb{Z}}(\pi_{\bullet}(X),\mathbb{Q}).$

This implies that $M(X)\cong
S(\operatorname{Hom}_{\mathbb{Z}}(\pi_{\bullet}(X),\mathbb{Q}))$.

###  4.1. Real algebraic models of manifolds

If $X$ is a connected manifold, then $\Omega^{\bullet}(X)$ is quasi-isomorphic
to $A_{\operatorname{PL}}(X)\otimes_{\mathbb{Q}}\mathbb{R}$ and defines the
real homotopy type of $X$, [FOT08, Section 2.4.1]. The _Sullivan minimal model
$M_{\mathbb{R}}(X)$ of $X$ over $\mathbb{R}$_ is then the minimal Sullivan
model of $\Omega^{\bullet}(X)$. Let us compute the real homotopy type, i.e.,
the Sullivan minimal model over the reals, of the sphere $S^{n}$ for every
$n\geq 1$. The Sullivan minimal model of $S^{n}$ over $\mathbb{Q}$ will be
exactly the same, except that the ground field will be changed to
$\mathbb{Q}$.

######  Example 4.1 (The real homotopy type of an odd-dimensional sphere).

We claim that the polynomial algebra
$\mathbb{R}[g_{n}]\cong\mathbb{R}\oplus\mathbb{R}g_{n}$ in a variable $g_{n}$
of degree $n$ with differential determined by $dg_{n}=0$ is a Sullivan minimal
model of $S^{n}$. Indeed, choose a differential $n$-form $\omega_{n}$ on
$S^{n}$ generating its cohomology $H^{n}(S^{n};\mathbb{R})$ and define an
algebra homomorphism

$\displaystyle\mathbb{R}[g_{n}]$ $\displaystyle\to\Omega^{\bullet}(S^{n}),$
$\displaystyle g_{n}$ $\displaystyle\mapsto\omega_{n},$

which obviously respects the grading and differentials. Moreover, it is clear
it is a quasi-isomorphism. $\mathbb{R}[g_{n}]$ is trivially Sullivan and
minimal. We conclude that $M_{\mathbb{R}}(S^{n})=\mathbb{R}[g_{n}]$ with
$\lvert g_{n}\rvert=n$ and $d=0$ for $n$ odd.

######  Example 4.2 (The real and rational homotopy type of an even-
dimensional sphere).

If $n\geq 2$ is even, then arguing similarly, we observe that the cohomology
algebra $(\mathbb{R}[g_{n}]/(g_{n}^{2}),0)$, admits a quasi-isomorphism to the
de Rham algebra $\Omega^{\bullet}(S^{n})$. The problem is that
$(\mathbb{R}[g_{n}]/(g_{n}^{2}),0)$ is not Sullivan, as it is not even
semifree. If we drop the relation $g_{n}^{2}=0$ to gain freeness, we will
acquire cohomology in each degree which is a multiple of $n$. To kill the
cohomology class $g_{n}^{2}$, we introduce a new generator $g_{2n-1}$ of
degree $2n-1$ with $dg_{2n-1}=g_{n}^{2}$ and get the semifree resolution
$\mathbb{R}[g_{n},g_{2n-1}]\xrightarrow{\operatorname{q-is}}\mathbb{R}[g_{n}]/(g_{n}^{2})$
with $dg_{n}=0$ and $dg_{2n-1}=g^{2}_{n}$ of
$(\mathbb{R}[g_{n}]/(g_{n}^{2}),0)$. This semifree resolution is simply
connected and minimal and thereby Sullivan. Thus,
$M_{\mathbb{R}}(S^{n})=\mathbb{R}[g_{n},g_{2n-1}]$ with $\lvert
g_{n}\rvert=n$, $\lvert g_{2n-1}\rvert=2n-1$, and $dg_{n}=0$,
$dg_{2n-1}=g_{n}^{2}$ for $n$ odd. Note that we could have set
$dg_{2n-1}=cg^{2}_{n}$ for any nonzero $c\in\mathbb{R}$. The resulting DGCA
would be isomorphic to the one with $c=1$. The choice of $c=-1/2$ looks more
like the Maurer-Cartan equation, $dg_{2n-1}+\tfrac{1}{2}g^{2}_{n}=0$, and is
particularly common in physics, see [Sat18].

Note that in both examples, the cohomology algebra is quasi-isomorphic to the
de Rham algebra. In such cases, we say that the manifold and its de Rham
algebra are _formal_. For a general DGCA $A$ or space $X$, we say it is
_formal_ , if $A$ or, respectively, $A_{\operatorname{PL}}(X)$, is quasi-
isomorphic to its cohomology DGCA (with the zero differential). Another famous
class of formal spaces are compact Kähler manifolds, as per the Deligne-
Griffiths-Morgan-Sullivan theorem [DGMS75].

######  Example 4.3 (Realization of $g_{3}$ for $S^{2}$ as a Chern-Simons
form).

For an even number $n\geq 2$, one may wonder where the element $g_{2n-1}$ for
the sphere $S^{n}$ lives. It is certainly not there in the cohomology or even
the de Rham algebra of $S^{n}$, as both vanish in degree $2n-1$. M. Kapranov
has suggested that $g_{3}$ for $S^{2}$ may be realized as a Chern-Simons form
on
$X=\mathbb{C}{\mathbb{P}}^{\infty}\setminus\mathbb{C}{\mathbb{P}}^{\infty-2}$,
which is another model of $S^{2}$:
$X=\\{[z_{0}:z_{1}:z_{2}:\dots]\in\mathbb{C}{\mathbb{P}}^{\infty}\mid
z_{0}\neq 0\text{ or }z_{1}\neq 0\\}$ deformation retracts to the subspace
$0=z_{2}=z_{3}=\dots$, which is homeomorphic to $\\{[z_{0}:z_{1}]\mid
z_{0}\neq 0\text{ or }z_{1}\neq 0\\}=\mathbb{C}{\mathbb{P}}^{1}\cong S^{2}$.
Let $\omega_{2}$ be the _curvature_ form of the Chern connection associated to
the standard Hermitian metric on $\mathcal{O}(1)$, the dual tautological line
bundle on $\mathbb{C}{\mathbb{P}}^{\infty}$. The cohomology class of
$\tfrac{i}{2\pi}\omega_{2}$ is the first Chern class $c_{1}(\mathcal{O}(1))$
of the line bundle $\mathcal{O}(1)$. The cohomology class of $\omega_{2}$
freely generates the real cohomology $\mathbb{R}[\omega_{2}]$ of
$\mathbb{C}{\mathbb{P}}^{\infty}$ and no power of $\omega_{2}$ vanishes on the
dense open $X\subset\mathbb{C}{\mathbb{P}}^{\infty}$. However, the cohomology
class of $\omega_{2}\wedge\omega_{2}$ in $H^{4}(X;\mathbb{R})\cong
H^{4}(S^{2};\mathbb{R})=0$ vanishes and hence there exists a 3-form
$\omega_{3}$ on $X$ such that $d\omega_{3}=\omega_{2}\wedge\omega_{2}$. In
this sense, $\omega_{3}$ is the rudimentary, abelian version of a Chern-Simons
form, being associated to a connection on a complex line bundle, rather than
an $\operatorname{SU}(n)$-bundle for $n\geq 2$. Note that the whole DGCA
$M(S^{2})=\mathbb{R}[g_{2},g_{3}]$ may be realized as the sub-DGCA
$\mathbb{R}[\omega_{2},\omega_{3}]$ of the de Rham algebra
$\Omega^{\bullet}(X)$, which is another algebraic model of $S^{2}\sim X$:

$M(S^{2})=\mathbb{R}[g_{2},g_{3}]\cong\mathbb{R}[\omega_{2},\omega_{3}]\xhookrightarrow{\operatorname{q-is}}\Omega^{\bullet}(X).$

###  4.2. Rational algebraic models of spaces

Here we define the polynomial de Rham algebra $A_{\operatorname{PL}}(X)$ for a
topological space $X$. We will do it by considering the simplicial set
$\operatorname{Sing}(X)$ of singular simplices of $X$ and gluing the algebra
out of polynomial differential forms on each singular simplex.

