A strictly commutative model for the cochain algebra of a space

The commutative differential graded algebra $A_{\mathrm{PL}}(X)$ of polynomial forms on a simplicial set $X$ is a crucial tool in rational homotopy theory. In this note, we construct an integral version $A^{\mathcal{I}}(X)$ of $A_{\mathrm{PL}}(X)$. Our approach uses diagrams of chain complexes indexed by the category of finite sets and injections $\mathcal{I}$ to model $E_{\infty}$ differential graded algebras by strictly commutative objects, called commutative $\mathcal{I}$-dgas. We define a functor $A^{\mathcal{I}}$ from simplicial sets to commutative $\mathcal{I}$-dgas and show that it is a commutative lift of the usual cochain algebra functor. In particular, it gives rise to a new construction of the $E_{\infty}$ dga of cochains. The functor $A^{\mathcal{I}}$ shares many properties of $A_{\mathrm{PL}}$, and can be viewed as a generalization of $A_{\mathrm{PL}}$ that works over arbitrary commutative ground rings. Working over the integers, a theorem by Mandell implies that $A^{\mathcal{I}}(X)$ determines the homotopy type of $X$ when $X$ is a nilpotent space of finite type.


Introduction
The cochains C(X; k) on a space X with values in a commutative ring k form a differential graded algebra whose cohomology is the singular cohomology H * (X; k) of X. The multiplication of C(X; k) induces the cup product on H * (X; k). Over the rationals, C(X; Q) is quasi-isomorphic to the commutative differential graded algebra (cdga) A PL (X) of polynomial forms on X, which is a very powerful tool in rational homotopy theory [Sul77,BG76]. The functor A PL has a contravariant adjoint, denoted by K • in [Hes07, Definition 1.23], and called the Sullivan realization in [FHT01,§17]. With the help of this adjoint pair of functors one can determine the homotopy type of rational nilpotent spaces of finite type (see [BG76,Chapter 9], or [Hes07, Theorem 1.25] for the simply connected case).
For a general commutative ring k, there is no cdga which is quasi-isomorphic to C(X; k), for example because the Steenrod operations witness the non-commutativity of C(X; F p ). However, C(X; k) is always commutative up to coherent homotopy. This can be encoded using the language of operads [May72]: the multiplication of C(X; k) extends to the action of an E ∞ operad in chain complexes turning C(X; k) into an E ∞ dga. This additional structure is important because Mandell showed that the cochain functor C(−; Z) to E ∞ dgas classifies nilpotent spaces of finite type up to weak equivalence [Man06,Main Theorem].
One can describe homotopy coherent commutative multiplications on chain complexes using diagram categories instead of operads. Let I be the category with objects the finite sets m = {1 . . . , m}, m ≥ 0, with the convention that 0 is the empty set. Morphisms in I are the injections. Concatenation in I and the tensor product of chain complexes of k-modules give rise to a symmetric monoidal product ⊠ on the category Ch I k of I-diagrams in Ch k . A commutative I-dga is a commutative monoid in (Ch I k , ⊠) or, equivalently, a lax symmetric monoidal functor I → Ch k . Equipped with suitable model structures, the category of commutative I-dgas, Ch I k [C], is Quillen equivalent to the category of E ∞ dgas [RS17,§9]. This is analogous to the situation in spaces, where commutative monoids in I-diagrams of spaces are equivalent to E ∞ spaces [SS12,§3].
Chasing the E ∞ dga of cochains C(X; k) on a space X through the chain of Quillen equivalences relating E ∞ dgas and commutative I-dgas shows that C(X; k) can be represented by a commutative I-dga. The purpose of this paper is to construct a direct point set level model A I (X) for the quasi-isomorphism type of commutative I-dgas determined by C(X; k) that should be viewed as an integral generalization of A PL (X). Despite the fact that A PL (X) was introduced more than 40 years ago and has been widely studied, it appears that a direct integral counterpart was neither known nor expected to exist.
If E is a commutative I-dga, then its Bousfield-Kan homotopy colimit E hI has a canonical action of the Barratt-Eccles operad, which is an E ∞ operad built from the symmetric groups. The commutative I-dga A I (X) thus gives rise to an E ∞ dga A I (X) hI which can be compared to the usual cochains without referring to model structures.
Theorem 1.1. The contravariant functors X → A I (X) hI and X → C(X; k) from simplicial sets to E ∞ dgas are naturally quasi-isomorphic.
We prove the theorem using Mandell's uniqueness result for cochain theories [Man02,Main Theorem]. Since the definition of A I does not rely on the existing constructions of E ∞ structures on cochains, the theorem implies that our approach provides an alternative construction of the E ∞ dga C(X; k). If k is a field of characteristic 0, then there is a natural quasi-isomorphism A I (X) hI → A PL (X) relating our approach to the classical polynomial forms (see Theorem 5.9).
The passage through commutative I-dgas has the advantage that we do not need to lift the action of the acyclic Eilenberg-Zilber operad to the action of an actual E ∞ operad as done by Mandell [Man02,§5] based on work of Hinich-Schechtman [HS87], and it also avoids the elaborate combinatorial arguments used by Berger-Fresse [BF04]. Another approach to capture the commutativity of C(X; k) has been pursued by Karoubi [Kar09] who introduces a notion of quasicommutative dgas that is based on a certain reduced tensor product, constructs a quasi-commutative model for the cochains, and uses Mandell's results to relate it to ordinary cochains.
