and
A Sullivan model of a rational space $X$ is a particularly well-behaved commutative dg-algebra quasi-isomorphic to the dg-algebra of Sullivan forms on $X$. These Sullivan algebras are precisely the cofibrant objects in the standard model structure on dg-algebras.
Sullivan models are a central tool in rational homotopy theory.
Sullivan models are particularly simple dg-algebras that are equivalent to the dg-algebras of Sullivan differrential forms on topological spaces. Conversely, every rational space can be obtained from a dg-algebra and the minimal Sullivan algebras provide convenient representatives that correspond bijectively to rational homtopy types under this correspondence.
Formally, (relative) Sullivan models are the (relative) cell complexes in the standard model structure on dg-algebras.
We now describe this in detail. First some notation and preliminaries:
(finite type)
A graded vector space $V$ is of finite type if in each degree it is finite dimensional. In this case we write $V^*$ for its degreewise dual.
A Grassmann algebra is of finite type if it is the Grassmann algebra $\wedge^\bullet V^*$ on a graded vector space of finite type
(the dualization here is just convention, that will help make some of the following constructions come out nicely).
A CW-complex is of finite type if it is built out of finitely many cells in each degree.
For $V$ a $\mathbb{N}$-graded vector space write $\wedge^\bullet V$ for the Grassmann algebra over it. Equipped with the trivial differential $d = 0$ this is a semifree dga $(\wedge^\bullet V, d=0)$.
With $k$ our ground field we write $(k,0)$ for the corresponding dg-algebra, the tensor unit for the standard monoidal structure on $dgAlg$. This is the Grassmann algebra on the 0-vector space $(k,0) = (\wedge^\bullet 0, 0)$.
(Sullivan algebras)
A relatived Sullivan algebra is a morphism of dg-algebras that is an inclusion
for $(A,d)$ some dg-algebra and for $V$ some graded vector space, such that
there is a well ordered set? $J$
indexing a basis $\{v_\alpha \in V| \alpha \in J\}$ of $V$;
such that with $V_{\lt \beta} = span(v_\alpha | \alpha \lt \beta)$ for all basis elements $v_\beta$ we have that
This is called a minimal relative Sullivan algebra if in addition the condition
holds. For a Sullivan algebra $(k,0) \to (\wedge^\bullet V, d)$ relative to the tensor unit we call the semifree dga $(\wedge^\bullet V,d)$ simply a Sullivan algebra. And a minimal Sullivan algebra if $(k,0) \to (\wedge^\bullet V, d)$ is a minimal relative Sullivan algebra.
See also the section Sullivan algebras at model structure on dg-algebras.
The special condition on the ordering in the relative Sullivan algebra says precisely that these morphisms are composites of pushouts of the generating cofibrations of the model structure on dg-algebras, which are the inclusions
where
is the dg-algebra on a single generator in degree $n$ with vanishing differential, and where
with $b$ an additional generator in degree $n-1$.
Therefore for $A \in dgAlg$ dg-algebra, a pushout
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 one of the old elements.
Notice that it follows in particular that the cofibrations in $dgAlg_{proj}$ are precisely all the retracts of relative Sullivan algebra inclusions.
($L_\infty$-algebras)
Because they are semifree dgas, Sullivan dg-algebras $(\wedge^\bullet V,d)$ are (at least for degreewise finite dimensional $V$) Chevalley-Eilenberg algebras of L-∞-algebras.
The co-commutative differential co-algebra encoding the corresponding L-∞-algebra is the free cocommutative algebra $\vee^\bullet V^*$ on the degreewise dual of $V$ with differential $D = d^*$, i.e. the one given by the formula
for all $\omega \in V$ and all $v_i \in V^*$.
(Sullivan models)
For $X$ a simply connected? topological space $X$, a Sullivan (minimal) model for $X$ is
a quasi-free dg-algebra $(\wedge^\bullet V^*, d_V)$ which is a (minimal) Sullivan algebra
such that there exists a quasi-isomorphism
(cofibrations are relative Sullivan algebras)
The cofibration in the standard model structure on dg-algebras $C dgAlg_{\mathbb{N}}$ are precisely the retracts of relative Sullivan algebras $(A,d) \to (A\otimes_k \wedge^\bullet V, d')$.
Accordingly, the cofibrant objects in $C dgAlg$ are precisely the Sullivan algebras $(\wedge^\bullet V, d)$
Rational homotopy types of simply connected spaces $X$ are in bijective corespondence with minimal Sullivan algebras $(\wedge^\bullet V,d)$
And homotopy classes of morphisms on both sides are in bijection.
This appears for instance as corollary 1.26 in
Write
for the Quillen adjunction induced by forming Sullivan differential forms, as discussed above.
Let $(\wedge^\bullet V^*, d_V)$ be a simply connected Sullivan algebra of finite type. Then
the unit of the adjunction $(\wedge^\bullet V^*, d_V) \to \Omega^\bullet_{Sul}(K(\wedge^\bullet, d_V) \rangle)$ is a quasi-isomorphism;
The elements of the homotopy groups of the rational space modeled by $(\wedge^\bullet V^*, d_V)$ are the generators in $V$:
there is an isomorphism of $\mathbb{N}$-graded vector spaces over $\mathbb{Q}$
This is recalled for instance as theorem 1.24 in
Sullivan mimimal models are unique up to isomorphism.
This appears for instance as prop 1.18 in
Rational homotopy types of simply connected spaces $X$ are in bijective corespondence with minimal Sullivan models $(\wedge^\bullet V,d)$
And homotopy classes of morphisms on both sides are in bijection.
This appears for instance as corollary 1.26 in
It follows that if (\wedge^^\bullet V^{*}, d)
is a minimal Sullivan model for $(\wedge^^X$, then the rational homotopy groups of $X$ can be read off from the generators $V$:
…
…
…
In
Sullivan algebras and minimal algebras appear in def 1.10