The _polynomial de Rham algebra_ or the _algebra of polynomial differential
forms on the standard $n$-simplex_

$\Delta^{n}\colonequals\left\\{(t_{0},\dots,t_{n})\in\mathbf{R}^{n+1}~{}\Bigg{|}~{}\sum_{i=0}^{n}t_{i}=1\text{
and }t_{i}\geq 0\text{ for }i=0,\ldots,n\right\\}$

is the DGCA

$\Omega^{\bullet}_{\text{poly}}([n])\colonequals\mathbb{Q}[t_{0},...,t_{n},dt_{0},...,dt_{n}]/(\sum
t_{i}-1,\sum dt_{i}).$

Varying $n$, we get a simplicial DGCA, $\Omega^{\bullet}_{\text{poly}}$. For
any simplicial set $K$, define the _polynomial de Rham algebra of $K$_ as

$A_{\operatorname{PL}}(K)\colonequals\operatorname{Hom}_{\operatorname{\mathbf{sSet}}}(K,\Omega^{\bullet}_{\text{poly}}).$

Here $\operatorname{Hom}_{\operatorname{\mathbf{sSet}}}$ stands for the set of
morphisms of simplicial sets, i.e., maps of sets
$K([n])\to\Omega^{\bullet}_{\text{poly}}([n])$ for each $n\geq 0$ respecting
the face and degeneracy maps, or equivalently, defining natural
transformations between the corresponding functors
$K\colon\thinspace\Delta\to\mathbf{Set}$ and
$\Omega^{\bullet}_{\text{poly}}\colon\thinspace\Delta\to\mathbf{Set}$ on the
simplex category $\Delta$, where $\mathbf{Set}$ is the category of sets. The
structure of a DGCA on $A_{\operatorname{PL}}(K)$ comes from that on each
$\Omega^{\bullet}_{\text{poly}}([n])$. Finally, using the simplicial set
$\operatorname{Sing}(X)$ of singular simplices of a space, we define the
_polynomial de Rham algebra of a space $X$_:

$A_{\operatorname{PL}}(X)\colonequals\operatorname{Hom}_{\operatorname{\mathbf{sSet}}}(\operatorname{Sing}(X),\Omega^{\bullet}_{\text{poly}}).$

For a path-connected space $X$, this DGCA is connected, called a _$($
rational$)$ model of $X$_, and its minimal Sullivan model is called the
_Sullivan minimal model $M(X)$ of $X$_. The minimal Sullivan model of a space
is unique up to isomorphism. For good enough spaces, (the isomorphism class
of) the Sullivan minimal model determines the space up to rational
equivalence, that is to say,

$X\sim_{\mathbb{Q}}Y\qquad\text{if and only if}\qquad M(X)\cong M(Y).$

Good enough spaces include spaces of finite $\mathbb{Q}$-type which are simply
connected, or, more generally, nilpotent, see definitions in Section 7. For
example, if the Sullivan minimal model of a simply connected space is
$\mathbb{Q}[g_{4},g_{7}]$ with $\lvert g_{4}\rvert=4$, $\lvert g_{7}\rvert=7$,
and $dg_{4}=0$, $dg_{7}=-\tfrac{1}{2}g_{4}^{2}$, we may rest assured that this
space is rationally equivalent to the sphere $S^{4}$, see Example 4.2.

###  4.3. Realizations

In the previous section, we associated a DGCA to every topological space. We
did that in two steps, going first from spaces to simplicial sets and then to
DGCAs. Here we will discuss a construction that produces a topological space
for a given DGCA, also in two steps. Let $A$ be a nonnegatively graded DGCA.
Its _simplicial realization_ is the simplicial set

$K_{\operatorname{SR}}(A)\colonequals\operatorname{Hom}_{\operatorname{\mathbf{DGCA}}}(A,\Omega^{\bullet}_{\text{poly}}),$

where $\operatorname{\mathbf{DGCA}}$ is the category of nonnegatively graded
DGCAs. This construction actually defines a (contravariant) functor
$K_{\operatorname{SR}}\colon\thinspace\operatorname{\mathbf{DGCA}}\to\operatorname{\mathbf{sSet}}$,
which turns out to be left adjoint for the (contravariant) polynomial de Rham
functor
$A_{\operatorname{PL}}\colon\thinspace\operatorname{\mathbf{sSet}}\to\operatorname{\mathbf{DGCA}}$
of the previous section. More about this is in Section 6, in which we discuss
model categories.

To pass to topological spaces, we can further apply the functor of _geometric
realization_ :

$K\mapsto\lvert
K\rvert\colonequals\operatorname*{colim}\left(\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
43.01953pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\coprod_{[n]\to[m]}K([m])\times\Delta^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
67.01953pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
67.01953pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
67.01953pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\coprod_{[n]}K([n])\times\Delta^{n}}$}}}}}}}\ignorespaces}}}}\ignorespaces\right).$

In other words, we take the disjoint union $\coprod_{n}K([n])\times\Delta^{n}$
of copies of the standard simplex $\Delta^{n}$, one for each $n$-simplex of
$K$, for all $n\geq 0$, and glue their faces as prescribed by the face maps of
$K$ and degenerate them as prescribed by the degeneracy maps of $K$. This
functor turns out to be left adjoint for the singular simplex functor
$\operatorname{Sing}$.

Finally, a _spatial realization_ of a DGCA $A$ is the geometric realization of
the simplicial realization:

$X_{\operatorname{SR}}(A)\colonequals\lvert K_{\operatorname{SR}}(A)\rvert.$

Unfortunately, this functor is not the left adjoint of the polynomial de Rham
functor $A_{\operatorname{PL}}(X)$: the contravariance of
$K_{\operatorname{SR}}$ messes things up, and we end up composing two
covariant functors, to be on the safe side:

$\textstyle{{\operatorname{\mathbf{DGCA}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{K_{\operatorname{SR}}}}$$\textstyle{{\operatorname{\mathbf{sSet}}}^{\operatorname{op}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\lvert-\rvert}^{\operatorname{op}}}$$\textstyle{{\operatorname{\mathbf{Top}}}^{\operatorname{op}},}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\operatorname{SR}}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\lvert
K_{\operatorname{SR}}(A)\rvert,}$

one which, $K_{\operatorname{SR}}$, is a left adjoint functor and the other
one, $\lvert-\rvert^{\operatorname{op}}$, is a right adjoint, being the
opposite of a left adjoint functor. The notation in the first row is spelled
out below in Section 6. However, the homotopy theory in
$\operatorname{\mathbf{sSet}}$ and $\operatorname{\mathbf{Top}}$ is just the
same, as the functors

$\textstyle{\lvert-\rvert\colon\thinspace\operatorname{\mathbf{sSet}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{\mathbf{Top}}:\operatorname{Sing}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

constitute a Quillen equivalence, and every topologist feels as comfortable in
$\operatorname{\mathbf{sSet}}$ as in $\operatorname{\mathbf{Top}}$, perhaps
even more so. The details are also explained in Section 6.

##  5\. Rationalization

A rationalization may be viewed as a rational topological model of a space,
categorically dual to the Sullivan models considered above. A path-connected
space is called _rational_ if the $\mathbb{Z}$-module $H_{n}(X;\mathbb{Z})$ is
a $\mathbb{Q}$-vector space for each $n\geq 1$. (This means unique
divisibility of each element by natural numbers.) A _rationalization_ of a
path-connected space $X$ is a rational space $X_{\mathbb{Q}}$ along with a
rational equivalence $X\to X_{\mathbb{Q}}$. A rationalization of a simply
connected space is unique up to weak homotopy equivalence and thereby called
_the_ rationalization. In a suitable model-category structure in which the
weak equivalences are the rational homotopy equivalences, the rationalization
$X_{\mathbb{Q}}$ is a fibrant replacement of $X$, see more on model categories
below. We will present a construction of rationalization in Remark 7.1.

##  6\. Model categories

W. G. Dwyer [Dwy08] taught us that ideally a homotopy theory is determined
just by defining the class of weak equivalences without invoking other
ingredients of a model category. Thus, if the reader prefers to keep the
technicalities at bay, they may wish to pay attention only to the classes of
weak equivalences in our model categories.

A _model-category_ structure on a category is characterized by the choice of
three classes of morphisms satisfying certain properties, see [DS95, Hov99]:
weak equivalences, fibrations, and cofibrations. These definitions are modeled
upon the classes of homotopy equivalences, Hurewicz fibrations, and
cofibrations in the category of topological spaces, but this is not the most
commonly used model-category structure on topological spaces and not the one
we will use here. The _standard model category $\operatorname{\mathbf{Top}}$
of topological spaces_ selects the following three classes of morphisms:

  * •

_weak equivalences_ : weak homotopy equivalences;

  * •

_fibrations_ : Serre fibrations;

  * •

_cofibrations_ : continuous maps which have the left lifting property (LLP)
with respect to Serre fibrations which are weak equivalences.

Here we use a more general notion of a weak homotopy equivalence than the one
defined earlier in Section 1 for simply connected spaces:

> For general spaces, we say that $f:X\to Y$ is a _weak $($homotopy$)$
> equivalence_ if the induced map $f_{*}:\pi_{n}(X,x)\to\pi_{n}(Y,f(x))$ is an
> isomorphism for each $n\geq 0$.

For simply connected spaces, this notion is equivalent to the one which
substitutes homology for homotopy, because of the relative Hurewicz theorem,
see the argument in [Hat02, Corollary 4.33].

On our “intermediary” category, $\operatorname{\mathbf{sSet}}$ of simplicial
sets, we impose a model category structure as follows:

  * •

_weak equivalences_ : simplicial maps whose geometric realization is a weak
homotopy equivalence;

  * •

_cofibrations_ : simplicial inclusions;

  * •

_fibrations_ : simplicial maps which have the right lifting (RLP) property
with respect to cofibrations which are weak equivalences.

Fibrations in $\operatorname{\mathbf{sSet}}$ are known as _Kan fibrations_ and
may be characterized by the RLP just with respect to inclusions of horns into
the simplicial representations of simplices, somewhat similar to Serre
fibrations.

Finally, the _model category $\operatorname{\mathbf{DGCA}}$_ of DGCAs is the
category of (by default, nonnegatively graded and over $\mathbb{Q}$) DGCAs
with

  * •

quasi-isomorphisms as _weak equivalences_ ;

  * •

surjective homomorphisms as _fibrations_ ;

  * •

homomorphisms which have the LLP with respect to fibrations which are weak
equivalences as _cofibrations_.