Since it is often easier to work with strictly commutative objects rather than E ∞ objects, we also expect that the commutative I-dga A I (X) will be a useful replacement of the E ∞ dga C(X; k) in applications. For instance, iterated bar constructions for E ∞ algebras as developed in [Fre11] are rather involved whereas iterated bar construction for commutative monoids are straightforward. Commutative I-dgas are tensored over simplicial sets whereas enrichments for E ∞ monoids are more complicated because the coproduct is not just the underlying monoidal product. This allows for constructions such as higher order Hochschild homology [Pir00] for commutative I-dgas.
Writing A I (X; Z) for A I (X) when working over k = Z, Theorem 1.1 leads to the following reformulation of the main theorem of Mandell [Man06] that highlights the usefulness of A I : Theorem 1.2. Two finite type nilpotent spaces X and Y are weakly equivalent if and only if A I (X; Z) and A I (Y ; Z) are weakly equivalent in Ch I Z [C].
1.3. Outline of the construction. Our chain complexes are homologically graded so that cochains are concentrated in non-positive degrees. We model spaces by simplicial sets and consider the singular complex of a topological space if necessary. The functor A PL : sSet op → cdga Q of polynomial forms used in rational homotopy theory (see e.g. [BG76,§1]) motivates our definition of A I . We recall that A PL arises by Kan extending the functor A PL,• : ∆ op → cdga Q sending [p] in ∆ to the algebra of polynomial differential forms A PL,p = Λ(t 0 , . . . , t p ; dt 0 , . . . , dt p )/(t 0 + · · · + t 1 = 1, dt 0 + · · · + dt p = 0) .
Here Λ is the free graded commutative algebra over Q, the generators t i have degree 0, and the dt i have degree −1 (in our homological grading). Setting d(t i ) = dt i extends to a differential that turns A PL,q into a commutative dga, and addition of the t i and insertion of 0 define the simplicial structure of A PL,• .
Let CD 0 be the free commutative Q-dga on the chain complex D 0 with (D 0 ) i = 0 if i = 0, −1 and d 0 : (D 0 ) 0 → (D 0 ) −1 being id Q . Moreover, let S 0 in Ch Q be the monoidal unit, i.e., the chain complex with a copy of Q concentrated in degree 0. Sending 1 ∈ (CD 0 ) 0 to either 1 or 0 in Q defines two commutative CD 0 -algebra structures on S 0 that we denote by S 0 0 and S 0 1 . One can now check that the simplicial Q-cdga A PL,• is isomorphic to the two sided bar construction whose face maps are provided by the algebra structures on S 0 1 and S 0 0 and the multiplication of CD 0 , and whose degeneracy maps are induced by the unit of CD 0 .
While polynomial differential forms appear to have no obvious counterpart in commutative I-dgas, their description in terms of a two sided bar construction easily generalizes to commutative I-dgas over an arbitrary commutative ground ring k. For this we consider the left adjoint A → m → s≥0 I(1 ⊔s ,m) A ⊗s / Σ s to the evaluation of a commutative I-dga at the object 1 in I and recall that the unit U I in Ch I k is the constant I-diagram on the unit S 0 in Ch k . As above, we form CF I 1 D 0 , observe that U I gives rise to two commutative CF I 1 D 0 algebras U I 0 and U I 1 , and define A I • : ∆ op → Ch I k [C] to be the two sided bar construction At this point it is central to work with strictly commutative objects since the multiplication map of an E ∞ object is typically not an E ∞ map. It is also important to use 1 rather than 0 in the above left adjoint since this ensures that A I p (m) is contractible. This is related to J. Smith's insight that one has to use positive model structures for commutative symmetric ring spectra.
Via Kan extension and restriction along the canonical functor ∆ op → sSet op , this A I • gives rise to functors A I : sSet op → Ch I k [C] and K I : Ch I k [C] op → sSet (see Definition 3.6). More explicitly, the evaluation of A I (X) at I-degree m and chain complex level q is the k-module of simplicial set morphisms sSet(X, A I • (m) q ). The functors A I and K I are contravariant right adjoint in the sense that there are natural isomorphisms Ch I k [C](E, A I (X)) ∼ = sSet(X, K I (E)). They are integral analogues of the functor of polynomial forms and of the Sullivan realization functor.  [Man02,§4] constructed an analogous adjunction between simplicial sets and E ∞ dgas using the E ∞ structure on cochains as input.
Since all simplicial sets are cofibrant, the statement of Theorem 1.5 implies that each A I (X) is positive fibrant. Writing I + for the full subcategory of I on objects m with |m| ≥ 1, this means that each morphism m → n in I + induces a quasiisomorphisms A I (X)(m) → A I (X)(n). Hence each chain complex A I (X)(m) with m in I + captures the quasi-isomorphism type of the cochains C(X; k). Since the positive I-model structure is the left Bousfield localization of a positive level model structure, it also follows that weak homotopy equivalences X → Y induce quasi- Analogous to the corresponding statement about A PL , the proof of the theorem is based on the observation that the the simplicial sets A I • (m) q obtained by fixing an m in I + and a chain level q are contractible.
Remark 1.6. After a first version of the present manuscript was made available, the authors learned from Dan Petersen that he recently found another construction of a commutative I-dga that models the cochain algebra of a space [Pet]. His approach applies to locally contractible topological spaces, uses sheaf cohomology, and has applications in the study of configuration spaces. 1.7. Notations and conventions. Throughout the paper, k denotes a commutative ring with unit, and Ch k denotes the category of unbounded homologically graded chain complexes of k-modules.
