model category

for ∞-groupoids

Contents

Idea

A model category structure on a category of dg-algebras tends to present an (∞,1)-category of ∞-algebras.

For dg-algebras bounded in negative or positive degrees, the monoidal Dold-Kan correspondence asserts that their model category structures are Quillen equivalent to the corresponding model structure on (co)simplicial algebras. This case plays a central role in rational homotopy theory.

The case of model structures on unbounded dg-algebras may be thought of as induced from this by passage to the derived geometry modeled on formal duals of the bounded dg-algebras. This is described at dg-geometry.

General

The category of dg-algebras is that of monoids in a category of chain complexes. Accordingly general results on a model structure on monoids in a monoidal model category apply.

Below we spellout special cases, such as restricting to commutative monoids when working over a field of characteristic 0, or restricting to non-negatively graded cochain dg-algebras.

Definition

Write $dgAlg$ for the category of cochain dg-algebras in non-negative degree over a field $k$ of characteristic 0. Write $C dgAlg \subset dgAlg$ for the subcategory of (graded-)commutative dg-algebras.

Definition

The projective model category structure on $C dgAlg$ and on $dgAlg$ is given by setting:

Proposition

This indeed defines a model category.

At least on $C dgAlg$ this is a cofibrantly generated model category.

Proof

See the references below.

Remark

(category of fibrant objects)

Evidently every object in $dgAlg$ and in $C dgAlg$ 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.

Properties

Cofibrations: Sullivan algebras

In this section we describe the cofibrations in the model structure on $C dgAlg_\mathbb{N}$ 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 $\wedge^\bullet V$ 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 $(k,0) = (\wedge^\bullet 0, 0)$.

Definition

(Sullivan algebras)

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

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

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

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

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

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 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.

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

($L_\infty$-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 $v_i \in V^*$.

Proposition

(cofibrations are relative Sullivan algebras)

The cofibration in $C dgAlg_{\mathbb{N}}$ 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 $(\wedge^\bullet V, d)$

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

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

and for $n \geq 1$ the semifree dga

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

Write

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

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 $(0,0)$).

For $n \gt 0$ write

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

(generating cofibrations)

The sets

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

and

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

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

Commutative vs. non-commutative dg-algebras

Observation
$F : CdgAlg \to dgAlg$

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

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

Theorem

Let the ground ring $k$ be a field of characteristic 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)

Unbounded dg-algebras

We discuss now the case of unbounded dg-algebras. For these there is no longer the monoidal Dold-Kan correspondence available. Instead, these can be understood as arising naturally as function $\infty$-algebras in the derived dg-geometry over formal duals of bounded dg-algebras, see function algebras on ∞-stacks.

In derived geometry two categorical gradings interact: a cohesive $\infty$-groupoid $X$ has a space of k-morphisms $X_k$ for all non-negative $k$, and each such has itself a simplicial T-algebra of functions with a component in each non-positive degree. But the directions of the face maps are opposite. We recall the grading situation from function algebras on ∞-stacks.

Functions on a bare $\infty$-groupoid $K$, modeled as a simplicial set, form a cosimplicial algebra $\mathcal{O}(K)$, which under the monoidal Dold-Kan correspondence identifies with a cochain dg-algebra (meaning: with positively graded differential) in non-negative degree

$\left( \array{ \vdots \\ \downarrow \downarrow \downarrow \downarrow \\ K_2 \\ \downarrow^{\partial_0} \downarrow^{\partial_1} \downarrow^{\partial_2} \\ K_1 \\ \downarrow^{\partial_0} \downarrow^{\partial_1} \\ K_0 } \right) \;\;\;\;\; \stackrel{\mathcal{O}}{\mapsto} \;\;\;\;\; \left( \array{ \vdots \\ \uparrow \uparrow \uparrow \uparrow \\ \mathcal{O}(K_2) \\ \uparrow^{\partial_0^*} \uparrow^{\partial_1^*} \uparrow^{\partial_2^*} \\ \mathcal{O}(K_1) \\ \uparrow^{\partial_0^*} \uparrow^{\partial_1^*} \\ \mathcal{O}(K_0) } \right) \;\;\;\;\; \stackrel{\sim}{\leftrightarrow} \;\;\;\;\; \left( \array{ \cdots \\ \uparrow^{\mathrlap{\sum_i (-1)^i \partial_i^*}} \\ A_2 \\ \uparrow^{\mathrlap{\sum_i (-1)^i \partial_i^*}} \\ A_1 \\ \uparrow^{\mathrlap{\sum_i (-1)^i \partial_i^*}} \\ A_0 \\ \uparrow \\ 0 \\ \uparrow \\ 0 \\ \uparrow \\ \vdots } \right) \,.$

