Contents

and

# Contents

## Idea

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.

## Definition

Sullivan models are particularly well-behaved differential graded-commutative algebras that are equivalent to the dg-algebras of piecewise polynomial differential 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 homotopy types under this correspondence.

Abstractly, (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:

###### Definition

(of 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 dgc-algebra $(\wedge^\bullet V, d=0)$.

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 $(k,0) = (\wedge^\bullet 0, 0)$.

###### Definition

(Sullivan algebras)

A relatived Sullivan algebra is a homomorphism of differential graded-commutative algebras that is an inclusion of the form

$(A,d) \hookrightarrow (A \otimes_k \wedge^\bullet V, d')$

for $(A,d)$ any dgc-algebra and for $V$ some graded vector space, such that

1. there is a well ordered set $J$ indexing a linear basis $\{v_\alpha \in V| \alpha \in J\}$ of $V$;

2. writing $V_{\lt \beta} \coloneqq span(v_\alpha | \alpha \lt \beta)$ then for all basis elements $v_\beta$ we have that

$d' v_\beta \in A \otimes \wedge^\bullet V_{\lt \beta} \,.$

This is called a minimal relative Sullivan algebra if in addition the condition

$(\alpha \lt \beta) \Rightarrow (deg v_\alpha \leq deg v_\beta)$

holds. For a Sullivan algebra $(k,0) \to (\wedge^\bullet V, d)$ relative to the tensor unit we call the semifree dgc-algebra $(\wedge^\bullet V,d)$ simply a Sullivan algebra, and we call it a minimal Sullivan algebra if $(k,0) \to (\wedge^\bullet V, d)$ is a minimal relative Sullivan algebra.

###### Remark

The special condition on the ordering in the relative Sullivan algebra says that these morphisms are composites of pushouts of the generating cofibrations for the model structure on dg-algebras, which are the inclusions

$S(n) \hookrightarrow D(n) \,,$

where

$S(n) = (\wedge^\bullet \langle c \rangle, d = 0)$

is the dg-algebra on a single generator in degree $n$ with vanishing differential, and where

$D(n) = (\wedge^\bullet (\langle b \rangle \oplus \langle c \rangle), d b = c, d c = 0)$

with $b$ an additional generator in degree $n-1$.

Therefore for $A \in dgcAlg$, a pushout

$\array{ S(n) &\stackrel{\phi}{\to}& A \\ \downarrow && \downarrow \\ D(n) &\to& (A \otimes \wedge^ \bullet \langle b \rangle, d b = \phi) }$

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.

Notice that it follows in particular that the cofibrations in $dgAlg_{proj}$ are precisely all the retracts of relative Sullivan algebra inclusions.

###### Remark

($L_\infty$-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

$\omega(D(v_1 \vee v_2 \vee \cdots v_n)) = - (d \omega) (v_1, v_2, \cdots, v_n)$

for all $\omega \in V$ and all $v_i \in V^*$.

###### Definition

(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

$(\wedge^\bullet V^*, d_V) \stackrel{\simeq}{\longrightarrow} \Omega^\bullet_{pwpoly}(X)$

to the dg-algebra of piecewise polynomial differential forms.

## Properties

### As cofibrations

###### Proposition

(cofibrations are relative Sullivan algebras)

The cofibrations in the projective model structure on differential graded-commutative algebras $(dgcAlg_{\mathbb{N}})_{proj}$ are precisely the retracts of relative Sullivan algebra inclusions (def. ).

Accordingly, the cofibrant objects in $(dgcAlg_{\mathbb{N}})_proj{}$ are precisely the retracts of Sullivan algebras.

###### Proposition

Minimal Sullivan models are unique up to isomorphism.

### Rationalization

###### Theorem

$Ho(Top) \simeq Ho(sSet) \underoverset {\underset{\mathbb{R} \Omega^\bullet_{poly}}{\longrightarrow}} {\overset{\mathbb{L} K_{poly} }{\longleftarrow}} {\bot} Ho( (dgcAlg_{\mathbb{Q}, \geq 0})^{op} )$

induced from the of the PL de Rham complex-Quillen adjunction

$(dgcAlg_{\mathbb{Q}, \geq 0}_{proj})^{op} \underoverset {\underset{K_{poly}}{\longrightarrow}} {\overset{\Omega^\bullet_{poly}}{\longleftarrow}} {\bot} sSet_{Quillen}$

(this theorem).

Then: On the full subcategory $Ho(Top_{\mathbb{Q}, \geq 1}^{fin})$ of nilpotent rational topological spaces of finite type this adjunction restricts to an equivalence of categories

$Ho(Top_{\mathbb{Q}, \gt 1}^{fin}) \underoverset {\underset{\mathbb{R} \Omega^\bullet_{poly}}{\longrightarrow}} {\overset{\mathbb{L} K_{poly} }{\longleftarrow}} {\simeq} Ho( (dgcAlg_{\mathbb{Q}, \gt 1}^{fin})^{op} ) \,.$

In particular the derived adjunction unit

$X \longrightarrow K_{poly}(\Omega^\bullet_{pwpoly}(X))$

exhibits the rationalization of $X$.

This is the fundamental theorem of dgc-algebraic rational homotopy theory, see there for more.

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:

###### Theorem

Let $(\wedge^\bullet V^*, d_V)$ be a minimal Sullivan model of a simply connected rational topological space $X$. Then there is an isomorphism

$\pi_\bullet(X) \simeq V$

between the homotopy groups of $X$ and the generators of the minimal Sullivan model.

### Relation to nilpotent $L_\infty$-algebras

Under the formal duality between $L_\infty$-algebras and their Chevalley-Eilenberg dgc-algebras, the connected Sullivan models correspond bijectively to connective nilpotent $L_\infty$-algebras (Berglund 2015, Thm. 2.3).

Dual interpretation as nilpotent $L_\infty$-algebras: