Contents

model category

for ∞-groupoids

# Contents

## Idea

A model category structure on a category of differential graded algebras or more specifically on a category of differential graded-commutative 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 spell out special cases, such as restricting to commutative monoids when working over a ground field of characteristic zero, or restricting to non-negatively graded cochain dg-algebras.

## Projective model structure on Non-negatively graded cochain dgc-algebras

We discuss the projective model structure on differential non-negatively graded-commutative algebras. This was originally introduced in Bousfield-Gugenheim 76 as a model category for Dennis Sullivan‘s approach to rational homotopy theory.

### Definition

###### Definition

For $k$ a field of characteristic zero, write

$dgcAlg^{\geq 0}_{k}$

for the category of differential graded-commutative algebras over $k$ in non-negative degrees, equivalently the category of commutative monoids in the symmetric monoidal category $Ch^{\geq 0}(k)$ of cochain complexes in non-negative degrees, equipped with the tensor product of chain complexes.

###### Definition

(finite type)

Say that a dgc-algebra $A \in dgcAlg^{\geq 0}_k$ (def. ) is of finite type if its underlying chain complex is in each degree of finite dimension as a $k$-vector space.

###### Definition

Write $(dgcAlg^{\geq 0}_k)_{proj}$ for the catgory of dgc-algebras from def. equipped with the following classes of morphisms:

###### Proposition

The category $(dgcAlg^{\geq 0}_k)_{proj}$ from def. is a model category, to be called the projective model structure.

###### Remark

(category of fibrant objects)

Evidently every object in $(dgcAlg^{\geq 0}_k)_{proj}$ (def. , prop. ) 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 and Sullivan algebras

###### 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 dgc-algebras

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

and for $n \geq 1$ the semifree dgc-algebras

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

for which the differential sends the generator of $k[n-1]$ to that of $k[n]$

Write

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

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

$j_n \colon k[0] \to D(n) \,.$
###### Proposition

(generating cofibrations)

The sets

$I = \{i_n \}_{n \geq 1} \cup \{k[0] \to S(0), S(0) \to k[0]\}$

and

$J = \{j_n \}_{n \gt 1}$

are sets of generating cofibrations and acyclic cofibrations, respectively, exhibiting the model category $(dgcAlg^{\geq 0}_k)_{proj}$ from prop. as a cofibrantly generated model category.

review includes (Hess 06, p. 6)

In this section we describe the cofibrations in the model structure on $(dgcalg^{\geq 0}_k)_{proj}$ (def. , prop. ). 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 relative 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 $(dgcAlg^{\geq 0}_{k})_{proj}$ are precisely the retracts of relative Sullivan algebras $(A,d) \to (A\otimes_k \wedge^\bullet V, d')$.

Accordingly, the cofibrant objects in $(dgcAlg^{\geq 0}_{k})_{proj}$ are precisely the Sullivan algebras $(\wedge^\bullet V, d)$

#### Simplicial hom-complexes

We discuss simplicial mapping spaces between dgc-algebras. These almost make the projective model structure $(dgcAlg^{\geq 0}_k)_{proj}$ from prop. into a simplicial model category, except that the tensoring/powering isomorphism holds only for finite simplicial sets or else on dgc-algebras of finite type. Still, this has useful implications, for instance it implies that the reduced suspension and loop space adjunction on [augmented algebras|augmented]] dg-algebras is a Quillen adjunction.

###### Definition

(simplicial mapping spaces)

For $A,B \in dgcAlg^{\geq 0}_k$ (def. ), let

$Maps(A,B) \in sSet$

be the simplicial set whose n-simplices are the dg-algebra homomorphisms from $A$ into the tensor product of $B$ with the de Rham complex of polynomial differential forms on the n-simplex $\Omega_{poly}^\bullet(\Delta^n)$.

$Maps(A,B)_n \;\coloneqq\; Hom_{dgcAlg^{\geq 0}_k} \left( A, \; \Omega^\bullet_{poly}(\Delta^n) \otimes_k B \right)$

and whose face and degeneracy maps are the obvious ones induced from the fact that $\Omega_{poly}^\bullet \colon \Delta^{op} \to dgcAlg^{\geq 0}_k$ is canonically a simplicial object in dgc-algebras.