On the other hand, a representable $X$ has itself a simplicial T-algebra of functions, which under the monoidal Dold-Kan correspondence also identifies with a cochain dg-algebra, but then necessarily in non-positive degree to match with the above convention. So we write

$\mathcal{O}(X) \;\;\;\;\; = \;\;\;\;\; \left( \array{ \mathcal{O}(X)_0 \\ \uparrow \uparrow \\ \mathcal{O}(X)_{-1} \\ \uparrow \uparrow \uparrow \\ \mathcal{O}(X)_{-2} \\ \uparrow \uparrow \uparrow \uparrow \\ \vdots } \right) \;\;\;\;\; \stackrel{\sim}{\leftrightarrow} \;\;\;\;\; \left( \array{ \vdots \\ \uparrow \\ 0 \\ \uparrow \\ 0 \\ \uparrow \\ \mathcal{O}(X)_0 \\ \uparrow \\ \mathcal{O}(X)_{-1} \\ \uparrow \\ \mathcal{O}(X)_{-2} \\ \uparrow \\ \vdots } \right) \,.$

Taking this together, for $X_\bullet$ a general ∞-stack, its function algebra is generally an unbounded cochain dg-algebra with mixed contributions as above, the simplicial degrees contributing in the positive direction, and the homological resolution degrees in the negative direction:

$\mathcal{O}(X_\bullet) \;\;\;\;\; = \;\;\;\;\; \left( \array{ \vdots \\ \uparrow \\ \bigoplus_{k-p = q} \mathcal{O}(X_k)_{-p} \\ \uparrow \\ \vdots \\ \uparrow^d \\ \mathcal{O}(X_1)_0 \oplus \mathcal{O}(X_2)_{-1} \oplus \mathcal{O}(X_3)_{-2} \oplus \cdots \\ \uparrow^{d} \\ \mathcal{O}(X_0)_0 \oplus \mathcal{O}(X_1)_{-1} \oplus \mathcal{O}(X_2)_{-2} \oplus \cdots \\ \uparrow^{d} \\ \mathcal{O}(X_0)_{-1} \oplus \mathcal{O}(X_1)_{-2} \oplus \mathcal{O}(X_2)_{-3}\oplus \cdots \\ \uparrow^{d} \\ \vdots } \right) \,.$

Definition

Theorem

For $k$ a field of characteristic 0 let

$cdgAlg = CMon(Ch_\bullet(k))$

be the category of undounded commutative dg-algebras. With fibrations the degreewise surjections and weak equivalences the quasi-isomorphisms this is a

which is

The existence of the model structure follows from the general discussion at model structure on dg-algebras over an operad.

Properness and combinatoriality is discussed in (ToënVezzosi):

• in lemma 2.3.1.1 they state that $cdgAlg_+$ constitutes the first two items in a triple which they call an HA context .

• this implies their assumption 1.1.0.4 which asserts properness and combinatoriality

Discussion of cofibrations in $dgAlg_{proj}$ is in (Keller).

Properties

Derived tensor product

Let $cdgAg_k$ be the projective model structure on commutative unbounded dg-algebras from above

Proposition

For cofibrant $A \in cdgAlg_k$, the functor

$A\otimes_k (-) : k Mod \to A Mod$

preserves quasi-isomorphisms.

