and
Rational homotopy theory is the homotopy theory of rational topological spaces, hence of rational homotopy types: simply connected topological spaces whose homotopy groups are vector spaces over the rational numbers.
Much of the theory is concerned with rationalization, the process that sends a general homotopy type to its closest rational approximation, in a precise sense. On the level of homotopy groups this means to retain precisely the non-torsion subgroups of the homotopy groups.
Two algebraic models of rational homotopy types exist, via differential graded-commutative algebras (Sullivan model) and via dg-Lie algebras (Quillen model).
This way rational homotopy theory connects homotopy theory and differential graded algebra. Akin to the Dold-Kan correspondence, the Sullivan construction in rational homotopy theory connects the conceptually powerful perspective of homotopy theory with the computationally powerful perspective of differential graded algebra.
Moreover, via the homotopy hypothesis the study of topological spaces is connected to that of ∞-groupoids, so that rational homotopy theory induces a bridge (the Sullivan construction) between ∞-groupoids and differential graded algebra. It was observed essentially by (Henriques 08, Getzler 09) that this bridge is Lie integration (see there) in the ∞-Lie theory of L-∞ algebras.
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There are two established approaches in rational homotopy theory for encoding rational homotopy types in terms of Lie theoretic data:
In the Sullivan approach (Sullivan 77) a 1-connected rational space, in its incarnation as a simplicial set, is turned into something like a piecewise smooth space by realizing each abstract $n$-simplex by the standard $n$-simplex in $\mathbb{R}^n$; and then a dg-algebra of differential forms on this piecewise smooth space is formed by taking on each simplex the dg-algebra of ordinary rational polynomial differential forms and gluing these dg-algebras all together.
In the Quillen approach (Quillen 69) the loop space of the rational space/simplicial set is formed and its H-space structure strictified to a simplicial group, of which then a dg-Lie algebra (a strict L-infinity-algebra) is formed by mimicking the construction of the Lie algebra of a Lie group from the primitive elements of its completed group ring: the group ring of the simplicial group here is a simplicial ring, whose degreewise primitive elements hence yield a simplicial Lie algebra. The Moore complex functor maps this to the dg-Lie algebra functor that models the rational homotopy type in the Quillen approach.
The connection between these two appoaches is discussed in (Majewski 00): The Sullivan dg-algebra of forms is the formal dual (the Chevalley-Eilenberg algebra) of an L-infinity algebra that may be rectified (see at model structure for L-infinity algebras) to a dg-Lie algebra, and that is the one from Quillen’s construction.
(Beware that – while both rational homotopy types as well as $L_\infty$-algebras are presented by formal duals of dg-algebras (via Sullivan construction and via forming Chevalley-Eilenberg algebras, respectively) – the class of weak equivalences in the former case strictly includes that in the latter. See this remark at model structure for L-∞ algebras.)
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Rational homotopy theory is mostly restricted to simply connected topological spaces. This is due to the existence of acyclic groups $\Gamma$ whose classifying space $B \Gamma$ is an “acyclic space” in that its ordinary cohomology vanishes in positive degrees. This means that Sullivan algebras do not distinguish classifying spaces of rational groups from contractible spaces. But by Hurewicz theorem asking that all spaces be simply connected precisely makes all “acyclic spaces” be contractible.
Let $X$ and $Y$ by simply connected topological spaces. Then a continuous function $f \colon X \to Y$ is a isomorphism on homotopy groups after tensor product with the rational numbes? $\mathbb{Q}$
\pi_\bullet(f) \otimes \mathbb{Q}
\;\colon\;
\pi_\bullet(X) \otimes \mathbb{Q}
\overset{\simeq}{\longrightarrow}
\pi_\bullet(Y) \otimes \mathbb{Q}$
precisely if it induces an isomorphism on ordinary homology with rational numbers coefficients:
A map $f$ satisfying the equivalent conditions of def. 1 is called a rational homotopy equivalence.
We review here the Sullivan approach to rational homotopy theory, where rational topological spaces are modeled by differential graded-commutative algebras over the rational numbers with good (cofibrant) representatives being Sullivan algebras which are formal duals to L-infinity algebras.
