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The Dold-Kan Correspondence and Coalgebra Structures

By using the Dold-Kan correspondence we construct a Quillen adjunction between the model categories of non-cocommutative coassociative simplicial and differential graded coalgebras over a field. We restrict to categories of connected coalgebras and prove a Quillen equivalence between them.


Introduction
The Dold-Kan correspondence establishes an equivalence between the category of simplicial objects and the category of non-negatively graded differential objects in abelian categories. In this paper we work with the abelian category of vector spaces over a fixed field K. We investigate the model categorical properties of the functors involved in the Dold-Kan correspondence on the level of the corresponding model categories of comonoids. We show that the Dold-Kan correspondence gives rise to Quillen adjunctions between the model categories of simplicial and nonnegatively graded differential counital coalgebras. We investigate some approaches to extending the resulting adjunctions to an equivalence on the level of homotopy categories.
Section 2 is devoted to some generalities about the monoidal categories of nonnegatively graded differential vector spaces, simplicial vector spaces and their corresponding categories of comonoids.
Section 3 is a dualization of the work by Schwede and Shipley in [SS03]. First, we show that the normalization functor N defines a functor from the category of simplicial coalgebras to the category of non-negatively differential graded coalgebras. Similar considerations hold for the functor Γ, however, as in [SS03], we point out that both functors are neither adjoint nor inverse to each other on the level of comonoids. Then, we construct functors that are right adjoint to the functors N and Γ on the level of comonoids. The adjoint pairs of functors obtained in this way turn out to be adjoint Quillen pairs. Finally, we discuss whether these Quillen adjunctions can be improved to Quillen equivalences via a categorical dualization of a result by Schwede-Shipley and a criterion by Hovey. Section 4 restricts to the categories of connected simplicial coalgebras and connected differential graded coalgebras. We investigate the completeness and cocompleteness properties of connected coalgebras. With these properties, we derive from [GG99], [Smi11] and [Goe95] model category structures for categories of connected coalgebras. We prove a Quillen equivalence between the categories of connected differential graded and simplicial coalgebras.
Section 5 closes this paper with an appendix on the category of connected differential graded algebras, which is helpful for constructing limits for connected differential graded coalgebras.

Preliminaries
2.1. Differential graded vector spaces. Let Vct be the category of vector spaces over a fixed field K. We denote by DGVct the category of differential graded K-vectors spaces which are concentrated in non-negative degrees and have differentials of degree −1. We denote by S n the n-sphere chain complex. This is the object of DGVct which has the field K in degree n and 0 in other degrees. All differentials in S n are trivial. Secondly, the n-disk, denoted by D n , is the object of DGVct which has the field K in degrees n and n − 1 and 0 elsewhere. The identity on K is its only non-trivial differential.
Recall that if X and Y are two objects in the category DGVct, the monoidal product ⊗ is given by ( The unit of this monoidal product is S 0 , the differential graded vector space concentred in degree 0 which we sometimes denote by K[0]. Notice that the monoidal product ⊗ is symmetric. The category DGVct endowed with the monoidal product ⊗ is closed. Given differential graded vector spaces (X, d X ) and (Y, d Y ), let Hom(X, Y ) be the object of DGVct with: The specified right adjoint of the functor − ⊗ Y : DGVct → DGVct is then given by the functor Hom(Y, −) : DGVct → DGVct.

2.2.
Simplicial vector spaces. We denote by SVct the category of simplicial vector spaces, that is, a functor X : ∆ op → Vct, where ∆ is the category of finite ordered sets.
Let X and Y be two objects in SVct. The monoidal product ⊗ is given by (X ⊗Y ) n = X n ⊗ K Y n with coordinatewise structure maps. The unit of the monoidal product ⊗ is the simplicial vector space that has K in each degree and identity maps on K as face and degeneracy operators. We denote this unit by I(K). This monoidal product ⊗ is symmetric and SVct is closed.
2.3. Differential graded coalgebras. We denote by DGcoAlg the category of counital coassociative differential graded K-coalgebras. In other words DGcoAlg is the category of comonoids the monoidal category (DGVct,⊗,K[0]). In the coming sections the symbols ⊓ and ⊔ stand respectively for the categorical product and coproduct in DGcoAlg.
Lemma 2.1. [GG99, Proposition 1.10]. The forgetful functor U d from the category of differential graded coalgebras to the category of differential graded vector spaces has a right adjoint S d .
Lemma 2.2. The forgetful functor U s from the category of simplicial coalgebras to the category of simplicial vector spaces has a right adjoint S s .
Here, the functor S s is obtained by extending degreewise the cofree coalgebra functor S from the category of vector spaces to the category of coalgebras as constructed in [Swe69, Theorem 6.4.1].