The axioms of a model category guarantee that every model category possesses
an initial object and a terminal object. An object $Q$ is called _cofibrant_
if the unique morphism from the initial object to $Q$ is a cofibration. An
object $R$ is called _fibrant_ if the unique morphism from $R$ to the final
object is a fibration. It is a fact that space is fibrant, every simplicial
set is cofibrant, and every DGCA is fibrant. A CW complex is an example of a
cofibrant topological space. Fibrant simplicial sets are exactly the _Kan
complexes_ ; they are characterized by the horn-filling condition. Sullivan
algebras are known to be cofibrant.

The _homotopy category_ of a model category ${\mathcal{M}}$ is the
localization
$\operatorname{\mathbf{Ho}}({\mathcal{M}})\colonequals{\mathcal{M}}[{\mathcal{W}}^{-1}]$
of ${\mathcal{M}}$ with respect to the class ${\mathcal{W}}$ of weak
equivalences. The localization formally adds inverses of weak equivalences to
the morphisms of ${\mathcal{M}}$. One can develop the notion of _homotopy
between morphisms_ in a model category and show that the homotopy category
$\operatorname{\mathbf{Ho}}({\mathcal{M}})$ is equivalent to the category
whose objects are fibrant cofibrant objects of ${\mathcal{M}}$ and whose
morphisms are homotopy classes of morphisms of ${\mathcal{M}}$, cf. [Hol21,
Theorem 3.29].

######  Theorem 3 ([Hov99, Proposition 3.1.5], [MP12, Theorem 17.5.2]).

The functor $\lvert-\rvert$ of geometric realization and the singular simplex
functor $\operatorname{Sing}$ define a Quillen equivalence

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
18.58333pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\lvert-\rvert\colon\thinspace\operatorname{\mathbf{sSet}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}}{\hbox{\kern
42.58339pt\raise 5.41406pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern
42.58333pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{\mathbf{Top}}:\operatorname{Sing}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
18.58275pt\raise-4.90477pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces}}}}\ignorespaces.$

The theorem means that these functors establish a Quillen adjunction and
induce an equivalence of the homotopy categories. The _Quillen adjunction_
part means it is an adjunction, i.e., there is a natural bijection

$\operatorname{Hom}_{\operatorname{\mathbf{Top}}}(\lvert
K\rvert,X)\xrightarrow{\sim}\operatorname{Hom}_{\operatorname{\mathbf{sSet}}}(K,\operatorname{Sing}(X))$

for simplicial sets $K$ and spaces $X$, and that the functor $\lvert-\rvert$
takes cofibrations to cofibrations and $\operatorname{Sing}$ takes fibrations
to fibrations. A Quillen adjunction always yields the adjunction of the
corresponding (left and right) _derived functors_ , in this case
$L\lvert-\rvert$ and $R\operatorname{Sing}$, that is to say, the functors
induced on the homotopy categories:

(3) $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
32.12503pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{L\lvert-\rvert\colon\thinspace\operatorname{\mathbf{Ho}}(\operatorname{\mathbf{sSet}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
56.11952pt\raise 5.80734pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern
56.12503pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{\mathbf{Ho}}(\operatorname{\mathbf{Top}}):R\operatorname{Sing}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
32.1148pt\raise-5.3655pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces}}}}\ignorespaces.$

The left derived functor is defined by taking a cofibrant replacement before
applying the left adjoint functor and a fibrant replacement before applying
the right adjoint functor. This works both at the level of objects and
morphisms. In our particular case, since all simplicial sets are cofibrant and
topological spaces are fibrant, we bypass the replacements at the level of
objects: $L\lvert K\rvert=\lvert K\rvert$ and
$R\operatorname{Sing}(X)=\operatorname{Sing}(X)$. We still need to replace
maps between simplicial sets by equivalent simplicial inclusions and maps
between spaces by Serre fibrations. All the above replacements exist by the
axioms of model categories.

The _Quillen equivalence_ part of the theorem statement says that the functors
between the original model categories establish an equivalence of their
homotopy categories. This means that the derived functors are not only
adjoint, but also full and faithful. All in all, this implies that homotopy
theory in $\operatorname{\mathbf{Top}}$ is the “same” as in
$\operatorname{\mathbf{sSet}}$. Choose whichever is more convenient!

##  7\. The fundamental theorem of RHT

The fundamental theorem of RHT in a dream world would have said that the
functor $X_{\operatorname{SR}}$ of spatial realization and the polynomial de
Rham algebra functor $A_{\operatorname{PL}}$ establish a Quillen equivalence
between the categories $\operatorname{\mathbf{Top}}$ and
$\operatorname{\mathbf{DGCA}}$. Unfortunately, this does not quite work,
because the spatial realization functor $X_{\operatorname{SR}}=\lvert
K_{\operatorname{SR}}\rvert$ is neither a left, nor a right adjoint functor.
There are also other technicalities which creep in the way, such as the
finiteness and nilpotency conditions. This is another realization, in addition
to the geometric, simplicial, and spatial ones: this time, it is a sad
realization. For this reason, we completely abandon the category of spaces for
the rest of this section and work with the category of simplicial sets
instead.

######  Theorem 4 ([BG76] and [Hol21, Lemma 6.17]).

The functor $K_{\operatorname{SR}}(A)$ of simplicial realization and the
polynomial de Rham algebra functor $A_{\operatorname{PL}}(K)$ form a Quillen
adjunction:

(4) $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
28.54723pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{K_{\operatorname{SR}}\colon\thinspace\operatorname{\mathbf{DGCA}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
52.55624pt\raise 5.24602pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern
52.54723pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{\mathbf{sSet}}^{\operatorname{op}}:A_{\operatorname{PL}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
22.28055pt\raise-4.91226pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces}}}}\ignorespaces.$

Thus, the theorem means that there is a natural bijection

$\operatorname{Hom}_{\operatorname{\mathbf{sSet}}}(L,K_{\operatorname{SR}}(B))\xrightarrow{\sim}\operatorname{Hom}_{\operatorname{\mathbf{DGCA}}}(B,A_{\operatorname{PL}}(L))$

for simplicial sets $L$ and DGCAs $B$ and the functors $K_{\operatorname{SR}}$
and $A_{\operatorname{PL}}$ take cofibrations to fibrations. This implies that
the above adjunction yields the adjunction of the corresponding (left and
right) derived functors $LK_{\operatorname{SR}}$ and $RA_{\operatorname{PL}}$,
that is to say, the functors induced on the homotopy categories:

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
42.08893pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{LK_{\operatorname{SR}}\colon\thinspace\operatorname{\mathbf{Ho}}(\operatorname{\mathbf{DGCA}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
66.09906pt\raise 5.65091pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern
66.08893pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{\mathbf{Ho}}(\operatorname{\mathbf{sSet}}^{\operatorname{op}}):RA_{\operatorname{PL}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
37.88608pt\raise-5.50037pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces}}}}\ignorespaces.$

Note that since the derived functors are the original functors precomposed
with cofibrant and fibrant replacements,
$LK_{\operatorname{SR}}(A)=K_{\operatorname{SR}}(M(A))$, where $M(A)\to A$ is
a minimal Sullivan (or any other cofibrant) model of a DGCA $A$, and
$RA_{\operatorname{PL}}(K)=A_{\operatorname{PL}}(K)$, just because fibrant
objects in the opposite category
$\operatorname{\mathbf{sSet}}^{\operatorname{op}}$ cofibrant in the original
category $\operatorname{\mathbf{sSet}}$, in which every object is cofibrant.

The following result “upgrades” the Quillen adjunction of the previous theorem
to something resembling a Quillen equivalence and may be viewed as the
fundamental theorem of RHT. However, it holds only for certain nice simplicial
sets and nice DGCAs, which we will describe now.

Let $\mathbb{Q}\operatorname{\mathbf{Nil}}^{\operatorname{fft}}$ be the full
subcategory of $\operatorname{\mathbf{sSet}}$ whose objects are connected,
nilpotent, rational Kan complexes of finite $\mathbb{Q}$-type. Let
$\operatorname{\mathbf{Ho}}(\mathbb{Q}\operatorname{\mathbf{Nil}}^{\operatorname{fft}})$
be the category with the same objects as
$\mathbb{Q}\operatorname{\mathbf{Nil}}^{\operatorname{fft}}$ and the morphisms
being homotopy classes of maps. Here _nilpotent_ means that the fundamental
group is nilpotent and acts nilpotently on higher homotopy groups. For
instance, all simply connected spaces and simplicial sets are trivially
nilpotent. Other examples include $S^{1}=K(\mathbb{Z},1)$ and more general
Eilenberg-MacLane spaces $K(A,n)$ with $A$ abelian, as well as path-connected
based loop spaces. On the other hand, $\mathbb{R}{\mathbb{P}}^{\infty}$ is not
nilpotent, despite an abelian fundamental group. A space or simplicial set is
of _finite $\mathbb{Q}$-type_ if its rational homology is finite-dimensional
in each degree.

On the algebraic side, let $\operatorname{\mathbf{DGCA}}^{\operatorname{cft}}$
denote the full subcategory of $\operatorname{\mathbf{DGCA}}$ whose objects
are cofibrant, connected differential graded commutative algebras of finite
type. Define
$\operatorname{\mathbf{Ho}}(\operatorname{\mathbf{DGCA}}^{\operatorname{cft}})$
to be the category whose objects are those of
$\operatorname{\mathbf{DGCA}}^{\operatorname{cft}}$ and whose morphisms are
homotopy classes of DGCA-homomorphisms. Sullivan algebras are known to be
cofibrant [Ber12, Theorem 8.11]. A DGCA is of _finite type_ if it is finite-
dimensional in each degree.