For $A,B \in cdgAlg_k$, their derived coproduct in $k Mod$ coincides in the homotopy category with the derived tensor product in $k Mod$: the morphism

$A \coprod_k^{L} B \stackrel{}{\to} A \otimes_k^L B$

is an isomorphism in $Ho(k Mod)$.

This follows by the above with (ToënVezzosi, assumption 1.1.0.4, and page 8).

Derived hom-functor

The model structure on unbounded dg-algebras is almost $sSet_{Quillen}$-enriched. See the section simplicial enrichment at model structure on dg-algebras over an operad for details.

Definition

Let $k$ be a field of characteristic 0. Let $\Omega^\bullet_{poly} : sSet \to (cdgAlg_k)^{op}$ be the functor that assigns polynomial differential forms on simplices.

For $A,B \in dgcAlg_k$ define the simplicial set

$cdgAlg_k(A,B) : ([n] \mapsto Hom_{cdgAlg_k}(A, B \otimes_k \Omega^\bullet_{poly}(\Delta[n])) \,.$

This extends to a functor

$cdgAlg_k(-,-) : cdgAlg_k^{op} \times cdgAlg_k \to sSet \,.$
Proposition

The functor $cdgAlg_k(-,-)$ satisfies the dual of the pushout-product axiom: for $i : A \to B$ any cofibration in $cdgAlg_k$ and $p : X \to Y$ any fibration, the canonical morphism

$(i^*, p_*) : cdgalg_k(A,B) \to cdgAlg_k(A,X) \times_{cdgAlg_k(A,Y)} cdgAlg_k(B,Y)$

is a Kan fibration, which is acyclic if $i$ or $p$ is.

This implies in particular that for $A$ cofibrant, $cdgAlg_k(A,B)$ is a Kan complex.

The proof works along the lines of (BousfieldGugenheim, prop. 5.3). See also the discussion at model structure on dg-algebras over an operad.

Proof

We give the proof for a special case. The general case is analogous.

We show that for $A$ cofibrant, and for any $B$ (automatically fibrant), $cdgAlg_k(A,B)$ is a Kan complex.

By a standard fact in rational homotopy theory (due to BousfieldGugenheim, discussed at differential forms on simplices) we have that $\Omega^\bullet_{poly} : sSet \to (cdgAlg^+_k)^{op}$ is a left Quillen functor, hence in particular sends acyclic cofibrations to acyclic cofibrations, hence acyclic monomorphisms of simplicial sets to acyclic fibrations of dg-algebras.

Specifically for each horn inclusion $\Lambda[n]_k \hookrightarrow \Delta[n]$ we have that the restriction map $\Omega^\bullet_{poly}(\Delta[n]) \to \Omega^\bullet_{poly}(\Lambda[n]_k)$ is an acyclic fibration in $cdgAlg_k^*$, hence in $cdgAlg_k$.

A $k$-horn in $cdgAlg_k(A,B)$ is a morphism $A \to B \otimes \Omega^\bullet_{poly}(\Lambda[n]_k)$. A filler for this horn is a lift $\sigma$ in

$\array{ && B \otimes \Omega^\bullet_{poly}(\Delta[n]) \\ & {}^{\mathllap{\sigma}}\nearrow & \downarrow \\ A &\to& B \otimes \Omega^\bullet_{poly}(\Lambda[n]_k) } \,.$

If $A$ is cofibrant, then such a lift does always exist.

Proposition

For $A \in cdgAlg$ cofibrant, $cdgAlg_k(A,B)$ is the correct derived hom-space

$cdgAlg_k(A,B) \simeq \mathbb{R}Hom(A,B) \,.$
Proof

By the assumption that $A$ is cofibrant and according to the facts discussed at derived hom-space, we need to show that

$s B : [n] \mapsto B\otimes_k \Omega^\bullet_{poly}(\Delta[n])$

is a resolution, or simplicial frame for $B$. (Notice that every object is fibrant in $cdgAlg_k$).

Since polynomial differential forms are acyclic on simplices (discussed here) it follows that

$const B \to s B$

is degreewise a weak equivalence. It remains to show that $s A$ is fibrant in the Reedy model structure $[\Delta^{op}, cdgAlg_k]_{Reedy}$.