We also call this the simplicial mapping space from $A$ to $B$. This construction naturally extends to a functor

$Maps(-,-) \;\colon\; (dgcAlg^{\geq 0}_k)^{op} \times dgcAlg^{\geq 0}_k \longrightarrow dgcAlg^{\geq 0}_k$

from the product category of the opposite category of dgc-algebras with the category itself.

Observe that

$Hom_{dgcAlg^{\geq 0}_k} \left( A, \; \Omega^\bullet_{poly}(\Delta^n) \otimes_k B \right) \;\simeq\; {}_{\Omega^\bullet_{poly}}Hom_{dgcAlg^{\geq 0}_k} \left( \Omega^\bullet_{poly}(\Delta^n) \otimes_k A \,,\, \Omega^\bullet_{poly}(\Delta^n) \otimes_k B \right) \,,$

where on the right we have those dg-algebra homomorphism which in addition preserves the left dg-module structure over $\Omega^\bullet_{poly}(\Delta^n)$. This induces for any three $A,B,C \in dgcAlg^{\geq 0}_k$ a composition homomorphism of simplicial sets out of the Cartesian product of mapping spaces

$\circ^{sSet}_{A,B,C} \;\colon\; Maps(A,B) \times Maps(B,C) \longrightarrow Maps(A,C) \,.$
###### Remark

The set of 0-simplices of of the mapping space $Maps(A,B)$ in def. is naturally isomorphic to the ordinary hom-set of dg-algebras:

$Maps(A,B)_0 \simeq Hom_{dgcAlg^{\geq 0}_k}(A,B)$

and under this identification the two notions of composition agree.

Definition makes $dgcAlg^{\geq 0}_k$ an sSet-enriched category (“simplicial category”). The follows says that it is also powered, not over all of $sSet$, but over finite simplicial sets:

###### Proposition

(powering over finite simplicial sets)

For $A, B \in dgcAlg^{\geq 0}_k$ and $S \in$ sSet, there is a natural transformation

$Hom_{dgcAlg^{\geq}_k}(A, \Omega^\bullet_{poly}(S) \otimes_k B) \longrightarrow Hom_{sSet}( S, Maps(A,B) )$

from the hom-set of dgc-algebras into the tensor product with the polynomial differential forms on n-simplices from def. to the hom-set in simplicial sets into the simplicial mapping space from def. .

Moreover, this morphism is an isomorphism if one of the following conditions holds:

###### Proposition

(pullback powering axiom)

Let $i \colon V \to W$ and $p \colon X \to Y$ be two morphisms in $dgcAlg^{\geq 0}_k$. Then their pullback power with respect to the simplicial mapping space functor (def. )

$p^i \;\colon\; Maps(W,X) \longrightarrow Maps(V,X) \underset{Maps(V,Y)}{\times} Maps(W,Y)$

is

1. a Kan fibration if $i$ is a cofibration and $p$ a fibration in the projective model category structure from prop. ;

2. in addition a weak homotopy equivalence (i.e. a weak equivalence in the classical model structure on simplicial sets) if at least one of $i$ or $p$ is a weak equivalence in the projective model structure from prop. .

###### Remark

Prop. would say that $(dgcAlg^{\geq 0}_k)_{proj}$ is a simplicial model category with respect to the simplicial enrichment from def. were it not for the fact that prop. gives the powering only over finite simplicial sets.

#### Relation to cosimplicial commutative algbras

The monoidal Dold-Kan correspondence gives a Quillen equivalence to the projective model structure on cosimplicial commutative algebras $(cAlg_k^{\Delta})_{proj}$.

#### Commutative vs. non-commutative dg-algebras

this needs harmonization

###### Proposition
$F dgcAlg_k \to dgAlg_k$

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

boundedness?

###### 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 zero. 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).

This is in (Kriz-May 95, II.1.5).

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.

###### Remark

Discussion of a restricted kind of homotopy-faithfulness of the forgetful functor from the homotopy theory of commutative to not-necessarily commutative dg-algebras is in (Amrani 14).

## 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

#### Properness

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

This is a proper model category. See MO discussion here.

#### 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 a simplicial model category. 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 (Bousfield-Gugenheim 76, 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 Bousfield-Gugenheim 76, 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 differential graded-commutative 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

• Aldridge 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

The relation between commutative and non-commutative dgas is further discussed in