1.8. Organization. In Section 2 we study homotopy colimits of commutative Idgas. Section 3 provides the construction of the functor A I . We review model structures on I-chain complexes and commutative I-dgas in Section 4. In Section 5 we establish the homotopical properties of A I , prove a comparison to the usual cochains disregarding multiplicative structures, and prove Theorem 1.5. In the final Section 6, we prove the E ∞ comparison from Theorem 1.1 as Theorem 6.2 and explain how to derive Theorem 1.2.

Homotopy colimits of I-chain complexes
Let I be the category with objects the finite sets m = {1, . . . , m} for m ≥ 0 and with morphisms the injective maps. In this section we study multiplicative properties of the homotopy colimit functor for I-diagrams of chain complexes. The functor F I 0 is isomorphic to the constant functor since 0 is initial in I.

Homotopy colimits.
Our next aim is to define Bousfield-Kan style homotopy colimits for I-diagrams of chain complexes. For the subsequent multiplicative analysis, we fix notation and conventions about bicomplexes.
Definition 2.3. Let Ch k (Ch k ) be the category of chain complexes in Ch k . Its objects are Z×Z-graded k-modules (Y p,q ) p,q∈Z with k-linear horizontal differentials, A morphism g : Y → Z in Ch k (Ch k ) is a family (g p,q : Y p,q → Z p,q ) p,q∈Z×Z of k-linear maps that commute with the horizontal and vertical differentials, i.e., Since we require horizontal and vertical differentials to commute, an additional sign is needed to form the total complex: Definition 2.4. Let Y be an object in Ch k (Ch k ). Its associated total complex Tot(Y ) is the chain complex with Tot(Y ) n = p+q=n Y p,q in chain degree n ∈ Z and with differential d Tot (y) = d h (y) + (−1) p d v (y) for every homogeneous y ∈ Y p,q .
Let sCh k be the category of simplicial objects in Ch k .
Definition 2.5. For A ∈ sCh k we denote by C * (A) the chain complex in chain complexes with (C * (A)) p,q = A p,q . We define the horizontal differential on C * (A), where the d i are the simplicial face maps of A. The vertical differential on C * (A) is given by the differential d A on A.
As the d i 's commute with d A , this gives indeed a chain complex in chain complexes whose horizontal part is concentrated in non-negative degrees.
Construction 2.6. Let P : I → Ch k be an I-chain complex. The simplicial replacement of P is the simplicial chain complex srep(P ) : ∆ op → Ch k given in simplicial degree [p] by The last face map sends the copy of P (n p ) indexed by (α 1 , . . . , α p ) via P (α p ) to the copy of P (n p−1 ) indexed by (α 1 , . . . , α p−1 ). The other face and degeneracy maps are induced by the identity on P (n p ) and corresponding simplicial structure maps of the nerve N (I) of I.
A bicomplex spectral sequence argument shows that P hI → Q hI is a quasiisomorphism if each P (m) → Q(m) is a quasi-isomorphism. There is a canonical map P hI → colim I P , and one can show by cell induction that it is a quasiisomorphism if P is cofibrant in the projective level model structure on Ch I k . Together this shows that P hI is a model for the homotopy colimit of P . A more elaborate argument that shows that P hI is a corrected homotopy colimit can be found in [RG14]. A version of the above homotopy colimit for functors with values in modules can be found in [DL98, Definition 3.13].

2.7.
Commutative I-dgas. The ordered concatenation of ordered sets m ⊔ n = m + n equips I with a symmetric strict monoidal structure that has 0 as a strict unit and the block permutations as symmetry isomorphisms. If P, Q : I → Ch k are I-chain complexes, then the left Kan extension of along ⊔ : I × I → I provides an I-chain complex P ⊠ Q. This defines a symmetric monoidal product ⊠ on Ch I k , the Day convolution product, with unit the constant : U for the adjunction with right adjoint the forgetful functor and left adjoint the free functor C given by The definition of ⊠ as a left Kan extension implies the existence of a natural . This shows that in the case P = F I 1 (A), we have an isomorphism F I 1 (A) ⊠s ∼ = F I 1 ⊔s (A ⊗s ) of Σ s -equivariant objects where Σ s acts on the target by permuting both the ⊗-powers of A and the index set of the sum. The commutative I-dga C(F I 1 (A)) will be of particular importance for us, and we note that the above implies 2.9. Homotopy colimits of commutative I-dgas. We will now construct an operad action on the homotopy colimit of a commutative I-dga. Our construction involves a symmetric monoidal structure on simplicial chain complexes: Definition 2.10. Let A and B be two simplicial chain complexes. Their tensor product A⊗B is the simplicial chain complex with ℓ+m=n A p,ℓ ⊗ B p,m in simplicial degree p and chain degree n. The simplicial structure maps act coordinatewise and the differential d⊗ is We denote by Σ s the translation category of the symmetric group Σ s . Its objects are elements σ ∈ Σ s and τ ∈ Σ s is the unique morphism from σ to τ • σ in Σ s . Since there is exactly one morphism between each pair of objects, we get a functor The action (2.4) is associative, unital, and symmetric. It turns the collection of categories ( Σ n ) n≥0 into an operad Σ in the category cat of small categories. For the next definition, we use that the nerve functor N : cat → sSet and the k-linearization k{−} : sSet → sMod k are strong symmetric monoidal and that the associated chain complex functor C * : sMod k → Ch k is lax symmetric monoidal (compare Proposition 2.16 below).