One finds that the matching object is given by

$(match s B)_k = B \otimes \Omega^\bullet_{poly}(\partial \Delta[k]) \,.$

Therefore $s B$ is Reedy fibrant if in each degree the morphism

$(s B_k \to (match s B)_k ) = (\Omega^\bullet_{poly}(\partial \Delta[k] \hookrightarrow \Delta[k]))$

is a fibration. But this follows from the fact that $\Omega^\bullet_{poly} : sSet \to cdgAlg_k^{op}$ is a left Quillen functor (as discussed at differential forms on simplices).

Derived copowering over $sSet$

We discuss a concrete model for the $(\infty,1)$-copowering of $(cdgAlg_k)^\circ$ over ∞Grpd in terms of an operation of $cdgAlg_k$ over sSet.

First notice a basic fact about ordinary commutative algebras.

Proposition

In $CAlg_k$ the coproduct is given by the tensor product over $k$:

$\left( \array{ A &\stackrel{i_A}{\to}& A \coprod B &\stackrel{i_B}{\leftarrow}& B } \right) \simeq \left( \array{ A &\stackrel{Id_A \otimes_k e_B}{\to}& A \otimes_k B & \stackrel{e_A \otimes Id_B}{\leftarrow}& B } \right)$
Proof

We check the universal property of the coproduct: for $C \in CAlg_k$ and $f,g : A,B \to C$ two morphisms, we need to show that there is a unique morphism $(f,g) : A \otimes_k B \to C$ such that the diagram

$\array{ A &\stackrel{Id_A \otimes e_B}{\to}& A \otimes_k B &\stackrel{e_A \otimes Id_B}{\leftarrow}& B \\ & {}_{\mathllap{f}}\searrow & \downarrow^{\mathrlap{(f,g)}} & \swarrow_{\mathrlap{g}} \\ && C }$

commutes. For the left triangle to commute we need that $(f,g)$ sends elements of the form $(a,e_B)$ to $f(a)$. For the right triangle to commute we need that $(f,g)$ sends elements of the form $(e_A, b)$ to $g(b)$. Since every element of $A \otimes_k B$ is a product of two elements of this form

$(a,b) = (a,e_B) \cdot (e_A, b)$

this already uniquely determines $(f,g)$ to be given on elements by the map

$(a,b) \mapsto f(a) \cdot g(b) \,.$

That this is indeed an $k$-algebra homomorphism follows from the fact that $f$ and $g$ are

Remark

For these derivations it is crucial that we are working with commutative algebras.

Corollary

We have that the copowering of $A$ with the map of sets from two points to the single point

$(* \coprod * \to *) \cdot A \simeq ( A \otimes_k A \stackrel{\mu}{\to} A )$

is the product morphism on $A$. And that the tensoring with the map from the empty set to the point

$(\emptyset \to *)\cdot A \simeq (k \stackrel{e_A}{\to} A)$

is the unit morphism on $A$. Generally, for $f : S \to T$ any map of sets we have that the tensoring

$(S \stackrel{f}{\to} T) \cdot A = A^{\otimes_k |S|} \to A^{\otimes_k |T|}$

is the morphism between tensor powers of $A$ of the cardinalities of $S$ and $T$, respectively, whose component over a copy of $A$ on the right corresponding to $t \in T$ is the iterated product $A^{\otimes_k |f^{-1}\{t\}|} \to A$ on as many tensor powers of $A$ as there are elements in the preimage of $t$ under $f$.

The analogous statements hold true with $CAlg_k$ replaced by $cdgAlg_k$: for $S \in sSet$ and $A \in cdgAlg_k$ we obtain a simplicial cdg-algebra

$S \cdot A \in cdgAlg_k^{\Delta^{op}}$

by the ordinary degreewise copowering over Set, using that $cdgAlg_k$ has coproducts (equal to the tensor product over $k$).