A comparison of coalgebras categories
3.1. Dold-Kan functors for coalgebras. The Dold-Kan correspondence asserts that SVct and DGVct are essentially the same. This equivalence of categories is achieved by the normalization functor N and its inverse Γ. We refer to [Wei94,8.8.4] for a detailed description of these functors. Moreover, it is well-known that the normalized versions of the Alexander-Whitney map AW : N (A ⊗B) → N A ⊗ N B and of the shuffle map ∇ : N A ⊗ N B → N (A ⊗B) both turn the normalization functor into a monoidal functor. From these observations, we derive that the normalization functor N and its inverse Γ pass on the level of comonoids as stated in propositions below.
is a differential graded coalgebra with a comultiplication given by the composition Proposition 3.2. If (B, ∆ B , ε B ) is a differential graded coalgebra, then (ΓB, ∆ ΓB , ε ΓB ) is a simplicial coalgebra with a comultiplication ∆ ΓB given by the following composition T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T ΓB ⊗ΓB ΓB ⊗ΓB and counit given by Γ(ε B ).
Notice that both propositions and their proofs are dual to results in [SS03, Section 2.3]. We refer to [Sor10, Section 5.1] for detailed proofs.
Dually to [SS03, Section 2.4], we point out that the coalgebra-valued functors N and Γ are neither adjoint nor inverse to each other. This fact is specified in the remark below.
Lemma 3.3. The adjunction counit ε : N Γ −→ Id is a comonoidal transformation. Let ψ X,Y be the composition of natural maps Proposition 3.4. The functor Γ : DGcoAlg −→ DGcoAlg is full and faithful and respects coalgebra structures. Moreover, the composite endofunctor N Γ is naturally isomorphic to the identity functor on the level of categories of comonoids.
Remark 3.5. The unit η : Id −→ ΓN does not have good comonoidal properties. More precisely, there are objects X and Y in the category ScoAlg such that the diagram does not commute. Indeed, consider for example X = Y = Γ(Z[1]) as in [SS03, Remark 2.14]. Since N is left inverse to Γ by the previous proposition, one has Therefore the lower composite map in the previous diagram vanishes in degree 1 since But in degree 1, the right map η X ⊗η Y is an isomorphism between free abelian groups of rank one since  [DS95,Chapter 7]. A map f in the category DGVct is a weak equivalence if H * f is an isomorphism, a cofibration if for each n ≥ 0, f n is injective, and a fibration if for each n ≥ 1, f n is surjective. The generating acyclic cofibrations are given by the maps 0 → D n | n ≥ 1 and the generating cofibrations by S n−1 → D n | n ≥ 1 .

3.2.2.
Model structure of DGcoAlg. The category DGcoAlg is endowed with a model structure in [GG99,Section 2]. A map f in the category DGcoAlg is a weak equivalence if H * f is an isomorphism, a cofibration if it is a degreewise injection of graded K-vector spaces, and a fibration if it has the right lifting property with respect to trivial cofibrations.

3.2.3.
Model structure of SVct. The category of simplicial vector spaces has a cofibrantly generated model structure proven in [Qui67, II.4, II.6]. The model structure comes from defining a map f in the category SVct to be a weak equivalence if π * f ∼ = H * N f is an isomorphism, a cofibration if it is a level-wise injection, and a fibration if it has the right lifting property with respect to trivial cofibrations. Since DGVct is cofibrantly generated, one can deduce that SVct is cofibrantly generated by applying the transfer result by Crans in [Cra95, Section 3] to the adjoint pair (Γ, N ) provided by the Dold-Kan correspondence. Hence in the category SVct, the generating acyclic cofibrations are given by the maps 0 → Γ(D n ) | n ≥ 1 and the generating cofibrations by Γ(S n−1 ) → Γ(D n ) | n ≥ 1 .