Definition 2.11. The Barratt-Eccles operad is the E ∞ operad E in Ch k with E n = C * (k{N ( Σ n )}) and operad structure induced by the functor (2.4).
The commutativity operad C in Ch k is the operad with C n = k concentrated in chain complex level 0. The operad E admits a canonical operad map E → C which is a quasi-isomorphism in each level. Moreover, E n is a free k[Σ n ]-module for each n.
Applying the nerve to Σ defines an operad in sSet that is more commonly referred to as the Barratt-Eccles operad. It is well known that the latter operad acts on the nerve of a permutative category [May74,Theorem 4.9]. The next lemma recalls the underlying action of Σ for the permutative category I.
Lemma 2.12. The operad Σ in cat acts on I. On objects σ in Σ n and m i in I, the action is given by Proof. This is a special case of [May74, Lemmas 4.3 and 4.4]. Functoriality in morphisms of Σ n uses the symmetry isomorphism of I while the functoriality in I is the evident one.
The next result is our main motivation for considering the Barratt-Eccles operad. It is analogous to the result about I-diagrams in spaces established in [Sch09, Proposition 6.5].
Theorem 2.13. For every commutative I-dga E, the chain complex E hI has a natural action of the Barratt-Eccles operad E.
Proof. We can view the simplicial k-module k{N ( Σ n )} as a simplicial chain complex concentrated in chain degree 0. The operad structure of Σ turns these simplicial k-modules into an operad in sMod k and in sCh k . We construct an action It is enough to specify the action of a q-simplex (σ 1 , . . . , σ q ) in N ( Σ s ) on a collection of elements (α i 1 , . . . , α i q ; x i ) in srep(E)[q] pi where α i j : n i j → n i j−1 is a map in I and x i is an element in E(n i q ) pi . On the indices (α i 1 , . . . , α i q ) for the sums in the simplicial replacement, we use the action of (σ 1 , . . . , σ q ) provided by the previous lemma. As . Since E is commutative, this does indeed define an operad action in sCh k . By Propositions 2.16 and 2.17 below, the composite Tot C * is lax symmetric monoidal. Hence it follows that E acts on E hI .
2.14. Monoidality of C * and Tot. It remains to verify the monoidal properties of C * and Tot that were used in the proof of Theorem 2.13.
Proposition 2.16. The functor C * : sCh k → Ch k (Ch k ) is lax symmetric monoidal.
As the simplicial structure maps of A and B commute with d A and d B , it follows that sh commutes with the vertical differential. The proof that the horizontal differential is compatible with sh is the same as for sh in the context of simplicial modules.
Proposition 2.17. The functor Tot is strong symmetric monoidal.
and we send a homogeneous element y ⊗ z ∈ Y r1,r2 ⊗ Z s1,s2 to the element This gives isomorphisms that are associative. It is clear that Tot respects the unit up to isomorphism. The maps ϕ Y,Z are compatible with the differential: Let y ⊗ z be a homogeneous element in Y r1,r2 ⊗ Z s1,s2 . The composition d Tot • ϕ applied to y ⊗ z gives First applying the differential to y ⊗ z and then ϕ yields thus both terms agree.
We denote the symmetry isomorphism in the category of chain complexes by χ. Then and this is equal to Remark 2.18. One can also consider a symmetric monoidal structure on Ch k (Ch k ) with the same underlying tensor product but with symmetry isomorphism for homogeneous elements y ⊗ z ∈ Y r1,r2 ⊗ Z s1,s2 . Then one can take ϕ in Proposition 2.17 to be the identity. However, this symmetry isomorphism is not compatible with the shuffle transformation from the proof of Proposition 2.16.
Remark 2.19. For a simplicial chain complex A one can also consider a normalized object N (A) ∈ Ch k (Ch k ) where one divides out by the subobject generated by degenerate elements. As the simplicial structure maps commute with the differential of A, this is well-defined, and the proof of Proposition 2.16 can be adapted as in [ML63,Corollary VIII.8.9] to show that the functor N : sCh k → Ch k (Ch k ) is also lax symmetric monoidal. Consequently, one can also use N instead of C * in the definition of the Barratt-Eccles operad E and the homotopy colimit P hI so that Theorem 2.13 remains valid.

Cochain functors with values in I-chain complexes
In this section we construct the functor A I discussed in the introduction and a version of the ordinary cochains with values in I-chain complexes.
3.1. Adjunctions induced by simplicial objects. We briefly recall an ubiquitous construction principle for adjunctions that we will later apply to simplicial objects in the categories of commutative I-dgas and I-chain complexes in order to define the commutative I-dga of polynomial forms on a simplicial set and an integral version of the Sullivan realization functor (see Definition 3.6). We note an immediate consequence of having the adjunction ( D, K D ).
Lemma 3.3. The functor D takes colimits in sSet to limits in D, and K D takes colimits in D to limits in sSet.
is a simplicial object in commutative I-dgas, we may apply Construction 3.2 both to D • and to its composite Since the extensions of D • and D ′ • to functors on sSet are defined by limit constructions and U commutes with limits, we have a natural isomorphism U (D(X)) ∼ = D ′ (X) for a simplicial set X. The adjoints K D and K D ′ are related by a natural isomorphism K D ′ ∼ = K D • C : (Ch I k ) op → sSet. An analogous remark applies to simplicial objects of algebras in Ch I k over a more general operad than the commutativity operad. For , the fact that Ch I k [C] → Set, E → E(m) q commutes with limits implies that the underlying set of D(X)(m) q is sSet(X, D • (m) q ). The pointwise k-module structure, differentials and multiplications on these sets give rise to the commutative I-dga structure on D(X).