First we discuss how to define an analog of the construction of the de Rham dg-algebra of a smooth manifold for topological spaces:
These dg-algebras of “piecewise polynomial” differential forms on a topological space are typically extremely large and unwieldy. Much more tractable dg-algebras are the minimal Sullivan algebra, which we discuss next:
The relation between these algebras is that the Sullivan algebras are the cofibrant resolutions of the larger dg-algebras. In order to make this precise, we next recall some basics of topological and algebraic homotopy theory in
Finally we state and discuss the main theorem, that the construction of dg-algebras of [piecewise polynomial differential forms]] on a topological space exhibits an equivalence between the homotopy theory of simply connected rational topological spaces of finite type and that of minimal Sullivan algebras:
A central tool in the study of rational topological spaces is an assignment that sends each topological space/simplicial set $X$ to a dg-algebra $\Omega^\bullet_{poly}(X)$ that behaves like the deRham dg-algebra of a smooth manifold. Instead of consisting of smooth differential forms, $\Omega^\bullet_{poly}(X)$ consists of piecewise linear polynomial differential forms , in a way described in detail now.
We first discuss this semi-formally in
and then in more detail in
The construction of $\Omega^\bullet_{poly}$ is a special case of the following general construction:
Let $C$ be any small category, write $PSh(C) = [C^{op}, Set]$ for its category of presheaves and let
be any functor to the category of dg-algebras. Following the logic of space and quantity, we may think of the objects of $C$ as being test spaces and the functor $\Omega^\bullet_C$ as assigning to each test space its deRham dg-algebra.
An example of this construction that is natural from the point of view of differential geometry appears in the study of diffeological spaces, where $C$ is some subcategory of the category Diff of smooth manifolds, and $\Omega^\bullet_C$ is the restriction of the ordinary assignment of differential forms to this. But in the application to topological spaces, in the following, we need a choice for $C$ and $\Omega^\bullet_C$ that is non-standard from the point of view of differential geometry. Still, it follows the same general pattern.
After postcomposing with the forgetful functor that sends each dg-algebra to its underlying set, the functor $\Omega^\bullet_C$ becomes itself a presheaf on $C$. For $X \in PSh(C)$ any other presheaf, we extend the notation and write
for the hom-set of presheaves. One checks that this set naturally inherits the structure of a dg-algebra itself, where all operations are given by applying “pointwise” for each $p : U \to X$ with $U \in C$ the operations in $\Omega^\bullet_C(U)$. This way we get a functor
to the opposite category of that of dg-algebras. We may think of $\Omega^\bullet_C(X)$ as the deRham complex of the presheaf $X$ as seen by the functor $\Omega^\bullet_C : C \to dgAlg^{op}$.
By the general discussion at nerve and realization this functor has a right adjoint $K_C : dgAlg^{op} \to PSh(C)$, that sends a dg-algebra $A$ to the presheaf
The adjunction
is an example for the adjunction induced from a dualizing object.
See differential forms on presheaves for more.
For the purpose of rational homotopy theory, consider the following special case of the above general discussion of differential forms on presheaves.
Recall that by the homotopy hypothesis theorem, Top is equivalent to sSet. In the sense of space and quantity, a simplicial set is a “generalized space modeled on the simplex category”: a presheaf on $\Delta$.
Therefore set in the above $C \coloneqq \Delta$.
Now, a simplicial set has no smooth structure in terms of which one could define differential forms globally, but of course each abstract $k$-simplex $\Delta[k]$ may be regarded as the standard $k$-simplex $\Delta^k_{Diff}$ in Diff, and as such it supports smooth differential forms $\Omega^\bullet_{deRham}(\Delta^k_{Diff})$.
The functor $\Omega^\bullet_{deRham}( \Delta_{Diff}^{(-)} ) : \Delta^{op} \to dgAlg$ obtained this way is almost the one that – after fed into the above procedure – is used in rational homotopy theory.
The only difference is that for the purposes needed here, it is useful to cut down the smooth differential forms to something smaller. Let $\Omega^\bullet_{poly}(\Delta^k_{Diff})$ be the dg-algebra of polynomial differential forms on the standard $k$-simplex. Notice that this recovers all differential forms after tensoring with smooth functions:
For more details see atdifferential forms on simplices.
We discuss the definition of polynomial differential forms on topological spaces in more detail.
(smooth $n$-simplex)
For $n \in \mathbb{N}$ the smooth n-simplex $\Delta^n_{smth}$ is the smooth manifold with boundary and corners defined, up to isomorphism, as the following locus inside the Cartesian space $\mathbb{R}^{n+1}$:
For $0 \leq i \leq n$ the function
which picks the $i$th component in the above definition is called the $i$th barycentric coordinate function.