######  Theorem 5 ([BG76] and [Hol21, Theorem 9.5]).

The Quillen adjunction (4) induces an equivalence of categories

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
45.27783pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{LK_{\operatorname{SR}}\colon\thinspace\operatorname{\mathbf{Ho}}(\operatorname{\mathbf{DGCA}}^{\operatorname{cft}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
69.27997pt\raise 5.6813pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern
69.27783pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{\mathbf{Ho}}(\mathbb{Q}\operatorname{\mathbf{Nil}}^{\operatorname{fft}})^{\operatorname{op}}:RA_{\operatorname{PL}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
40.05594pt\raise-5.50037pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces}}}}\ignorespaces.$

Here the homotopy categories are the categories with the same objects but
morphisms replaced with homotopy classes of morphisms.

######  Corollary 6 ([Hol21, Corollary 10.2], [Ber12, Corollary 8.47]).

Let $M(K)\xrightarrow{\operatorname{q-is}}A_{\operatorname{PL}}(K)$ be a
Sullivan minimal model of a connected nilpotent simplicial set $K$ of finite
$\mathbb{Q}$-type. Then the adjoint map $K\to K_{\operatorname{SR}}(M(K))$ is
a rationalization $K\to K_{\mathbb{Q}}$ of $K$.

######  Remark 7.1.

Similarly, given a path-connected nilpotent CW complex $X$ of
$\mathbb{Q}$-finite type, take the adjoint $\operatorname{Sing}(X)\to
K_{\operatorname{SR}}(M(X))$ of the Sullivan minimal model
$M(X)\xrightarrow{\operatorname{q-is}}A_{\operatorname{PL}}(X)=A_{\operatorname{PL}}(\operatorname{Sing}(X))$.
From the derived equivalence (3) between singular sets and spaces, we get a
weak equivalence $\lvert\operatorname{Sing}(X)\rvert\to X$ (the _counit of
adjunction_), which is a homotopy equivalence by Whitehead’s theorem. We can
compose a homotopy inverse to get a continuous map
$X\to\lvert\operatorname{Sing}(X)\rvert\to X_{\operatorname{SR}}(M(X))$, which
will be a rationalization $X\to X_{\mathbb{Q}}$ of $X$.

##  8\. Quillen minimal models of spaces

The advantage of Sullivan models over historically earlier Quillen’s models
derives from the fact that Sullivan models are more user-friendly: they are
based on more familiar constructions, such as algebras and differential forms,
both known since the late 19th century. Quillen minimal models use coalgebras
and free Lie algebras, or in modern interpretation, $L_{\infty}$-algebras, all
of which are less ingrained in the brain of a practitioner and thereby more
awkward to work with.

In the foundational paper [Qui69], Quillen defines a sequence of left and
right Quillen equivalences, carefully avoiding calling them that way. At the
end of the day, these functors assign to a simply connected topological space
$X$ a differential graded Lie algebra (DGLA) $\lambda X$,

$X\mapsto\lambda X,$

defining a functor

(5) $\lambda:\operatorname{\mathbf{Top}}_{1}\to\operatorname{\mathbf{DGLA}},$

where $\operatorname{\mathbf{Top}}_{1}$ is the category of simply connected
spaces and $\operatorname{\mathbf{DGLA}}$ is the category of _positively
graded DGLAs_ $L=\bigoplus_{n>0}L_{n}$, with the bracket $[-,-]:L_{m}\otimes
L_{n}\to L_{m+n}$ being graded antisymmetric,

$[y,x]=-(-1)^{\lvert x\rvert\cdot\lvert y\rvert}[x,y],$

and satisfying a graded version of the Jacobi identity, with a derivation of
degree $-1$:

$d[x,y]=[dx,y]+(-1)^{\lvert x\rvert}[x,dy].$

Quillen constructs $\lambda X$ as the following mouthful:

$\lambda X=N\mathcal{P}\widehat{\mathbb{Q}}G\operatorname{Sing}(X),$

where $G$ associates Kan’s loop group to a simplicial set, a simplicial
version of the based loop space, $\widehat{\mathbb{Q}}G$ is a completed group
algebra, $\mathcal{P}$ takes the simplicial Lie algebra of primitive elements,
and N maps simplicial Lie algebras to DGLAs by taking the normalized chains.

Then Quillen uses a similar machinery to that of DGCAs to define the minimal
model of a (reduced) DGLA and shows that its isomorphism class determines the
rational homotopy type of $X$. Here we will use the same $\lambda X$ but
assign to it a minimal $L_{\infty}$-algebra $Q(X)$, rather than a minimal
DGLA. We will explain in which way this is more in line with Sullivan’s
approach.

First, we need a number of algebraic definitions and constructions. A
_$L_{\infty}$ -algebra_ is a graded vector space
$L=\bigoplus_{n\in\mathbb{Z}}L_{n}$ with a codifferential $D$ on the _cofree
coaugemented cocommutative coalgebra_ , i.e., the graded symmetric coalgebra

$C(L):=S(L[-1])=\bigoplus_{n\geq 0}S^{n}(L[-1]),$

where $L[-1]=\bigoplus_{n\geq 2}L[-1]_{n}$ with $L[-1]_{n}\colonequals
L_{n-1}$ is the suspension of $L$. The comultiplication $\Delta:S(L[-1])\to
S(L[-1])\otimes S(L[-1])$ is the standard shuffle comultiplication, which is
coassociative and graded cocommutative. The coaugmentation is the coalgebra
homomorphism $k=S^{0}(L[-1])\hookrightarrow S(L[-1])$ and the _codifferential_
$D:S(L[-1])\to S(L[-1])$ is a graded degree-$(-1)$ coderivation,

$\Delta(D(x))=D(x_{(1)})\otimes x_{(2)}+(-1)^{\lvert
x_{(1)}\rvert}x_{(1)}\otimes D(x_{(2)}),$

using _Sweedler’s notation_ $x_{(1)}\otimes x_{(2)}\colonequals\Delta(x)$,
such that $D(1)=0$ and $D^{2}=0$. The codifferential on a cofree coalgebra is
determined by the projection onto the cogenerating space $L[-1]$:

$S(L[-1])\xrightarrow{D}S(L[-1])\to L[-1],$

and the restriction of this projection to $S^{n}(L[-1])$ is the _$n$ -ary
bracket_ $l_{n}$ of the $L_{\infty}$-algebra $L$:

$l_{n}:S^{n}(L[-1])\hookrightarrow S(L[-1])\xrightarrow{D}S(L[-1])\to
L[-1],\qquad n\geq 1.$

An $L_{\infty}$-algebra is a generalization of a DGLA: a DGLA $(L,[-,-],d)$
may be regarded as an $L_{\infty}$-algebra with $l_{1}=d$, $l_{2}=[-,-]$, and
$l_{n}=0$ for $n\geq 0$. The condition $D^{2}=0$ is equivalent to the graded
Jacobi equation for the bracket $[-,-]$ and the derivation property of $d$,
along with the equation $d^{2}=0$. When higher brackets $l_{n}$ are present,
the equation $D^{2}$ gives the Jacobi identity for $l_{2}$ up to homotopy
given by $l_{3}$, and similar higher Jacobi identities. The unary bracket
$l_{1}$ is in fact a linear map $L\to L$ of degree $-1$ such that
$(l_{1})^{2}=0$, i.e., a differential on the underlying graded vector space,
denoted by $d\colonequals l_{1}$.

An _$L_{\infty}$ -homomorphism_ $L_{1}\to L_{2}$ is defined as a homomorphism
of the corresponding coaugmented dg-coalgebras $C(L_{1})\to C(L_{2})$. An
_$L_{\infty}$ -isomorphism_ is an invertible $L_{\infty}$-homomorphism, and an
_$L_{\infty}$ -quasi-isomorphism_ is an $L_{\infty}$-homomorphism which
induces an isomorphism on homology of the differentials $d$ on $L_{1}$ and
$L_{2}$.

We say that an $L_{\infty}$-algebra is _minimal_ , if $d=0$. We say that an
$L_{\infty}$-algebra is _contractible_ if $l_{n}=0$ for $n\geq 2$ and
$H_{\bullet}(L,d)=0$ for the homology of $d$. Note that this is not an
$L_{\infty}$-invariant notion, but it is invariant under _linear
$L_{\infty}$-isomorphisms_, i.e., those which are given by linear maps of the
corresponding coalgebras. An _$L_{\infty}$ -model_ of an $L_{\infty}$-algebra
$L$ is an $L_{\infty}$-algebra $L^{\prime}$ with an $L_{\infty}$-quasi-
isomorphism $L^{\prime}\xrightarrow{\operatorname{q-is}}L$. As in the theory
of minimal Sullivan models, minimal $L_{\infty}$-models play a crucial role.

The theory of minimal $L_{\infty}$-models for $L_{\infty}$-algebras, developed
by M. Kontsevich in a couple of impressionistic brush strokes [Kon03], is
parallel to the theory of minimal Sullivan models for DGCAs, but actually
simpler.

The following theorem is parallel to 2 for DGCAs.

