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As with many other parts of homotopy theory one can view rational homotopy theory in two ways:
either one looks at a subcategory of topological spaces for which a particular homotopy functor? gives exceptionally good results;
or alternatively one looks at fairly arbitrary nice spaces, but collects up and uses only the information available using some particular type of homotopy functor?.
From the first viewpoint, rational homotopy theory studies special topological spaces called rational spaces: simply connected spaces whose homotopy groups are vector spaces over the rational numbers.
The point of this is that
every (simply connected) topological space can be approximated by a rational space;
rational topological spaces are entirely encoded in terms of differential graded algebra.
Alternatively, looking at simply connected spaces, we can concentrate on what happens if we tensor their homotopy groups with the field of rationals. What information can be gleaned from these homotopy functors? The answer is ‘an incredible amount’ – namely all non-torsion information – and results in a very rich theory.
In these ways, rational homotopy theory connects and unifies two large areas of mathematics, 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 infinity-groupoids, so that rational homotopy theory induces a bridge between infinity-groupoids and differential graded algebra. It was observed essentially by Ezra Getzler that this bridge is nothing but higher Lie theory of L-infinity-algebras.
There are two main approaches in rational homotopy theory for encoding rational homotopy types in terms of Lie theoretic data:
In the Sullivan approach 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.
This goes back to
In the Quillen approach 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.
This goes back to (Quillen 69).
The connection between these two appoaches is discussed in
the Sullivan dg-algebra of forms is dual to an L-infinity algebra and may be strictified to a dg-Lie algebra, and this is equivalent to the dg-Lie algebra obtained from Quillen’s construction.
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.
The construction of $\Omega^\bullet_{poly}$ is a special case of the following general construction:
See differential forms on presheaves for more.
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 general abstract nonsense 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.
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 $C := \Delta$.
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
So we have a functor $\Omega^\bullet_{polynomial} : \Delta \to dgAlg^{op}$. Feeding that into the above general machinery produces a pair of adjoint functors
This is a Quillen adjunction with respect to the standard model structure on simplicial sets on the left, and the standard model structure on dg-algebras on the right.
The original proof in the literature is apparently the one in section 8 of
A review is on page 9
of
See Sullivan model.
The following sequence of six consecutive functors, each of which is a Quillen equivalence, take 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 $1$. Then one applies the Kan loop group functor (the simplicial analogue of the based loop space functor) to $S(X)$, obtaining an honest simplicial group
Then one takes the 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 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.
A useful introduction to rational homotopy theory is
A standard textbook is
Early original articles include:
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