For
a morphism of finite non-empty linear orders $[n] \coloneqq \{0 \lt 1 \lt \cdots \lt n\}$, let
be the smooth function defined by $x_i \mapsto x_{f(i)}$.
For definiteness, we write
for the category of differential graded-commutative algebras over the complex numbers in non-negative $\mathbb{Z}$-degree, i.e. $\mathbb{N}$-graded.
(smooth differential forms on the smooth $n$-simplex)
For $k \in \mathbb{N}$ then a smooth differential k-form on the smooth $n$-simplex (def. 2) is a smooth differential form in the sense of smooth manifolds with boundary and corners. Explicitly this means the following.
Let
be the affine plane in $\mathbb{R}^{n+1}$ that contains $\Delta^n_{smth}$ in its defining inclusion from def. 2. This is a smooth manifold diffeomorphic to the Cartesian space $\mathbb{R}^{n}$.
A smooth differential form on $\Delta^n_{smth}$ of degree k$ is a collection of linear functions
out of the $k$-fold skew-symmetric tensor power of the tangent space of $F^n$ at some point $x$ to the real numbers, for all $x \in \Delta^n_{smth}$ such that this extends to a smooth differential $k$-form on $F^n$.
Write $\Omega^\bullet(\Delta^n_{smth})$ for the graded real vector space defined this way. By definition there is then a canonical linear map
from the de Rham complex of $F^n$ and there is a unique structure of a differential graded-commutative algebra on $\Omega^\bullet(\Delta^n_{smth})$ that makes is a homomorphism of dg-algebras form the de Rham algebra of $F^n$. This is the de Rham algebra of smooth differential forms on the smooth $n$-simplex.
For $f \colon [n_1] \to [n_2]$ a homomorphism of finite non-empty linear orders with $\Delta_{smth}(f) \colon \Delta^{n_1}_{smth} \to \Delta^{n_2}_{smth}$ the corresponding smooth function according to def. 2, there is the induced homomorphism of differential graded-commutative algebras
induced from the usual pullback of differential forms on $F^n$. This makes smooth differential forms on smooth simplices be a simplicial object in differential graded-commutative algebras (def. 3):
For $n \in \mathbb{N}$ write
for the quotient of the $\mathbb{Z}$-graded symmetric algebra over the rational numbers on $n+1$ generators $t_i$ in degree 0 and $n+1$ generators $d t_i$ of degree 1.
In particular in degree 0 this are called the polynomial functions
due to the canonical inclusion
into the smooth functions on the $n$-simplex according to def. 4, obtained by regarding the generator $t_i$ as the $i$th barycentric coordinate function.
Observe that the tensor product of the polynomial differential forms over these polynomial functions with the smooth functions on the $n$-simplex, is canonically isomorphic to the space $\Omega^\bullet(\Delta^n_{smth})$ of smooth differential forms, according to def. 4:
where moreover the generators $d t_i$ are identified with the de Rham differential of the $i$th barycentric coordinate functions.
This defines a canonical inclusion
and there is uniquely the structure of a differential graded-commutative algebra on $\Omega^\bullet_{poly}(\Delta^n)$ that makes this a homomorphism of dg-algebras. This is the dg-algebra of polynomial differential forms.
For $f \colon [n_1] \to [n_1]$ a morphism of finite non-empty linear orders, let
be the morphism of dg-algebras given on generators by
This yields a simplicial differential graded-commutative algebra
which is a sub-simplicial object of that of smooth differential form
Consider the simplicial differential graded-commutative algebra of polynomial differential forms from def. 5, equivalently a cosimplicial object in the opposite category of differential graded-commutative algebras (def. 3):
By the general discussion at nerve and realization, this induces a pair of adjoint functors between the opposite category of differential graded-commutative algebras $(dgcAlg_{\mathbb{Q}, \geq 0})^{op}$ (def. 3) and the category sSet of simplicial sets:
Here the left adjoint is the left Kan extension of $\Omega \bullet_{poly}$ along the Yoneda embedding $\Delta \hookrightarrow sSet$, which we denote by the same symbols.