######  Theorem 7 ([Kon03, Lemma 4.9], [BFM13, Proposition 2.6]).

Every $L_{\infty}$-algebra is $L_{\infty}$-isomorphic to the direct sum of a
minimal $L_{\infty}$-algebra and a contractible one.

###### Proof.

The proof of this theorem is also simpler than that of its DGCA version. If
the fact that in Kontsevich’s paper [Kon03], it takes only seven lines is not
convincing, let us indicate the idea. Since we are working over a field, the
underlying complex (i.e., the dg-space) $(L,d)$ of an $L_{\infty}$-algebra $L$
is isomorphic to the direct sum of its homology $H_{\bullet}(L,d)$ and an
acyclic complex. The resulting projection of the complex $(L,d)$ onto its
homology will then be a chain homotopy equivalence. Then the homotopy transfer
theorem, also known as the homological perturbation lemma, which uses the fact
that the $L_{\infty}$-operad is cofibrant in a suitable model category of
operads, see for example, [Ber12, Theorem 1.23], gives an $L_{\infty}$
structure on $H_{\bullet}(L,d)$ so that the inclusion to $L$ and projection
onto $H_{\bullet}(L,d)$ are $L_{\infty}$-quasi-isomorphisms. ∎

This theorem and a rather obvious inverse mapping theorem [Kon03, Section
4.1], which says that an $L_{\infty}$-homomorphism whose linear part is an
isomorphism of graded vector spaces is an $L_{\infty}$-isomorphism,
immediately implies the following analogue of 1.

######  Theorem 8 ([Kon03, Section 4.5.1], [BFM13, Theorem 2.7]).

  1. $(1)$

Every $L_{\infty}$-quasi-isomorphism between minimal $L_{\infty}$-algebras is
an $L_{\infty}$-isomorphism.

  2. $(2)$

Every DGLA has a minimal $L_{\infty}$-model, unique up to isomorphism.

Now back to spaces. Let us use the functor (5) constructed by Quillen. Given a
simply connected space $X$, let us call a minimal $L_{\infty}$-model
$Q(X)\xrightarrow{\operatorname{q-is}}\lambda X$ of the DGLA $\lambda X$ the
_Quillen minimal model_ of $X$. It is defined up to $L_{\infty}$-isomorphism
by the theorem above. Note that traditionally, the Quillen model is defined
differently: it is a DGLA which is free as a graded Lie algebra and with a
decomposable differential, see for example, [BS23, Section 1.9]. In this way,
Quillen’s original approach is also parallel to Sullivan’s theory, but the
structure Theorems 2 and 7 suggest that Koszul duality between DGCAs and
$L_{\infty}$-algebras may be a better alternative to the parallel between
semifree DGCAs and semifree DGLAs.

The following analogue of 5 works under weaker assumptions and deals with
covariant, rather than contravariant functors. Consider the composition of
functors

$\operatorname{\mathbf{Top}}_{1}\xrightarrow{\lambda}\operatorname{\mathbf{DGLA}}\hookrightarrow\mathbf{L_{\infty}}\text{-}\mathbf{Alg}\xrightarrow{C}\operatorname{\mathbf{DGCC}}_{1}.$

Even though Quillen does not include the category
$\mathbf{L_{\infty}}\text{-}\mathbf{Alg}$ of _positively graded
$L_{\infty}$-algebras with $L_{\infty}$-morphisms_, by our definition above,
it is equivalent to a full subcategory of the category
$\operatorname{\mathbf{DGCC}}_{1}$ of _simply connected coaugmented dg-
cocommutative coalgebras_ , those with $C_{0}=\mathbb{Q}$ and $C_{1}=0$, the
full subcategory given by the cofree objects. structures under which

######  Theorem 9 ([Qui69, Theorem I]).

The functor $\lambda$ composed with the functors
$\operatorname{\mathbf{DGLA}}\to\mathbf{L_{\infty}}\text{-}\mathbf{Alg}\linebreak[0]\to\operatorname{\mathbf{DGCC}}_{1}$
induces equivalences of categories

$\operatorname{\mathbf{Ho}}_{\mathbb{Q}}(\operatorname{\mathbf{Top}}_{1})\xrightarrow{\sim}\operatorname{\mathbf{Ho}}(\operatorname{\mathbf{DGLA}})\xrightarrow{\sim}\operatorname{\mathbf{Ho}}(\mathbf{L_{\infty}}\text{-}\mathbf{Alg})\xrightarrow{\sim}\operatorname{\mathbf{Ho}}(\operatorname{\mathbf{DGCC}}_{1}),$

where
$\operatorname{\mathbf{Ho}}_{\mathbb{Q}}(\operatorname{\mathbf{Top}}_{1})$ is
the localization of the category $\operatorname{\mathbf{Top}}_{1}$ by rational
equivalences and, in the other cases, $\operatorname{\mathbf{Ho}}$ stands for
the localization of the corresponding category with respect to quasi-
isomorphisms.

This theorem implies that the isomorphism class of the Quillen minimal model
$Q(X)$ is a complete invariant of the rational homotopy type of a simply
connected space $X$: such spaces $X$ and $Y$ are rationally equivalent if and
only if their Quillen minimal models $Q(X)$ and $Q(Y)$ are
$L_{\infty}$-isomorphic.

One feature of Quillen’s functor $\lambda$ is that the rational homology of
the DGLA $\lambda X$ is naturally isomorphic to the rational homotopy of a
simply connected space $X$, with a shift:

$H_{\bullet}(\lambda X,d)\cong\pi_{\bullet}(X)\otimes\mathbb{Q}[1].$

Therefore, the shifted rational homotopy groups
$\pi_{\bullet}(X)\otimes\mathbb{Q}[1]$ acquire a graded Lie bracket. Quillen
shows that this bracket is the Whitehead product [Qui69, Theorem I].

By definition, the Quillen minimal model $Q(X)$ is an $L_{\infty}$-algebra
based on the graded $\mathbb{Q}$-vector space

$Q(X)\cong H_{\bullet}(\lambda X,d)\cong\pi_{\bullet}(X)\otimes\mathbb{Q}[1],$

cf. (1) and (2) for the Sullivan minimal model. The binary bracket of this
$L_{\infty}$ structure is the Whitehead product, while the higher $L_{\infty}$
brackets have been shown to be the higher Whitehead products, see [BBMFM17].

For a simply connected space $X$ of finite $\mathbb{Q}$-type, the graded
linear dual $C(Q(X))^{*}$ of the dg-cocommutative coalgebra $C(Q(X))$ of the
Quillen minimal model $Q(X)$ is a DGCA. The fact that the differential on the
$L_{\infty}$-algebra $Q(X)$ vanishes implies that the linear part of the
differential, the dual of $l_{1}$, on $C(Q(X))^{*}$ vanishes, i.e., the
differential is decomposable. Also, under the finiteness assumption, the
graded commutative algebra $C(Q(X))^{*}$ is isomorphic to $S(Q(X)^{*})$. This
implies that $C(Q(X))^{*}$ is a minimal Sullivan algebra. Thus, a space $X$ of
the above type gets two minimal Sullivan algebras: $C(Q(X))^{*}$ and the
Sullivan minimal model $M(X)$. Two natural constructions rarely happen to be
different. This is not the case here, either, as the following interpretation
of Majewski’s result shows.

######  Theorem 10 (Majewski [Maj00, Theorem 4.90], cf. [BFM13, Remark 2.9]
and [BS23, Remark 1.40]).

The two minimal Sullivan algebras associated to a simply connected space of
finite $\mathbb{Q}$-type, $C(Q(X))^{*}$ and the Sullivan minimal model $M(X)$
are isomorphic.

###### Proof.

Under the assumptions, the graded dual $M(X)^{*}\cong
S(\pi_{\bullet}(X)\otimes\mathbb{Q})$, see (2), is a dg-cocommutative
coalgebra, just like $C(Q(X))\cong S(\pi_{\bullet}(X)\otimes\mathbb{Q})$. The
corresponding $L_{\infty}$ structures on
$\pi_{\bullet}(X)\otimes\mathbb{Q}[1]$ are $L_{\infty}$-isomorphic. Indeed,
the first one is $L_{\infty}$-quasi-isomorphic to its double bar construction,
which is the bar construction ${\mathcal{L}}^{*}(M(X))$ of $M(X)$ (the free
Lie algebra on $M(X)^{*}[1]$ with the differential incorporating the DGCA
structure on $M(X)$). By the very definition of $Q(X)$, the second
$L_{\infty}$ structure on $\pi_{\bullet}(X)\otimes\mathbb{Q}[1]$ is
$L_{\infty}$-quasi-isomorphic to $\lambda X$, which is quasi-isomorphic as a
DGLA to Quillen’s original minimal model $L_{X}$ of $X$ (the minimal semifree
DGLA model of $\lambda X$). Majewski [Maj00, Theorem 4.90] proves that the
DGLAs ${\mathcal{L}}^{*}(M(X))$ and $L_{X}$ are quasi-isomorphic. Therefore,
so are the two $L_{\infty}$ structures on
$\pi_{\bullet}(X)\otimes\mathbb{Q}[1]$. Since both are minimal, they must be
$L_{\infty}$-isomorphic by 8. By definition of an $L_{\infty}$-isomorphism,
this means the corresponding dg-cocommutative coalgebras $M(X)^{*}$ and
$C(Q(X))$ are isomorphic. Thus, the DGCAs $M(X)$ and $C(Q(X))^{*}$ are
isomorphic. ∎

##  9\. Some physical applications

###  9.1. Hypothesis H

Sati [Sat18] made a remarkable observation that the equations of motion (EOMs)
of supergravity in an 11-dimensional ($d=11$) spacetime are essentially the
equations for the differential in the Sullivan minimal model of $S^{4}$, see
Example 4.2. Supergravity is the low-energy, infrared limit of M-theory, which
is supposed to be the “mother of them all:” by combining S- and T-dualities,
one can arrive at type I, IIA, and IIB string theories as well as various
flavors of heterotic string theories. Dimensional reductions of $d=11$
supergravity lead to supergravity theories in lower dimensions.

