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model structure on dg-algebras

Idea

By the dual Dold-Kan correspondence cochain complexes in non-negative degree are equivalent to cosimplicial abelian groups. Moreover, the monoidal Dold-Kan correspondence maps cosimplicial algebras to dg-algebras, but this is no longer an equivalence of ordinary categories. It should, however, be an equivalence of the full (∞,1)-categories of these objects. This, in turn, should be modeled by a model category structures.

The model structure on dg-algebras is such a model.

Definition

Write $\mathrm{dgAlg}$ for the category of dg-algebras over a field $k$ of characteristic 0. Write $C\mathrm{dgAlg}\subset \mathrm{dgAlg}$ for the subcategory of (graded-)commutative dg-algebras.

Definition

The projective model category structure on $C\mathrm{dgAlg}$ and on $\mathrm{dgAlg}$ is given by setting:

• weak equivalences are the quasi-isomorphisms

• fibrations are the degreewise surjections.

Proposition

This indeed defines a model category.

At least on $C\mathrm{dgAlg}$ this is a cofibrantly generated model category.

Proof

See the references below.

Remark

(category of fibrant objects)

Evidently every object in $\mathrm{dgAlg}$ and in $C\mathrm{dgAlg}$ is fibrant. Therefore these model categories structures are in particular also structures of a category of fibrant objects.

The nature of the cofibrations is discussed below.

Cofibrations: Sullivan algebras

In this section we describe the cofibrations in the model structure on $C{\mathrm{dgAlg}}_{\mathbb{N}}$ of non-negatively graded dg-algebras. Notice that it is these that are in the image of the dual monoidal Dold-Kan correspondence.

Before we characterize the cofibrations, first some notation.

For $V$ a $\mathbb{Z}$-graded vector space write ${\wedge }^{•}V$ for the Grassmann algebra over it. Equipped with the trivial differential $d=0$ this is a semifree dga $({\wedge }^{•}V,d=0)$.

With $k$ our ground field we write $(k\mathrm{,0})$ for the corresponding dg-algebra, the tensor unit for the standard monoidal structure on $\mathrm{dgAlg}$. This is the Grassmann algebra on the 0-vector space $(k\mathrm{,0})=({\wedge }^{•}0,0)$.

Definition

(Sullivan algebras)

A relatived Sullivan algebra is a morphism of dg-algebras that is an inclusion

$$(A,d)\to (A{\otimes }_k{\wedge }^{•}V,d\prime )$$(A,d) \to (A \otimes_k \wedge^\bullet V, d')

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 $\{v_{\alpha }\in V\mid \alpha \in J\}$ of $V$;

• such that with $V_{<\beta }=\mathrm{span}(v_{\alpha }\mid \alpha <\beta )$ for all basis elements $v_{\beta }$ we have that

  $$d\prime v_{\beta }\in A\otimes {\wedge }^{•}{V}_{<\beta }.$$d' v_\beta \in A \otimes \wedge^\bullet V_{\lt \beta} \,.

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

$$(\alpha <\beta )\Rightarrow (\mathrm{deg}{v}_{\alpha }\le \mathrm{deg}{v}_{\beta })$$(\alpha \lt \beta) \Rightarrow (deg v_\alpha \leq deg v_\beta)

holds. For a Sullivan algebra $(k\mathrm{,0})\to ({\wedge }^{•}V,d)$ relative to the tensor unit we call the semifree dga $({\wedge }^{•}V,d)$ simply a Sullivan algebra. And a minimal Sullivan algebra if $(k\mathrm{,0})\to ({\wedge }^{•}V,d)$ is a minimal relative Sullivan algebra.

Remark

Sullivan algebras were introduced by Dennis Sullivan in his development of rational homotopy theory. This is one of the key application areas of the model structure on dg-algebras.

Remark

($L_{\mathrm{\infty }}$-algebras)

Because they are semifree dgas, Sullivan dg-algebras $({\wedge }^{•}V,d)$ 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 ${\vee }^{•}{V}^{*}$ on the degreewise dual of $V$ with differential $D=d^{*}$, i.e. the one given by the formula

$$\omega (D(v_1\vee v_2\vee \cdots v_n))=-(d\omega )(v_1,v_2,\cdots ,v_n)$$\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 $v_i\in V^{*}$.

Proposition

(cofibrations are relative Sullivan algebras)

The cofibration in $C{\mathrm{dgAlg}}_{\mathbb{N}}$ are precisely the retracts of relative Sullivan algebras $(A,d)\to (A{\otimes }_k{\wedge }^{•}V,d\prime )$.