This adjunction is an algebraic analog of the singular simplicial complex construction, which we briefly recall (for detailed exposition see at geometry of physics -- homotopy types):
For $n \on \mathbb{N}$, write $\Delta^n_{top}$ for the n-simplex canonically regarded as a topological space, i.e. the topological space underlying the smooth simplex $\Delta^n_{smth}$ from def. 2. As in def. 5 this is a cosimplicial object
now in the category Top of topological spaces.
This also induces a nerve and realization adjunction
where the left adjoint $Sing$ is the singular simplicial complex functor and the right adjoint ${\vert- \vert}$ is the geometric realization functor.
Combining the functors from def. 6 and def. 7 we finally obtain a functorial association of differential forms to a topological space
(foring “piecewise polynomial differential forms”) and a functorial operation that turns every differential graded-commutative algebra into a topological space
(Sullivan algebras)
A relative Sullivan algebra is a homomorphism of differential graded-commutative algebras in non-negative degrees, hence a morphism in $dgcAlg_{\geq 0}$, that is an inclusion of the form
for $(A,d) \in dgcAlg_{\geq 0}$ any dgc-algebra and for $V$ some graded vector space, such that
there is a well ordered set $J$ indexing a linear basis $\{v_\alpha \in V| \alpha \in J\}$ of $V$;
writing $V_{\lt \beta} \coloneqq span(v_\alpha | \alpha \lt \beta)$ then for all basis elements $v_\beta$ we have that
(See remark 1 below for what this means.)
Such a relative Sullivan algebra if called minimal if in addition the degrees of these basis elements increase monotonicly:
If $A \in dgcAlg_{\geq 0}$ is such that the unique homomorphism
is a (minimal) relative Sullivan algebra in the above sense, then $A$ is simply called a (minimal) Sullivan algebra. In particular this means that $A = (\wedge^\bullet, d)$ is a semifree dgc-algebra.
Let $\mathfrak{g}$ be a finite dimensional Lie algebra and write
for its Chevalley-Eilenberg algebra. This CE-algebra is a Sullivan algebra in the sense of def. 9, precisely if $\mathfrak{g}$ is a nilpotent Lie algebra.
For $n \in \mathbb{N}$ let
be the dgc-algebra on a single generator in degree $n$ with vanishing differential.
For $n \geq 1$ let
be the dgc-algebra generated by an additional generator in degree $n-1$ such that the differential takes this to the previous generator.
Then the canonical inclusions
and for $n \geq 1$
are relative Sullivan algebras according to 9.
These are to be called the generating cofibrations for the projective model structure on dgc-algebras below in theorem 4.
Moreover, the inclusions
for $n \geq 1$ are relative Sullivan algebras.
These are to be called the generating acyclic cofibrations for the projective model structure on dgc-algebras below in theorem 4.
The examples in 2 are trivial, but they generate all examples of relative Sullivan algebras:
The special condition on the ordering in the relative Sullivan algebra in def. 9 says that these morphisms are transfinite compositions of pushouts of the generating cofibrations in def. 2:
For $A \in dgcAlg_{\geq 0}$ any dgc-algebra, then a pushout of the form
is precisely a choice $\phi \in A$ of a $d_A$-closed element in degree $n$ and results in adjoining to $A$ the element $b$ whose differential is $d b = \phi$.
This gives the condition in the above definition: the differential of any new element has to be a sum of wedge products of the old elements.
(Sullivan models)
For $X$ a simply connected topological space $X$, a Sullivan (minimal) model for $X$ is a Sullivan (minimal) algebra (\wedge^\bullet V^^ \ast, d_V)
equipped with a quasi-isomorphism
to the dg-algebra of piecewise polynomial differential forms.
Minimal Sullivan models (def. 10) are unique up to isomorphism.
e.g Hess 06, prop 1.18.
We briefly recall classical statement of the equivalene of the homotopy theories of topological spaces and of simplicial sets (simplicial homotopy theory), i.e. the “homotopy hypothesis”. For full exposition see at geometry of physics -- homotopy types.
Say that a continuous function, hence a morphism in the category Top of topological spaces is
a weak equivalence if it is a weak homotopy equivalence;
a fibration if it is a Serre fibration
a cofibration if it is the retract of a relative cell complex inclusion.
These classes of morphisms make the category Top into a model category, the classical model structure on topological spaces, to be denoted $Top_{Quillen}$.