The EOMs of $d=11$ supergravity are the equations

(6) $\displaystyle dG_{4}=0,\qquad dG_{7}=-\frac{1}{2}G_{4}\wedge G_{4},$
$\displaystyle G_{7}=*G_{4}$

on two fields $G_{4}$ and $G_{7}$, which are just differential forms on the
11-dimensional spacetime $X$:

$G_{4}\in\Omega^{4}(X),\qquad G_{7}\in\Omega^{7}(X).$

Here $*$ is the Hodge star, derived from a given metric on $X$. Returning to
the following Sullivan minimal model of $S^{4}$:
$M_{\mathbb{R}}(S^{4})=\mathbb{R}[g_{4},g_{7}]$ with $\lvert g_{4}\rvert=4$,
$\lvert g_{7}\rvert=7$, and $dg_{4}=0$, $dg_{7}=-\tfrac{1}{2}g_{4}^{2}$, we
see that there is a DGCA homomorphism

(7) $\displaystyle M_{\mathbb{R}}(S^{4})$
$\displaystyle\to\Omega^{\bullet}(X),$ $\displaystyle g_{4}$
$\displaystyle\mapsto G_{4},$ $\displaystyle g_{7}$ $\displaystyle\mapsto
G_{7}.$

Moreover, the RHT of the four-sphere encompasses the topological part (6) of
the EOMs, called the _duality-symmetric EOMs_. The equation $G_{7}=*G_{4}$ is
important, but it is geometric, rather than topological. In a way, the four-
sphere carries in its rational homotopy type information about the duality-
symmetric dynamics of any $d=11$ supergravity theory. RHT gives even more
content to this observation, a continuous map

(8) $X\to S^{4}_{\mathbb{R}}$

to the _rationalization of $S^{4}$ over the reals_, which may be defined as
the spatial realization $X_{\operatorname{SR}}(M_{\mathbb{R}}(S^{4}))$.
($S^{4}_{\mathbb{R}}$ has the same real homotopy type as $S^{4}$ via a real
homotopy equivalence $S^{4}\to S^{4}_{\mathbb{R}}$.) The above map is obtained
from the adjoint of (7), see Remark 7.1. It intertwines the fields $G_{4}$ and
$G_{7}$ on spacetime $X$ with the generators $g_{4}$ and $g_{7}$ of the
Sullivan minimal model of $S^{4}$. Thus, $S^{4}_{\mathbb{R}}$ becomes the
universal target space of $d=11$ supergravity in any spacetime. These
observations is the essence of Sati’s _Hypothesis H_ [Sat18].

Supergravity in 11 dimensions may be reduced to lower dimensions, and
Hypothesis H extends to establish similar ties between the EOMs and the
Sullivan minimal models of certain loop spaces of $S^{4}$, along with similar
maps.

The $(11-k)$-dimensional reduction of 11-dimensional supergravity is obtained
under the assumption that there is a $k$-torus $(S^{1})^{k}$ acting on
spacetime $Y$ of $\dim Y=11$, so that the new spacetime is the quotient
$Y/(S^{1})^{k}$. Let us look at the case $k=1$, as this case is easily
generalized to higher $k$’s. There is an adjunction

(9)
$\operatorname{Hom}_{/BS^{1}}(Y/S^{1},\mathcal{L}_{c}Z)\xrightarrow{\;\;\sim\;\;}\operatorname{Hom}(Y,Z),$

where $Y$ is a space with a free action of $S^{1}$, $Z$ is another topological
space, $\mathcal{L}_{c}Z\colonequals\operatorname{Map}(S^{1},Z)/\\!\\!/S^{1}$
is the _cyclic $($or equivariant$)$ loop space of $Z$_, the left-hand side is
the set of morphisms in the category $/BS^{1}$ of spaces over $BS^{1}$
(equivalent to the category of principal $S^{1}$-bundles, such as $Y\to
Y/S^{1}$ and $\mathcal{L}Z\to\mathcal{L}_{c}Z$), and the right-hand side is
the set of continuous maps $Y\to Z$; see [FSS18b][BMSS19, Theorem 2.44]. This
adjunction produces a map

(10) $Y/S^{1}\longrightarrow\mathcal{L}_{c}S^{4}_{\mathbb{R}}$

from the map (8). If the action of $S^{1}$ on $Y$ is not free, one has to
replace the naive quotient $Y/S^{1}$ by the homotopy quotient $Y/\\!\\!/S^{1}$
in the above. Roughly speaking, thinking of
$\mathcal{L}_{c}S^{4}_{\mathbb{R}}$ as the space of unparameterized (or
equivariant) free loops in $S^{4}_{\mathbb{R}}$, a map like (8) from an
$S^{1}$-space $Y$ will produce a map (10) which assigns to a point $y\in
Y/S^{1}$ the map that takes the $S^{1}$-orbit in $Y$ over $y$ to
$S^{4}_{\mathbb{R}}$ by the given map $Y\to S^{4}\mathbb{R}$.

10-Dimensional supergravity (corresponding to type IIA string theory), is
based on the spacetime $Y/S^{1}$ and, apart from the above map (10), it is
related to the rational homotopy type of $\mathcal{L}_{c}S^{4}_{\mathbb{R}}$
via the (duality-symmetric) EOMs, which explicitly match the equations for the
differential of the Sullivan minimal model $M(\mathcal{L}_{c}S^{4})$:

$\displaystyle
M(\mathcal{L}_{c}S^{4})=(\mathbb{R}[g_{4},g_{7},sg_{4},sg_{7},w],d)\;,$
$\displaystyle dg_{4}=sg_{4}\cdot w,\qquad
dg_{7}=-\tfrac{1}{2}g_{4}^{2}+sg_{7}\cdot w\;,$ $\displaystyle dsg_{4}=0,\quad
dsg_{7}=sg_{4}\cdot g_{4},\quad dw=0\;,$

where $\deg w=2$ and, for each generator $x$ of $M(S^{4})$, $sx$ is a new
generator of $\deg sx=\deg x-1$. The Sullivan minimal model
$M(\mathcal{L}_{c}Z)$ of the cyclic loop space of a path-connected, nilpotent
space $Z$ is obtained via a simple algebraic procedure developed in the good
old times of RHT, a theorem of Vigué-Poirrier and Burghelea [VPB85]. This
matching of equations of $d=10$, type IIA supergravity with the Sullivan
minimal model of $M(\mathcal{L}_{c}S^{4})$ was observed and developed by D.
Fiorenza, U. Schreiber, and Sati [FSS15, FSS17, FSS18a] and provided a
confirmation of Hypothesis H: it is compatible with dimensional reduction.

The papers [SV23b, SV23a] extended Hypothesis H to further dimensional
reductions of supergravity. Namely, the same pattern persists for the values
of $k$ through $k=10$, which is a matter of explicit computation of
$M(\mathcal{L}_{c}^{k}S^{4})$ by iterating Vigué-Poirrier and Burghelea’s
theorem and comparing it to the known EOMs of dimensionally reduced
supergravity theories. We may summarize this application of RHT to
Mathematical Physics as follows:

> The iterated cyclic loop space $\mathcal{L}_{c}^{k}S^{4}_{\mathbb{R}}$
> admits a map from the $(11-k)$-dimensional supergravity spacetime, thus
> making $\mathcal{L}_{c}^{k}S^{4}_{\mathbb{R}}$ a universal target for all
> possible spacetimes $Y/(S^{1})^{k}$. The Sullivan minimal model
> $M(\mathcal{L}_{c}^{k}S^{4})$ determines the duality-symmetric equations of
> motion of $(11-k)$-dimensional supergravity reduced from 11-dimensional
> supergravity.

This matching also implies the following general principle:

> Any feature of or statement about the Sullivan minimal model of an iterated
> cyclic loop space $\mathcal{L}_{c}^{k}S^{4}$ (or the rational homotopy type
> thereof) may be translated into a feature of or statement about the
> compactification of M-theory on the $k$-torus.

###  9.2. Mysterious Triality and cyclic loop spaces of $S^{4}$

Mysterious Duality has been tantalizing the world of mathematical physics
since 2001, when A. Iqbal, A. Neitzke, and C. Vafa [INV01] discovered a
convincing, yet mysterious correspondence between certain symmetry patterns in
the geometry of del Pezzo surfaces, see [Man86], and in toroidal
compactifications of M-theory, both governed by the root system series
$E_{k}$.