Accordingly, the cofibrant objects in $C\mathrm{dgAlg}$ are precisely the Sullivan algebras $({\wedge }^{•}V,d)$

Definition

(sphere and disk algebras)

Write $k[n]$ for the graded vector space which is the ground field $k$ in degree $n$ and 0 in all other degrees. For $n\in \mathbb{N}$, consider the semifree dgas

$$S(n):=({\wedge }^{•}k[n],0)$$S(n) := (\wedge^\bullet k[n], 0)

and for $n\ge 1$ the semifree dga

$$D(n):=\{\begin{array}{cc}0& (n=0)\\ ({\wedge }^{•}(k[n+1]\oplus k[n]),0)& (n>0)\end{array}.$$D(n) := \left\lbrace \array{ 0 & (n = 0) \\ (\wedge^\bullet (k[n+1] \oplus k[n]), 0) & (n \gt 0) } \right. \,.

Write

$$i_n:S(n)\to D(n)$$i_n : S(n) \to D(n)

for the obvious morphism that takes the generator in degree $n$ to the generator in degree $n$ (and for $n=0$ it is the unique morphism from the initial object $(\mathrm{0,0})$).

For $n>0$ write

$$j_n:0\to D(n).$$j_n : 0 \to D(n) \,.

Definition

(generating cofibrations)

The sets

$$I=\{i_n{\}}_n\cup \{S(0)\to 0\}$$I = \{i_n \}_n \cup \{S(0) \to 0\}

and

$$J=\{j_n{\}}_{n>0}$$J = \{j_n \}_{n \gt 0}

are sets of generating cofibrations and acyclic cofibrations, respectively, exhibiting $C\mathrm{dgAlg}$ as a cofibrantly generated model category.

Commutative vs. non-commutative dg-algebras

Observation

The forgetful functor

$$F:\mathrm{CdgAlg}\to \mathrm{dgAlg}$$F : CdgAlg \to dgAlg

from (graded-)commutative dg-algebras to dg-algebras is the right adjoint part of a Quillen adjunction

$$\mathrm{Ab}:\mathrm{dgAlg}\overleftarrow{\to }\mathrm{CdgAlg}:F$$Ab : dgAlg \stackrel{\leftarrow}{\to} CdgAlg : F

Proof

The forgetful functor clearly preserves fibrations and cofibrations. It has a left adjoint, the free abelianization functor $\mathrm{Ab}$, which sends a dg-algebra $A$ to its quotient $A/[A,A]$.

Theorem

Let the ground ring $k$ be a field of characteritsic 0. Then every dg-algebra $A$ which has the structure of an algebra over the E-∞ operad has a dg-algebra morphism $A\to A_c$ to a commutative dg-algebra $A_c$ which is

• a morphism of E-∞ algebras (where $A_c$ has the obvious E-∞ algebras structure)

• a weak weak equivalence in the model structure on dg-algebras (i.e. a quasi-isomorphism of the underlying cochain complexes).

So this says that the weak equivalence classes of the commutative dg-algebras in the model category of all dg-algebras already exhaust the most general non-commutative but homotopy-commutative dg-algebras.

Proof

This is in II.1.5 of

• Igor Kriz and Peter May, Operads, algebras, modules and motives , Astérisque No 233 (1995)

References

The cofibrantly generated model structure on commutative dg-algebras is surveyed usefully for instance on p. 6 of

• Kathryn Hess, Rational homotopy theory: a brief introduction (arXiv)

This makes use of the general discussion in section 3 of

• Paul Goerss, Kirsten Schemmerhorn, Model categories and simplicial methods Notes from lectures given at the University of Chicago, August 2004 (arXiv)

that obtains the model structure from the model structure on chain complexes.

A standard textbook reference is section V.3 of

• Sergei Gelfand, Yuri Manin, Methods of homological algebra, transl. from the 1988 Russian (Nauka Publ.) original. Springer 1996. xviii+372 pp. 2nd corrected ed. 2002.

An original reference seems to be

• A. Bousfield, V. Gugenheim, On PL deRham theory and rational homotopy type Memoirs of the AMS 179 (1976)

For general non-commutative (or rather: not necessarily graded-commutative) dg-algebras a model structure is given in

• J.F. Jardine, A Closed Model Structure for Differential Graded Algebras , Cyclic Cohomology and Noncommutative Geometry, Fields Institute Communications, Vol. 17, AMS (1997), 55-58.

This is also the structure used in

• J.L. Castiglioni G. Cortiñas, Cosimplicial versus DG-rings: a version of the Dold-Kan correspondence (arXiv)

where aspects of its relation to the model structure on cosimplicial rings is discussed. (See monoidal Dold-Kan correspondence for more on this).