Say that a morphism of simplicial sets is
a weak equivalence if its geometric realization (def. 7) is a weak homotopy equivalence of topological spaces;
a fibration if it is a Kan fibration;
a cofibration if it is a monomorphism (hence degreewise an injection).
These classes of morphisms make the category sSet of simplicial sets into a model category, the classical model structure on simplicial sets, to be denoted $sSet_{Quillen}$.
The singular nerve and realization adjunction from def. 7 is a Quillen equivalence between the classical model structure on topological spaces (theorem 1) and the classical model structure on simplicial sets (theore 2):
Say that a homomorphism of differential graded-commutative algebras in non-negative degrees is
a weak equivalence if its underlying chain map is a quasi-isomorphism;
a fibration if it is degreewise a surjection
a cofibration if it is the retract of a relative Sullivan algebra inclusion (def. 9).
These classes of morphisms make the category of differential graded-commutative algebras over the rational numbers and in non-negative degree into a model category, to be called the projective model structure on differential graded-commutative algebras, $(dcgAlg_{\mathbb{Q} ,\geq 0})_{proj}$.
This is a cofibrantly generated model category, with generating (acyclic) cofibrations the morphisms from example 2.
The adjunction of def. 6 is a Quillen adjunction with respect to the classical model structure on simplicial sets on the left (theorem 1), and the opposite model structure of the projective model structure on differential graded-commutative algebras on the right (theorem 4):
This is due to (Bousfield-Gugenheim 76, section 8) Review includes (Hess 06, p. 9).
(subcategories of nilpotent objects of finite type)
Write
$Ho(Top_{\mathbb{Q},nil,fin}) \hookrightarrow Ho(Top)$ for the full subcategory on those topological spaces $X$ which are
rational: their homotopy groups are uniquely divisible;
nilpotent: their fundamental group is a nilpotent group;
finite type: the $\mathbb{Q}$-vector space $H_\bullet(X,\mathbb{Q})$ are of finite dimension.
$Ho(dgcAlg^{fin}_{\mathbb{Q},nil})$ for the full subcategory on the differential graded-commutative algebras which are equivalent to minimal Sullivan models $(\wedge^\bullet V, d)$ (def. 9) for which the graded vector space $V$ is of finite type, i.e. is degreewise of finite dimension over $\mathbb{Q}$.
Bousfield-Gugenheim 76, p. viii
=–
Consider the adjunction of derived functors
induced from the Quillen adjunction from theorem 5.
On the full subcategories of nilpotent objects of finite type, def. 11, this adjunction restricts to an equivalence of categories
In particular for such spaces the adjunction unit
exhibits the rationalization of $X$.
Bousfield-Gugenheim 76, p. viii
e. g. Hess 06, corollary 1.26.
It follows that the cochain cohomology of the cochain complex of piecewise polynomial differential forms on any topological, hence equivalently that of any of its Sullivan models, coincides with its ordinary cohomology with coefficients in the rational numbers:
But more is true, also the rationalization of the homotopy groups may be read off from any minimal Sullivan model:
Let $(\wedge^\bullet V^*, d_V)$ be a minimal Sullivan model of a simply connected rational topological space $X$. Then there is an isomorphism
between the homotopy groups of $X$ and the generators of the minimal Sullivan model.
e.g. Hess 06, theorem 1.24.
The need to restrict to simply connected topological spaces in theorem 6 is due to the existence of acyclic groups. This are discrete groups $\Gamma$ such that their classifying space $B \Gamma$ has trival ordinary cohomology in positive degree
Therefore, by corollary 1, its dg-algebra of piecewise polynomial differential forms do not distinguish such spaces from contractible topological spaces. But, unless $\Gamma = 1$ is in fact the trivial group, $B \Gamma$ is not contractible, instead it is the Eilenberg-MacLane space $K(\Gamma,1)$ with nontrivial fundamental group $\pi_1(B \Gamma) \simeq \Gamma$. However, by the Hurewicz theorem, this fundamental group is the only obstruction to contractibility.
We discuss the minimal Sullivan models of rational n-spheres.