In the context of algebraic geometry, the $E_{k}$ root system appears out of
intersection theory on del Pezzo surfaces by a simple algebraic argument:
given a _Lorentzian_ , _i.e_., of index $(1,k)$, lattice of rank $k+1$ with
$k\leq 8$, let $-K$ be the vector $-K=(3,-1,-1,\dots,-1)$ with respect to a
Lorentzian basis; then the set of lattice vectors of inner square $-2$ and
orthogonal to $-K$ form a root system that may be shown to be the root system
of type $E_{k}$. What is remarkable in the geometry of del Pezzo surfaces is
that all this data is natural: the lattice is the second cohomology group
$H^{2}(\mathbb{B}_{k};\mathbb{Z})$ of the del Pezzo surface $\mathbb{B}_{k}$,
the Lorentzian inner product is the intersection form, and $-K$ is the
anticanonical class, the first Chern class of the holomorphic tangent bundle
of $\mathbb{B}_{k}$.

In [SV23b], it was observed that the same data (a Lorentzian lattice and
distinguished vector) as for del Pezzo surfaces above naturally appears in the
rational homotopy theory models for the iterated cyclic loop spaces of
$4$-spheres $S^{4}$:

$S^{4},\mathcal{L}_{c}S^{4},\mathcal{L}_{c}^{2}S^{4},\dots,\mathcal{L}_{c}^{8}S^{4}.$

We assume that if $\mathcal{L}_{c}Z$ on this list happens not to be connected,
we keep only the connected component of the constant loop $S^{1}\to Z$ and
slightly abuse notation by using the same symbol $\mathcal{L}_{c}Z$ for the
connected component.

Here is how the $E_{k}$ root system shows up: it is naturally associated to
the infinitesimal symmetries of the Sullivan minimal model
$M(\mathcal{L}_{c}^{k}S^{4})$. The paper [SV23b] shows that that a maximal
$\mathbb{Q}$-split torus $T$ in the automorphism group
$\operatorname{Aut}M(\mathcal{L}_{c}^{k}S^{4})$ of
$M(\mathcal{L}_{c}^{k}S^{4})$ is isomorphic to $(\mathbb{Q}^{\times})^{k+1}$,
where
$\mathbb{Q}^{\times}:=\mathbb{Q}\setminus\\{0\\}=\mathbb{G}_{m}(\mathbb{Q})$
is the _multiplicative group_ of the rationals, and so the Lie algebra
$\mathfrak{h}=\operatorname{Lie}(T)$ of this maximal split torus $T$ is a
$(k+1)$-dimensional rational vector space. The Lie algebra $\mathfrak{h}$ is
canonically equipped with a Lorentzian metric and a distinguished Lorentzian
basis. Indeed, this data is equivalent to the data of two distinguished bases,
one for $\mathfrak{h}$ and one for the dual vector space $\mathfrak{h}^{*}$.
For the dual space $\mathfrak{h}^{*}$, a canonical basis comes from the weight
lattice for the $\mathfrak{h}$-action on $M(\mathcal{L}_{c}^{k}S^{4})$. A
distinguished basis of $\mathfrak{h}$ comes directly from the rational
homotopy theory of $\mathcal{L}_{c}^{k}S^{4}$. In the rational homotopy
category, both the degree-$n$ folding maps of the source $S^{1}$’s and of the
target $S^{4}$ for $\mathcal{L}_{c}^{k}S^{4}$ are invertible as long as $n\neq
0$. Taking into account their inverses, one has a natural action (in the
rational homotopy category) of $k+1$ copies of $\mathbb{Q}^{\times}$ on
$\mathcal{L}_{c}^{k}S^{4}$, one for each of the source circles $S^{1}$ and one
for the target $S^{4}$. And since the action of the folding maps on the loop
space is by precomposition in the sources and postcomposition in the target,
one gets from this that the weights for the infinitesimal generators of the
folding actions are $-1$ for the infinitesimal source foldings and $+1$ for
the target folding, thus giving a natural Lorentzian metric. Finally, the
distinguished vector $-K$ is also quite natural: it is a unique element
$-K\in\mathfrak{h}$ which acts as the degree operator on the Quillen minimal
model $\mathbb{Q}(\mathcal{L}_{c}^{k}S^{4})$ of $\mathcal{L}_{c}^{k}S^{4}$.
Here, the Quillen minimal model is understood in the sense of Section 8 as a
minimal $L_{\infty}$ structure on
$\pi_{\bullet}(\mathcal{L}_{c}^{k}S^{4})\otimes\mathbb{Q}[1]$. Because the
differential of the Sullivan minimal model $M(\mathcal{L}_{c}^{k}S^{4})$
happens to be purely quadratic, the Quillen minimal model
$Q(\mathcal{L}_{c}^{k}S^{4})$ is just a graded Lie algebra, being defined by
the Sullivan minimal model $M(\mathcal{L}_{c}^{k}S^{4})$ via the
Sullivan/Quillen-model duality 10.

The above sketchy argument may be summarized in the following theorem.

######  Theorem 11.

  1. (a)

The maximal $\mathbb{Q}$-split torus of the rational algebraic group
$\operatorname{Aut}M(\mathcal{L}_{c}^{k}S^{4})$ for $k\geq 0$ is a
$(k+1)$-dimensional torus $T^{k+1}$ canonically isomorphic to
$\mathbb{G}_{m}^{k+1}$ over $\mathbb{Q}$.

  2. (b)

The action of the maximal split torus $\mathbb{G}_{m}^{k+1}$ on
$M(\mathcal{L}^{k}_{c}S^{4})$ may be lifted to an action on the space
$\mathcal{L}_{c}^{k}S^{4}$ in the rational homotopy category. In this way, the
last factor $\mathbb{G}_{m}$ of $\mathbb{G}_{m}^{k+1}$ acts via self-maps of
the target $S^{4}$ and the first $k$ factors act via self-maps of the source
$S^{1}$’s.

  3. (c)

The $(k+1)$-dimensional rational abelian Lie algebra
$\mathfrak{h}_{k}=\operatorname{Lie}(T^{k+1})\subseteq\linebreak[1]\operatorname{Der}\linebreak[0]M(\mathcal{L}_{c}^{k}S^{4}),$
which plays the role of a Cartan subalgebra, of the maximal $\mathbb{Q}$-split
torus $T=T^{k+1}$ of the rational algebraic group
$\operatorname{Aut}M(\mathcal{L}_{c}^{k}S^{4})$ has an explicit canonical
basis. So does the linear dual $\mathfrak{h}_{k}^{*}$, which plays the role of
a weight space.

  4. (d)

There is a unique element $-K$ of the Lie algebra $\mathfrak{h}_{k}$, which
acts on the Quillen minimal model $Q(\mathcal{L}_{c}^{k}S^{4})$ as the degree
operator.

  5. (e)

For each $k$, $0\leq k\leq 8$, the basis of $\mathfrak{h}_{k}$ from c and
element $-K$ from d above form a Lorentzian lattice of rank $k+1$ with a
distinguished element, giving rise to the exceptional root system $E_{k}$.
This data, extracted from the cyclic loop space $M(\mathcal{L}^{k}_{c}S^{4})$,
replicates the root data determined by the del Pezzo surface $\mathbb{B}_{k}$.
The construction of the root data extends to $k\geq 9$ to the root systems of
Kac-Moody algebras of type $E_{k}$.

  6. (f)

$27$ Lines via rational homotopy of $6$-fold cyclic loop space: In the weight-
space decomposition

$\pi_{\bullet}(\mathcal{L}_{c}^{6}S^{4})\otimes\mathbb{Q}=\bigoplus_{\alpha\in\mathfrak{h}_{6}^{*}}\left(\pi_{\bullet}(\mathcal{L}_{c}^{6}S^{4})\otimes\mathbb{Q}\right)_{\alpha}$

corresponding to the $7$-torus action on the Quillen minimal model
$Q(\mathcal{L}_{c}^{6}S^{4})\linebreak[0]=\pi_{\bullet}(\mathcal{L}_{c}^{6}S^{4})\otimes\mathbb{Q}[1]$,
the $27$ exceptional vectors $\alpha_{i}\in\mathfrak{h}_{6}^{*}$,
$i=1,\dots,27$, single out precisely the second rational homotopy group
$\pi_{2}(\mathcal{L}_{c}^{6}S^{4})\otimes\mathbb{Q}$:

$\pi_{2}(\mathcal{L}_{c}^{6}S^{4})\otimes\mathbb{Q}=\bigoplus_{i=1}^{27}\left(\pi_{\bullet}(\mathcal{L}_{c}^{6}S^{4})\otimes\mathbb{Q}\right)_{\alpha_{i}}\;.$

Moreover,

$\dim\left(\pi_{\bullet}(\mathcal{L}_{c}^{6}S^{4})\otimes\mathbb{Q}\right)_{\alpha_{i}}=1$

for each $i=1,\dots,27$, which means there are $27$ canonically defined,
linearly independent lines in the $\mathbb{Q}$-vector space
$\pi_{2}(\mathcal{L}_{c}^{6}S^{4})\otimes\mathbb{Q}$ and
$\dim\pi_{2}(\mathcal{L}_{c}^{6}S^{4})\otimes\mathbb{Q}=27$.

###  9.3. Other applications to physics

Sati and M. Wheeler [SW18] studied rationalizations of string, fivebrain, and
ninebrain structures on the tangent space of a manifold. These are liftings of
the standard structure group $O(n)$ of the tangent bundle of an $n$-manifold
to the groups $\operatorname{String}(n)$, $\operatorname{Fivebrain}(n)$, and
$\operatorname{Ninebrain}(n)$, respectively, which are, in a certain way,
higher generalizations of the spin group $\operatorname{Spin}(n)$.
Rationalization allowed for considerable simplification of these structures,
and RHT was applied to describe the space of these rationalized structures.
Sati and Wheeler [SW19] used these results to describe the topological parts
of action functionals in string and M-theories.