The commutative I-dga version of polynomial forms.
Composing the left adjoints in the adjunctions (F I 1 , Ev 1 ) and (C, U ) introduced in (2.1) and (2.2) provides a left adjoint CF I 1 : Ch k → Ch I k [C] made explicit in (2.3). We are particularly interested in the commutative I-dga CF I 1 (D 0 ). For an element i ∈ k, the k-module map k → k = Ev 1 (U I ) 0 determined by 1 → i gives rise to a map ε i : CF I 1 (D 0 ) → U I . We write U I 0 and U I 1 for the two commutative CF I 1 (D 0 )algebras resulting from the elements 0, 1 ∈ k.
Definition 3.5. We let A I • : ∆ op → Ch I k [C] be the simplicial commutative I-dga given by the two-sided bar construction As with the space level version (see e.g. [May72]), the outer face maps are provided by the algebra structures of U I 0 and U I 1 , the inner face maps come from the multiplication of CF I 1 (D 0 ), and the degeneracy maps are induced by its unit. To make this simplicial object more explicit, we write D 0 r for the chain complex with copies of k on generators r in degree 0 and on dr in degree −1 and 0 elsewhere. Its non-zero differential is d(a·r) = a·dr. Since CF I 1 is left adjoint and U I is the unit for ⊠, commuting CF I 1 with coproducts provides an isomorphism of commutative I-dgas ) where the generators r 1 (p), . . . , r p (p) correspond to the p copies of CF I 1 (D 0 ). By adjunction, maps f : CF I 1 (D 0 r1(p) ⊕ · · · ⊕ D 0 rp(p) ) → E in Ch I k [C] correspond to families of elements f (r 1 (p)), . . . , f (r p (p)) ∈ E(1) 0 .
We now set r 0 (p) = 0 and define r p+1 (p) to be the image of 1 under the map k = U I (1) → CF I 1 (D 0 r1(p) ⊕ · · · ⊕ D 0 rp(p) )(1) induced by the unit. With this notation, the simplicial structure maps of the two sided bar construction (3.2) are determined by requiring Applying Construction 3.2, we obtain the following pair of adjoint functors.
The presence of extra degeneracies implies that Z is contractible to 0 (in the sense that Z → 0 is a weak equivalence in sMod k ) since the maps (−1) p+1 s p+1 define a contracting homotopy for the chain complex C * (Z).
The following lemma is the technical backbone for our homotopical analysis of the prolongation A I of A I • in Section 5. It is analogous to [BG76, Proposition 1.1]. Lemma 3.7. For all q ∈ Z and all positive objects m in I, the simplicial k-module A I •,q (m) is contractible to 0. Remark 3.8. The statement of the lemma does not hold for m = 0 since A •,0 (m) is the constant simplicial object on k, which is not contractible.
Proof of Lemma 3.7. We decompose the sum over s in (2.2) as C(P ) = U I ⊕ N(P ) where N(P ) = s≥1 P ⊠s /Σ s is the free non-unital I-dga on the I-chain complex P . For P = F I 1 (D 0 r1(p) ⊕ · · · ⊕ D 0 rp(p) ), this gives a decomposition of A I p that we use to define the extra degeneracy s p+1 . Restricted to the summand N(P ), the maps s p+1 for varying m will form a map of non-unital I-dgas. By the universal property of the free non-unital I-dgas, this map is determined by setting s p+1 (r j (p)) = r j (p+1) for all 1 ≤ j ≤ p. On the summand U I in the decomposition of C(P ), the map s p+1 will neither be a chain map nor be functorial in I. To define s p+1 on the copy of k in I-degree m, we let r p+1 (p) be its generator, choose a map ι : 1 → m, and define s p+1 (r p+1 (p)) = r ι p+1 (p + 1) where the index of the latter generator indicates that it lives in the summand with indices s = 1 and ι ∈ I(1 ⊔1 , m) of N(P ) = s≥1 I(1 ⊔s ,m) (D 0 r1(p) ⊕ · · · ⊕ D 0 rp(p) ) ⊗s / Σ s . It remains to check that the s p+1 provide extra degeneracies. In simplicial degree 0, the relation d 0 s 1 = 0 holds because d 0 sends the generator r ι 1 (1) to 0. Now assume p ≥ 0. We again use the sum decomposition C(P ) = U I ⊕ N(P ) and the universal property of the free non-unital I-dga to see that it is enough to check the compatibilites on the generators. The relation d p+1 s p+1 = id holds on the generators r 1 (p), . . . , r p (p) since d p+1 s p+1 r j (p) = d p+1 r j (p + 1) = r j (p).
Here we use that the restriction of d p+1 to the s = 1 summand is the sum over all α : 1 → m of the maps that send the generator r α p+1 (p) of the respective copy of k to 1 = r p+1 (p). Now let p ≥ 1 and 0 ≤ i ≤ p . For r 1 (p), . . . , r p (p), the relations d i s p+1 = s p d i hold since s p+1 only raises the index p in generators by 1 while d i does not depend on p. For r p+1 (p), we have d i s p+1 (r p+1 (p)) = d i (r ι p+1 (p + 1)) = r ι p (p) = s p (r p (p − 1)) = s p d i (r p+1 (p)). 3.9. Ordinary cochains. Let C(X; k) be the cochains with values in k on the simplicial set X, viewed as a homologically graded chain complex concentrated in non-positive degrees. (At this point, we disregard its cup product structure.) So for q ≥ 0, we have C −q (X; k) = Set(X q , k) with the pointwise k-module structure and differential induced by the face maps of X. The cochains on the standard n-simplices assemble to a functor C • :  Proof. For (i), we note that the description of the extension as lim ∆ p →X C(∆ p ; k) implies that there is a natural map from C(X; k). Writing X as a colimit of representable functors over its category of elements, the evaluation of this map at q is a bijection since taking maps into k turns colimits into limits.