This is equivalently a commutative monoid in simplicial unbounded chain complexes

$cdgAlg_k^{\Delta^{op}} \simeq CMon(Ch^\bullet(k)^{\Delta^{op}}) \,.$

By the logic of the monoidal Dold-Kan correspondence the symmetric lax monoidal Moore complex functor (via the Eilenberg-Zilber map) sends this to a commutative monoid in non-positively graded cochain complexes in unbounded cochain complexes

$C^\bullet(S \cdot A) \in CMon(Ch^\bullet_-(Ch^\bullet(k))) \,.$

Since the total complex functor $Tot : Ch^\bullet(Ch^\bullet(k)) \to Ch^\bullet(k)$ is itself symmetric lax monoidal (…), this finally yields

$Tot C^\bullet(S \cdot A) \in CMon(Ch^\bullet(k)) \simeq cdgAlg_k$
Definition

Define the functor

$CC : sSet \times cdgAlg \to cdgAlg$

by

$CC(S,A) := Tot C^\bullet(S \cdot A) \,.$
Remark

We have

$CC(Y,A)^n := \bigoplus_{k \geq 0} (A^{\otimes_k |Y_k| })_{n+k}$

This appears essentially (…) as (GinotTradlerZeinalian, def 3.1.1).

Proposition

The (∞,1)-copowering of $(dgcAlg_k)^\circ$ over ∞Grpd is modeled by the derived functor of $CC$.

This follows from (GinotTradlerZeinalian, theorem 4.2.7), which asserts that the derived functor of this tensoring is the unique (∞,1)-functor, up to equivalence, satisfying the axioms of $(\infty,1)$-copowering.

Proposition

The functor

$CC : sSet \times cdgAlg_k \to cdgAlg_k$

preserves weak equivalences in both arguments.

This is essentially due to (Pirashvili). The full statement is (GinotTradlerZeinalian, prop. 4.2.1).

Remark

This means that the assumption for the copowering models of higher order Hochschild cohomology are satsified in $cdgAlg_k$ which are described in the section Pirashvili’s higher Hochschild homology is satisfied:

this means that for $A \in cdgAlg$ and $S \in sSet$, $CC(S,A)$ is a model for the function $\infty$-algebra on the free loop space object of $Spec A$. See the section Higher order Hochschild homology modeled on cdg-algebras for more details.

Derived powering over $sSet$

Claim

Let $S \in \infty Grpd$ be presented by a degreewise finite simplicial set (which we denote by the same symbol).

Then the homotopy limit in $cdgAlg_k$ over the $S$-shaped diagram constant on $k$ is given by $\Omega^\bullet_{poly}(S)$.

$\mathbb{R}{\lim_{\leftarrow}}_S const k \simeq \Omega^\bullet_{poly}(S) \,.$
Proof

We show dually that for degreewise finite $S$ the assignment $(S, Spec A) \mapsto Spec (\Omega^\bullet_{poly}(S) \otimes A)$ models the $\infty$-copowering in $cdgAlg_k^{op}$.

By the discussion at (∞,1)-copowering it is sufficient to to establish an equivalence

$(dgcAlg_{k}^{op})^\circ(Spec (\Omega^\bullet_{poly}(S) \otimes A), Spec B) \simeq \infty Grpd(S, (dgcAlg_{k}^{op})^\circ(Spec A, Spec B))$

natural in $B$. Consider a cofibrant model of $B$, which we denote by the same symbol. The we compute with 1-categorical end/coend calculus

\begin{aligned} sSet(S, cdgAlg_k^{op}(Spec A,Spec B)) & \simeq \int^{[r] \in\Delta} \Delta[r] \cdot Hom_{sSet}(S \times \Delta[r], cdgAlg_k^{op}(Spec A, Spec B)) \\ & \simeq \int^{[r] \in\Delta} \Delta[r] \cdot \int_{[k] \in \Delta} Hom_{Set}(S_k \times \Delta[k,r], Hom_{cdgAlg_k^{op}}(Spec \Omega^\bullet_{poly}(\Delta^k) \times Spec A, Spec B)) \\ & \simeq \int^{[r] \in\Delta} \Delta[r] \cdot \int_{[k] \in \Delta} Hom_{cdgAlg_k^{op}}((S_k \times \Delta[k,r]) \cdot Spec \Omega^\bullet_{poly}(\Delta^k) \times Spec A, Spec B)) \\ & \simeq \int^{[r] \in\Delta} \Delta[r] \cdot Hom_{cdgAlg_k^{op}}(\int^{[k] \in \Delta} (S_k \times \Delta[k,r]) \cdot Spec \Omega^\bullet_{poly}(\Delta^k) \times Spec A, Spec B)) \\ & \simeq \int^{[r] \in\Delta} \Delta[r] \cdot Hom_{cdgAlg_k^{op}}(Spec \Omega^\bullet_{poly}(S \times \Delta[r]) \times Spec A, Spec B)) \end{aligned} \,,