The minimal Sullivan model of a sphere $S^{2k+1}$ of odd dimension is the dg-algebra with a single generator $\omega_{2k+1}$ in degre $2k+1$ and vanishing differential
The minimal Sullivan model of a sphere $S^{2k}$ of even dimension, for $k \geq 1$. is the dg-algeba with a generator $\omega_{2k}$ in degree $2k$ and another generator $\omega_{4k-1}$ in degree $4k+1$ with the differential defined by
One may understand this form theorem 7: an $n$-sphere has rational cohomology concentrated in degree $n$. Hence its Sullivan model needs at least one closed generator in that degree. In the odd dimensional case one such is already sufficient, since the wedge square of that generator vanishes and hence produces no higher degree cohomology classes. But in the even degree case the wedge square $\omega_{2k}\wedge \omega_{2k}$ needs to be canceled in cohomology. That is accomplished by the second generator $\omega_{4k-1}$.
Again by theorem 7, this now implies that the rational homotopy groups of spheres are concentrated, in degree $2k+1$ for the odd $(2k+1)$-dimensional spheres, and in degrees $2k$ and $4k-1$ in for the even $2k$-dimensional spheres.
For instance the 4-sphere has rational homotopy in degree 4 and 7. The one in degree 7 being represented by the quaternionic Hopf fibration.
We briefly review Quillen’s approach to rational homotopy theory (Quillen 69), see for instance (Griffith-Morgan 13, chapter 17) .
The following sequence of six consecutive functors, each of which is a Quillen equivalence, takes one from a 1-connected rational space $X$ to a dg-Lie algebra.
One starts with the singular simplicial set
and throws away all the simplices except the basepoint in degrees $0$ and $1$, to get a reduced simplicial set. Then one applies the Kan loop group functor (the simplicial analogue of the based loop space functor, see here) to $S(X)$, obtaining an a simplicial group
Then one forms its group ring
and completes it with respect to powers of its augmentation ideal, obtaining a “reduced, complete simplicial Hopf algebra”,
which happens to be cocommutative, since the group ring is cocommutative. Taking degreewise primitives, one then gets a reduced simplicial Lie algebra
At the next stage, the normalized chains functor is applied, to get Quillen’s dg-Lie algebra model of $X$:
Finally, to get a cocommutative dg coalgebra model for $X$, one uses a slight generalization of a functor first defined by Koszul for computing the homology of a Lie algebra, which always gives rise to a cocommutative dg coalgebra.
One may think of this procedure as doing the following: we are taking the Lie algebra of the “group” $\Omega X$ which is the loop space of $X$. From a group we pass to the enveloping algebra, i.e. the algebra of distributions supported at the identity, completed. The topological analog of distributions are chains (dual to functions = cochains), so Quillen’s completed chains construction is exactly the completed enveloping algebra. From the (completed) enveloping algebra we recover the Lie algebra as its primitive elements.
Precursors include
The original articles are
Dan Quillen, Rational homotopy theory, The Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 (JSTOR, pdf)
Dennis Sullivan, Infinitesimal computations in topology, Publications mathématiques de l’ I.H.É.S. tome 47 (1977), p. 269-331. (pdf)
Aldridge Bousfield, V. K. A. M. Gugenheim, On PL deRham theory and rational homotopy type , Memoirs of the AMS, vol. 179 (1976)
Joseph Neisendorfer, Lie algebras, coalgebras and rational homotopy theory for nilpotent spaces, Pacific J. Math. Volume 74, Number 2 (1978), 429-460. (euclid)
Flavio da Silveira, Rational homotopy theory of fibrations, Pacific Journal of Mathematics, Vol. 113, No. 1 (1984) (pdf)
Survey and review includes
Kathryn Hess, Rational homotopy theory: a brief introduction (arXiv:math.AT/0604626)
Yves Félix, Steve Halperin and J.C. Thomas, Rational Homotopy Theory, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000.
Martin Majewski, Rational homotopy models and uniqueness , AMS Memoir (2000):
Phillip Griffiths, John Morgan, Rational Homotopy Theory and Differential Forms, Progress in Mathematics Volume 16, Birkhauser (2013)
Review that makes the L-infinity algebra aspect completely manifest is in
More on the relation to Lie theory is in:
The above description of the Quillen approach draws on blog comments by Kathryn Hess here and by David Ben-Zvi here.
Discussion from the point of view of (∞,1)-category theory and E-∞ algebras is in
An extension of rational homotopy theory to describe (some) non-simply connected spaces is given, using derived algebraic geometry, in
See in particular Cor. 2.4.11, Cor. 2.5.3 and Cor. 2.5.4, and the MathOverflow answer MO/79309/2503 by Denis-Charles Cisinski.