For (ii), we only need to consider the case q ≤ 0, set n = −q and define For later use, we lift C • to I-chain complexes by defining . Corollary 3.11. (i) The extension C I of C I • to a functor sSet → Ch I k resulting from Construction 3.2 is naturally isomorphic to X → F I 0 C(X; k). (ii) For all q ∈ Z and m in I, the simplicial k-module C I •,q (m) = F I 0 (C(∆ • ; k) q ) is contractible to 0.

Homotopy theory of I-chain complexes and commutative I-dgas
In this section we review basic results about model category structures on I-chain complexes and commutative I-dgas. Much of this is motivated by (and analogous to) the corresponding results for space valued functors developed in [SS12,§3].
We continue to consider the category of unbounded chain complexes Ch k . For q ∈ Z, we as usual write S q for the chain complex with k concentrated in degree q, and D q for the chain complex with (D q ) i = k if i ∈ {q, q − 1}, with (D q ) i = 0 for all other i, and with d q = id k . We equip Ch k with the projective model structure whose weak equivalences are the quasi-isomorphisms and whose fibrations are the  The cofibrations in these level model structures are the retracts of relative cell complexes built out of cells of the form F I m (S q−1 ֒→ D q ) with m in I (resp. I + ) and q ∈ Z. Here F I m is the free functor defined in (2.1). 4.3. I-model structures. We now again use the homotopy colimit P hI from Construction 2.6. A map P → Q in Ch I k is an I-equivalence if it induces a quasiisomorphism P hI → Q hI . Moreover, an I-chain complex P is absolute (resp. positive) I-fibrant if α * : P (m) → P (n) is a quasi-isomorphism for all α : m → n in I (resp. I + ). Corollary 4.6. Let f : P → Q be a map between positive I-fibrant objects.
(i) If f is an I-equivalence, then it is also a positive level equivalence.
(ii) If f is a positive level fibration, then it is also a positive I-fibration.
We also note that the quasi-isomorphism type of P hI can easily be read off for positive I-fibrant P : Lemma 4.7. If P is positive I-fibrant in Ch I k and m is positive, then the inclusion of m in I induces a natural quasi-isomorphism P (m) → P hI .
Proof. This follows for example from [Dug01, Proposition 5.4] since the inclusion I + → I is homotopy cofinal and I + has contractible classifying space.
As another consequence of [Dug01, Theorem 5.2], we note that the adjunction colim I : Ch I k ⇄ Ch k : const I is a Quillen equivalence when Ch I k is equipped with the absolute or positive I-model structure. In particular, the composite of (4.1) is always a quasi-isomorphism, and each P in Ch I k is related by a zig-zag of Iequivalences (4.2) const I colim I (P cof ) ← P cof → P to a constant I-diagram colim I (P ) cof where P cof → P is a cofibrant replacement. We record the following lemma for later use.
Lemma 4.8. If (P j ) j∈J is a family of I-chain complexes, then the canonical map is a quasi-isomorphism provided that all the P j 's are positive I-fibrant.
Proof. Arbitrary products of weak equivalences between fibrant objects in a model category are weak equivalences. Therefore, using that (4.2) is a zig-zag of Iequivalences between positive I-fibrant objects under our assumptions allows us to assume that each P j is of the form const I A j . Forming the adjoint of the isomorphism j∈J const I A j ∼ = − → const I j∈J A j under the Quillen equivalence (colim I , const I ) shows that the composite in is a quasi-isomorphism. Since the second map is a product of quasi-isomorphisms, the claim follows by two-out-of-three. 4.9. Commutative I-dgas. Although essentially only our formulation of Theorem 1.5 depends on the existence of a lifted model structure on Ch I k [C], the following result is the main motivation for working with commutative I-dgas. Proof. An I-chain complex X admits a bar resolution X → X defined by X(n) = hocolim I↓n (X • π) where π : I ↓ n → I is the canonical projection from the overcategory forgetting the augmentation to n. The inclusion of the terminal object in I ↓ n induces a map X → X which is a level equivalence by a homotopy cofinality argument. The bar resolution has the property colim I X ∼ = X hI . When M is an E-algebra in Ch I k , then M inherits an E-algebra structure with diagonal E-action (compare the Theorem 2.13 and an analogous space level statements in [Sch09, Lemma 6.7]). When M is a commutative I-dga, then the E-algebra structure on colim I (M ) resulting from this observation and the strong monoidality of colim I coincides with the one on M hI provided by Theorem 2.13. We also note that if X is a cofibrant I-chain complex, then the map colim I X → colim I X is a quasi-isomorphism. This can be checked directly on free I-chain complexes, and the general case follows because both sides preserve colimits and send generating cofibrations to levelwise injections.
To prove the proposition, we note that the chain of Quillen equivalences from Theorem 4.10 sends a commutative I-dga M to colim I M cof , the colimit over I of a cofibrant replacement of M in Ch I k [E]. This colimit is related to M hI by a natural zig-zag of E-algebra maps where the first map is a quasi-isomorphism since the cofibrant replacement is an Iequivalence and the last map is a quasi-isomorphism by the above discussion since M cof is a cofibrant I-chain complex by [PS18,Theorem 4.4].