where all steps are isomorphisms and the dot denotes the ordinary 1-categorical copowering of the 1-category $cdgAlg^{op}$ over Set. In the last step we are using that the tensor product commutes with finite limits of dg-algebras. (This is where the finiteness assumption is needed).

Now we use that $\Omega^\bullet_{poly}$ preserves products up to quasi-isomorphism (as discussed here)

$\Omega^\bullet_{poly}(S \times \Delta[r]) \simeq \Omega^\bullet_{poly}(S) \otimes \Omega_{poly}^\bullet(\Delta[r]) \,.$

This being a weak equivalence between fibrant objects and since $B$ is assumed cofibrant, we have by the above discussion of the derived hom-functor (and using the factorization lemma) a weak equivalence

$\cdots \simeq \int^{[r] \in\Delta} \Delta[r] \cdot Hom_{cdgAlg_k^{op}}(Spec \Omega^\bullet_{poly}(S) \times Spec \Omega^\bullet_{poly}\Delta[r]) \times Spec A, Spec B)) \,.$

Since all this is natural in $B$, this proves the claim.

Path objects

Proposition

For $A \in cdgAlg_k$, a path object

$A \stackrel{\simeq}{\to} P(A) \stackrel{fib}{\to} A \times A$

for $A$ is given by

$P(A) := A \otimes_k \Omega^\bullet_{poly}(\Delta[1])$

This follows along the above lines. The statement appears for instance as (Behrend, lemma 1.19).

Relation to $H \mathbb{Z}$-algebra spectra

For every ring spectrum $R$ there is the notion of algebra spectra over $R$. Let $R := H \mathbb{Z}$ be the Eilenberg-MacLane spectrum for the integers. Then unbounded dg-algebras (over $\mathbb{Z}$) are one model for $H \mathbb{Z}$-algebra spectra.

Proposition

There is a Quillen equivalence between the standard model category structure for $H \mathbb{Z}$-algebra spectra and the model structure on unbounded differential graded algebras.

See algebra spectrum for details.

Relation to $\mathbb{E}_\infty$-algebras

Commutative dg-algebras over a field $k$ of characteristic 0 constitute a presentation of E-infinity algebras over $k$ ([Lurie, prop. A.7.1.4.11]).

References

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

This makes use of the general discussion in section 3 of

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

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).

Disucssion of the model structure on unbounded dg-algebras over a field of characteristic 0 is in

A general discussion of algebras over an operad in unbounded chain complexes is in

A survey of some useful facts with an eye towards dg-geometry is in

Discussion of cofibrations in unbounded dg-algebras are in

• Bernhard Keller, $A_\infty$-algebras, modules and functor categories (pdf)

The derived copowering of unbounded commutative dg-algebras over $sSet$ is discussed (somewhat implicitly) in

• Grégory Ginot, Thomas Tradler, Mahmoud Zeinalian, Derived higher Hochschild homology, topological chiral homology and factorization algebras, (arxiv/1011.6483)

The commutative product on the dg-algebra of the higher order Hochschild complex is discussed in

• Grégory Ginot, Thomas Tradler, Mahmoud Zeinalian, A Chen model for mapping spaces and the surface product (pdf)

The relation to E-infinity algebras is discussed in section 7.1 of

For more see model structure on dg-algebras over an operad.

Revised on April 4, 2014 23:09:29 by Tim Porter (2.26.36.99)