Surveys of some other applications of RHT to Mathematical Physics may be found
in [FSS19] and [SS24].

## References

  * [BBMFM17] F. Belchí, U. Buijs, J. M. Moreno-Fernández, and A. Murillo, _Higher order Whitehead products and $L_{\infty}$ structures on the homology of a DGL_, Linear Algebra Appl. 520 (2017), 16–31.
  * [Ber12] A. Berglund, _Rational homotopy theory_ , Preprint on webpage at https://staff.math.su.se/alexb/papers.html, November 2012.
  * [Ber15] by same author, _Rational homotopy theory of mapping spaces via Lie theory for $L_{\infty}$-algebras_, Homology Homotopy Appl. 17 (2015), no. 2, 343–369.
  * [BFM13] U. Buijs, Y. Félix, and A. Murillo, _$L_{\infty}$ rational homotopy of mapping spaces_, Rev. Mat. Complut. 26 (2013), no. 2, 573–588.
  * [BFMT20] U. Buijs, Y. Félix, A. Murillo, and D. Tanré, _Lie models in topology_ , Progress in Mathematics, vol. 335, Birkhäuser/Springer, Cham, [2020] ©2020.
  * [BG76] A. K. Bousfield and V. K. A. M. Gugenheim, _On ${\rm PL}$ de Rham theory and rational homotopy type_, Mem. Amer. Math. Soc. 8 (1976), no. 179, ix+94.
  * [BM13] U. Buijs and A. Murillo, _Algebraic models of non-connected spaces and homotopy theory of $L_{\infty}$ algebras_, Adv. Math. 236 (2013), 60–91.
  * [BMSS19] V. Braunack-Mayer, H. Sati, and U. Schreiber, _Gauge enhancement of super M-branes via parametrized stable homotopy theory_ , Comm. Math. Phys. 371 (2019), no. 1, 197–265.
  * [BS23] A. Berglund and R. Stoll, _Higher structures in rational homotopy theory_ , 2023, arXiv:2310.11824 [math.AT].
  * [DGMS75] P. Deligne, P. Griffiths, J. Morgan, and D. Sullivan, _Real homotopy theory of Kähler manifolds_ , Invent. Math. 29 (1975), no. 3, 245–274.
  * [DS95] W. G. Dwyer and J. Spaliński, _Homotopy theories and model categories_ , Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 73–126.
  * [Dwy08] B. Dwyer, _Homotopy theory and classifying spaces_ , Preprint on webpage at https://ncatlab.org/nlab/files/Dwyer_HomotopyTheoryOfClassifyingSpaces.pdf, June 2008.
  * [FHT01] Y. Félix, S. Halperin, and J.-C. Thomas, _Rational homotopy theory_ , Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001.
  * [FOT08] Y. Félix, J. Oprea, and D. Tanré, _Algebraic models in geometry_ , Oxford Graduate Texts in Mathematics, vol. 17, Oxford University Press, Oxford, 2008.
  * [FSS15] D. Fiorenza, H. Sati, and U. Schreiber, _The Wess-Zumino-Witten term of the M5-brane and differential cohomotopy_ , J. Math. Phys. 56 (2015), no. 10, 102301, 10.
  * [FSS17] by same author, _Rational sphere valued supercocycles in $M$-theory and type IIA string theory_, J. Geom. Phys. 114 (2017), 91–108.
  * [FSS18a] by same author, _T-duality from super Lie $n$-algebra cocycles for super $p$-branes_, Adv. Theor. Math. Phys. 22 (2018), no. 5, 1209–1270, arXiv:1611.06536 [math-ph].
  * [FSS18b] by same author, _T-duality in rational homotopy theory via $L_{\infty}$-algebras_, 2018, arXiv:1712.00758 [math-ph].
  * [FSS19] by same author, _The rational higher structure of M-theory_ , Fortschr. Phys. 67 (2019), no. 8-9, 1910017, 28.
  * [FSS20] by same author, _Twisted cohomotopy implies M-theory anomaly cancellation on 8-manifolds_ , Comm. Math. Phys. 377 (2020), no. 3, 1961–2025.
  * [FSS21] by same author, _Twisted cohomotopy implies level quantization of the full 6d Wess-Zumino term of the M5-brane_ , Comm. Math. Phys. 384 (2021), no. 1, 403–432.
  * [Get09] E. Getzler, _Lie theory for nilpotent $L_{\infty}$-algebras_, Ann. of Math. (2) 170 (2009), no. 1, 271–301.
  * [GK94] V. Ginzburg and M. Kapranov, _Koszul duality for operads_ , Duke Math. J. 76 (1994), no. 1, 203–272.
  * [GS21] D. Grady and H. Sati, _Differential cohomotopy versus differential cohomology for M-theory and differential lifts of Postnikov towers_ , J. Geom. Phys. 165 (2021), Paper No. 104203, 24.
  * [Hat02] A. Hatcher, _Algebraic topology_ , Cambridge University Press, Cambridge, 2002\.
  * [Hes07] K. Hess, _Rational homotopy theory: a brief introduction_ , Interactions between homotopy theory and algebra, Contemp. Math., vol. 436, Amer. Math. Soc., Providence, RI, 2007, pp. 175–202.
  * [Hin97] V. Hinich, _Descent of Deligne groupoids_ , Internat. Math. Res. Notices (1997), no. 5, 223–239.
  * [Hol21] J. Holstein, _Rational homotopy theory_ , Preprint on webpage at https://www.math.uni-hamburg.de/home/holstein/lehre/rht20.html, March 2021\.
  * [Hov99] M. Hovey, _Model categories_ , Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999.
  * [INV01] A. Iqbal, A. Neitzke, and C. Vafa, _A mysterious duality_ , Adv. Theor. Math. Phys. 5 (2001), no. 4, 769–807.
  * [Kon03] M. Kontsevich, _Deformation quantization of Poisson manifolds_ , Lett. Math. Phys. 66 (2003), no. 3, 157–216.
  * [LM15] A. Lazarev and M. Markl, _Disconnected rational homotopy theory_ , Adv. Math. 283 (2015), 303–361.
  * [Maj00] M. Majewski, _Rational homotopical models and uniqueness_ , Mem. Amer. Math. Soc. 143 (2000), no. 682, xviii+149.
  * [Man86] Yu. I. Manin, _Cubic forms_ , second ed., North-Holland Mathematical Library, vol. 4, North-Holland Publishing Co., Amsterdam, 1986, Algebra, geometry, arithmetic, Translated from the Russian by M. Hazewinkel.
  * [Men15] L. Menichi, _Rational homotopy—Sullivan models_ , Free loop spaces in geometry and topology, IRMA Lect. Math. Theor. Phys., vol. 24, Eur. Math. Soc., Zürich, 2015, pp. 111–136.
  * [Moe17] I. Moerdijk, _Rational homotopy theory_ , Preprint on webpage at https://www.uu.nl/medewerkers/RestApi/Public/GetFile?Employee=9611&l=EN&id=478&t=000000, April 2017.
  * [MP12] J. P. May and K. Ponto, _More concise algebraic topology_ , Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2012, Localization, completion, and model categories.
  * [Qui69] D. Quillen, _Rational homotopy theory_ , Ann. of Math. (2) 90 (1969), 205–295.
  * [RNV20] D. Robert-Nicoud and B. Vallette, _Higher Lie theory_ , 2020, arXiv:2010.10485 [math.AT].
  * [Sat18] H. Sati, _Framed M-branes, corners, and topological invariants_ , J. Math. Phys. 59 (2018), no. 6, 062304, 25.
  * [SS12] M. Schlessinger and J. Stasheff, _Deformation theory and rational homotopy type_ , December 2012, arXiv:1211.1647 [math.QA].
  * [SS24] H. Sati and U. Schreiber, _Flux quantization_ , February 2024, arXiv:2402.18473 [hep-th], in this volume.
  * [Sul77] D. Sullivan, _Infinitesimal computations in topology_ , Inst. Hautes Études Sci. Publ. Math. (1977), no. 47, 269–331.
  * [SV23a] H. Sati and A. A. Voronov, _Mysterious triality and M-theory_ , Preprint IPMU23-0001, Kavli IPMU, 2023, arXiv:2212.13968 [hep-th].
  * [SV23b] by same author, _Mysterious triality and rational homotopy theory_ , Comm. Math. Phys. 400 (2023), no. 3, 1915–1960.
  * [SW18] H. Sati and M. Wheeler, _Variations of rational higher tangential structures_ , J. Geom. Phys. 130 (2018), 229–248.
  * [SW19] by same author, _Topological actions via gauge variations of higher structures_ , Phys. Lett. B 789 (2019), 114–118.
  * [VPB85] M. Vigué-Poirrier and D. Burghelea, _A model for cyclic homology and algebraic $K$-theory of $1$-connected topological spaces_, J. Differential Geom. 22 (1985), no. 2, 243–253.
  * [VPS76] M. Vigué-Poirrier and D. Sullivan, _The homology theory of the closed geodesic problem_ , J. Differential Geometry 11 (1976), no. 4, 633–644.
  * [Wan19] H. Wang, _Rational homotopy theory_ , Preprint on webpage at https://hewang.sites.northeastern.edu/research/, 2019.