For later use we note that the commutative I-dga CF I 1 (A) from (2.3) has the following homotopical feature: Lemma 4.12. Let A be a cofibrant acyclic chain complex. Then each (CF I 1 (A))(m) is cofibrant in Ch k , and the unit U I → CF I 1 (A) is an absolute level equivalence. Proof. We show that the summands in (2.3) indexed by s ≥ 1 are acyclic and cofibrant. When |m| < s, the indexing set I(1 ⊔s , m) is empty. If |m| ≥ s, the Σ s -action on the set of injections I(1 ⊔s , m) is free. If R is a set of representatives of the orbits, we get an isomorphism R A ⊗s ∼ = ( I(1 ⊔s ,m) A ⊗s )/ Σ s , and the claim follows because each A ⊗s is cofibrant and acyclic.

Comparison of cochain functors
We now define a simplicial I-chain complex B I • by setting B I p = A I p ⊠ C I p in simplicial level p and using the ⊠-products of the simplicial structure maps of A I and C I as simplicial structure maps for B I • . There is a natural isomorphism (5.1) We prove the theorem at the end of the section.
Corollary 5.2. If X → Y is a weak homotopy equivalence of simplicial sets, then A I (Y ) → A I (X) is a positive level equivalence between positive I-fibrant objects.
Proof. The map C I (Y ) → C I (X) is an I-equivalence since C(Y ) → C(X) is a quasi-isomorphism of chain complexes by the homotopy invariance of singular homology. By the theorem, the claim about A I (Y ) → A I (X) follows.
Lemma 5.3. The maps in (5.2) are absolute level equivalences between absolute I-fibrant objects when evaluated in simplicial degree p.
Proof. Let m be an object in I. By Lemma 4.12 the map S 0 = U I (m) → A I p (m) is a quasi-isomorphism between cofibrant and fibrant objects and thus even a chain homotopy equivalence. The map S 0 → C(∆ p ) = C p is a quasi-isomorphism by the known computation of H * (∆ p ; k). Applying F I 0 , it provides an absolute level equivalence U I → C I p . By (5.1), we can decompose U I (m) → B I p (m) as We already showed that the first map is is a quasi-isomorphism. The last one is a quasi-isomorphism since − ⊗ C p preserves chain homotopy equivalences. The I-chain complexes A I p , B I p , and C I p are absolute I-fibrant for each p ≥ 0 since they absolute level equivalent to U I and U I = const I S 0 is absolute I-fibrant. Lemma 5.5. Let D • : ∆ op → Ch I k be a simplicial object in I-chain complexes such that for all q ∈ Z and all positive objects m in I, the simplicial k-module (D •,q )(m) is contractible to 0. Then for all p ≥ 0, the boundary inclusion ∂∆ p → ∆ p induces a positive level fibration D(∆ p ) → D(∂∆ p ).

Proof. A map in Ch I
k is a positive level fibration if and only if it has the right lifting property against the maps (U → V ) = F I m (0 → D q ) with m positive and q ∈ Z. By the adjunction (3.1), the lifting property for U → V and D(∆ p ) → D(∂∆ p ) is equivalent to the lifting property for ∂∆ p → ∆ p and K D (V ) → K D (U ). Inspecting the definition of K D , it follows that asking the latter lifting property for all p ≥ 0 is equivalent to asking the map of simplicial sets Ch I k (V, D • ) → Ch I k (U, D • ) to be an acyclic Kan fibration. Since (F I m , Ev m ) is an adjunction and since morphisms in Ch k out of D q correspond to level q elements, the assumption that U → V is The source of this map is contractible by assumption and a Kan complex because it is the underlying simplicial set of a simplicial k-module. Hence (D •,q )(m) → 0 is an acyclic Kan fibration.
Remark 5.6. When (D •,0 )(0) is not contractible and U → V is F I 0 (0 → D 0 ), the map Ch I k (V, D • ) → Ch I k (U, D • ) considered in the previous proof is not an acyclic Kan fibration. Thus D(∆ p ) → D(∂∆ p ) is not an absolute level fibration. In view of Remark 3.8, this shows that A I (∆ p ) → A I (∂∆ p ) is not an absolute level fibration.
• be a natural transformation of functors ∆ op → Ch I k . Suppose that for all p ≥ 0, the map D p → D ′ p is a positive level equivalence between positive I-fibrant objects and that for all q ∈ Z and all positive objects m in I, the simplicial k-modules (D •,q )(m) and (D ′ •,q )(m) are contractible. Then for every simplicial set X, the map D(X) → D ′ (X) is a positive level equivalence between positive I-fibrant objects.
Proof. As usual, this is proved by cell induction. Let us first assume that for all p ≥ 0, the map D(∂∆ p ) → D ′ (∂∆ p ) is a positive level equivalence between positive I-fibrant objects. Any simplicial set X can be written as a cell complex X = colim λ<κ X λ built from attaching cells of the form ∂∆ p → ∆ p . The functor D takes the inclusion ∂∆ p → ∆ p to a positive level fibration by Lemma 5.5. Since we assume that D(∂∆ p ) and D p ∼ = D(∆ p ) are positive I-fibrant, it follows from Corollary 4.6 that D(∆ p ) → D(∂∆ p ) is a positive I-fibration. The same holds for D ′ . Since both D and D ′ take colimits to limits by Lemma 3.3, the coglueing lemma in the positive level model structure and the fact that base change preserves I-fibrations shows that D(X) → D ′ (X) arises as a limit of pointwise positive level equivalences between inverse systems of positive I-fibrations. Hence it is a positive level equivalence between positive I-fibrant objects.
Since ∂∆ p only has non-degenerate simplices in dimensions strictly less than p, an analogous induction over the dimension of ∂∆ p shows the remaining claim about D(∂∆ p ) → D ′ (∂∆ p ).
Proof of Theorem 5.1. Combining Lemma 3.7, Corollary 3.11(ii), Lemma 5.4, and Lemma 5.3, the two maps A I → B I and C I → B I satisfy the hypotheses of Proposition 5.7.
We can now also prove Theorem 1.5 from the introduction: Proof of Theorem 1.5. The adjunction (A I , K I ) arises from A I • by applying Construction 3.2. Lemma 3.7 and Lemma 5.5 show that A I sends cofibrations to positive level fibrations and thus to positive I-fibrations. Corollary 5.2 implies that A I sends weak homotopy equivalences to positive level equivalences and thus to I-equivalences. The rest is an immediate consequence of the self-duality of model structures with respect to the passage to opposite categories and the adjunction isomorphisms (3.1). 5.8. The relation to polynomial forms. By adjunction, the canonical map D 0 → (const I CD 0 )(1) induces a map CF I 1 (D 0 ) → const I CD 0 of commutative I-dgas. Using the description of A PL,• as a two sided bar construction outlined in the introduction, it induces a map A I • → const I A PL,• in Ch I k [C] and thus a natural map A I (X) → const I A PL (X) on the extensions to simplicial sets.
Theorem 5.9. Let k be a field of characteristic 0. Then A I (X) → const I A PL (X) is a positive level equivalence. It induces a quasi-isomorphism A I (X) hI → A PL (X) that is an E-algebra map if we view the cdga A PL (X) as an E-algebra by restricting along the canonical operad map from E to the commutativity operad.
Proof. In characteristic zero the homology groups of (D 0 ) ⊗n /Σ n are isomorphic to the coinvariants H * (D 0 ) ⊗n /Σ n and the latter term is trivial for n ≥ 1 because D 0 is acyclic. Therefore CF I 1 (D 0 ) → const I CD 0 is a positive level equivalence. The claim about general X follows from Proposition 5.7 and the contractibility property of A PL,• established in [BG76, Proposition 1.1]. Applying (−) hI to this positive level equivalence and composing with the natural quasi-isomorphism (4.1) gives the quasi-isomorphism A I (X) hI → A PL (X). To see that it is an E-algebra map, we note that it follows from the definitions that (4.1) is an E-algebra map when evaluated on a cdga.

Comparison of E ∞ structures
Let E be the Barratt-Eccles operad introduced in Definition 2.11. We now define A : sSet op → Ch k [E] to be the composite A = (A I ) hI of the functor A I from the previous section and the functor (−) hI : Ch I k [C] → Ch k [E] resulting from Theorem 2.13. The following proposition shows that A is a cochain theory in the sense of [Man02].
Proposition 6.1. The functor A : sSet op → Ch k [E] has the following properties.
(i) It sends weak equivalence of simplicial sets to quasi-isomorphisms.
(ii) For a sub-simplicial set Y ⊆ X, the induced map from hofib(A(X/Y ) → A( * )) to hofib(A(X) → A(Y )) is a quasi-isomorphism. (iii) For a family (X j ) j∈J of simplicial sets indexed by a set J, the canonical map A( j∈J X j ) → j∈J A(X j ) is a quasi-isomorphism. (iv) It satisfies H 0 (A( * )) ∼ = k and H n (A( * )) ∼ = 0 if n = 0.
Proof. Part (i) follows from Corollary 5.2, part (iv) is an immediate consequence of Theorem 5.1, and part (iii) follows from Lemma 4.8 because A I takes coproducts in sSet to products of fibrant objects in Ch I k . For (ii), we view X/Y as the pushout of * ← Y → X. The functor A I sends this pushout to a pullback diagram (6.1) We need to show that we get a homotopy cartesian square after applying (−) hI to (6.1). Since all objects in (6.1) are positive fibrant, it follows from Lemma 4.7 that the resulting square of homotopy colimits over I is quasi-isomorphic to the square obtained from (6.1) by evaluating at 1. The latter square is homotopy cartesian since it is a pullback in which the vertical maps are fibrations. Remark 6.3. It is well-known how to express the cup-i products on the singular cohomology of spaces using the Barratt-Eccles operad, see for instance [BF04, Theorem 2.1.1]. This way the E-algebra structure on A(X) = A I (X) hI gives rise to cup-i products, and the previous theorem shows that they are equivalent to the usual cup-i products on the cochain algebra.
Theorem 6.2 also allows us to express Mandell's theorem [Man06] using A I : Proof of Theorem 1.2. Let X and Y be two finite type nilpotent spaces. By Proposition 4.11, the commutative I-dgas A I (X; Z) and A I (Y ; Z) are I-equivalent in Ch I Z [C] if and only if A I (X; Z) hI and A I (Y ; Z) hI are quasi-isomorphic in Ch Z [E], which is in turn equivalent to being quasi-isomorphic in Ch Z [E cof ]. By Theorem 6.2, this holds if an only if C * (X; Z) and C * (Y ; Z) are quasi-isomorphic in Ch Z [E cof ]. By [Man06,Main Theorem], this is the case if and only if X and Y are weakly equivalent.