3.2.4.
Model structure of ScoAlg. The category of cocommutative simplicial coalgebras has a model structure proven in [Goe95, Section 3]. But one can adapt the arguments therein to the non-cocommutative case as well. A map f in ScoAlg is a weak equivalence if π * f ∼ = H * N f is an isomorphism, a cofibration if it is a levelwise inclusion, and a fibration if it has the right lifting property with respect to trivial cofibrations.
Goerss' main line of argumentation remains unchanged for the case of non cocommutative simplicial coalgebras. The only slight difference is concerned with the lemma recalled below.
where i is a cofibration and p is an acyclic fibration.
In the proof of this lemma we may replace the cocommutative cofree functor by its non-cocommutative version S s : SVct −→ ScoAlg. In this way, the arguments of Goerss transfer to the non-cocommutative setting since only the cofreeness property is required.
3.3. Quillen adjunctions for coalgebras. We compare ScoAlg and DGcoAlg in terms of Quillen setting. The situation may be illustrated in the diagram where N stands for the coalgebra-valued normalization functor.
Proposition 3.7. In the above situation the functor N has a right adjoint R com . Moreover the adjoint pair ( N , R com ) is a Quillen pair.
Proof. First, let V be a differential graded vector space and S d (V ) its differential graded cofree coalgebra. We set Indeed, the various adjoint pairs (U s , S s ), (N, Γ), (U d , S d ) and the identity N U s = U d N respectively, yield the following bijections This implies that the functor R com is right adjoint to N for cofree coalgebras. Now we notice that the adjunction (U d , S d ) defines a monad S d U d over the category DGcoAlg. Thus, if C is a differential graded coalgebra, it can be writtten as the equalizer of the diagram Since the functor R com should be a right adjoint it has to preserve limits. Therefore defining R com (C) as the equalizer of the maps R com (d 0 ) and R com (d 1 ) yields the desired right adjoint. Finally we observe that the cofibrations and acyclic cofibrations in SVct and DGVct match with those of their respective categories of comonoids ScoAlg and DGcoAlg. Since the functor N is a left Quillen functor the identity N U s = U d N ensures that the functor N is a left Quillen functor.
Remark 3.8. We mention that instead of N , we could consider the coalgebravalued functor Γ. By similar techniques, it is possible to construct a right adjoint functor R com to Γ and show that ( Γ, R com ) is a Quillen pair. In the propositions below, we find good candidates for, respectively, generating acyclic fibrations and generating fibrations in DGVct.
and only if f has the left lifting property with respect to any map in Q.
Proof. One direction is obvious since maps D n → 0 are acyclic fibrations in the standard model structure of DGVct. For the other direction, suppose f has the left lifting property with respect to Q. Assume that there is an x n ∈ X n with x n = 0 and f n (x n ) = 0. Since x n = 0 there is a linear map α n : X n → K such that α n (x n ) = 1.
(i) If x n is not a cycle then define α n−1 (dx n ) = 1. This gives a chain map α : X → D n and by assumption a lift ζ exists in the following diagram which is the desired contradiction. Therefore f n (x n ) = 0 and the chain map f is injective. (ii) Let x n be a cycle and a boundary. That is, d(x n ) = 0 and there is an x n+1 ∈ X n+1 such that d(x n+1 ) = x n . Then define α n+1 (x n+1 ) = 1. Note that x n+1 is not a boundary. This gives a chain map α : X → D n+1 and a contradiction follows as in case (i). (iii) Let x n be a cycle and not a boundary. As we are working with field coefficients, x n then generates a sphere sub-complex S n of X. This sub-complex S n can be mapped to D n+1 and the above argument works again.
Proposition 3.10. Define P to be the set {D n → 0, D n → S n | n ≥ 1}. A chain map f is in P -proj if and only if it is injective and induces an isomorphism in homology.
Proof. One direction is obvious since every map in P is a fibration. For the other direction, injectivity comes from P -proj ⊂ Q-proj (since Q ⊂ P ) and the previous lemma. We show that every map in P -proj induces isomorphism in homology. Let f : A → B be a map in P -proj. As f is injective, we have the short exact sequence 0 → A → B → cokerf → 0. As this sequence induces a long exact sequence in homology, it suffices to show that cokerf has no homology. The object cokerf which is built degreewise is also a pushout / / cokerf We deduce that the map 0 → cokerf is also in P -proj since P -proj is closed under pushouts. Let us now assume that cokerf is not acyclic. Then there is an element x n ∈ (cokerf ) n with d(x n ) = 0 and x n not a boundary. Note that x n generates a subcomplex S n of cokerf and that cokerf ∼ = S n ⊕ cokerf /S n as chain complexes. But the map 0 → S n is not in P -proj since 0 / / D n S n S n does not admit a lift. Therefore the map 0 → cokerf cannot be in P -proj either which is the desired contradiction.
Unfortunately, the codomains S n in the set P are not cosmall. Indeed, consider a cofiltered sequence of maps in the category DGVct does not need to be a bijection: for instance, consider Y α to be S n and all maps Y α → Y α−1 to be trivial.
Moreover, we lose closedness property of the category ((DGVct) op , ⊗) since the functor − ⊗ Y : (DGVct) op −→ (DGVct) op does not have a right adjoint. To conclude, although the opposite category (DGcoAlg) op inherits a model structure in the usual way, this model structure does not arise from an application of the transfer theorem.
3.4.2. Hovey criterion. The Quillen adjunction ( N , R com ) would be a Quillen equivalence if N reflected weak equivalences between cofibrant objects and if for every fibrant differential graded coalgebra Y , the map N R com Y cof → Y was a weak equivalence in DGcoAlg. Notice that since every object in ScoAlg is cofibrant, the cofibrant replacement R com Y cof can be taken to be R com Y . Hence, we are reduced to studying the map λ Y : N R com Y → Y for fibrant coalgebras Y .
Definition 3.11. A fibrant differential graded coalgebra Y satisfies the Hovey criterion if λ Y is a weak equivalence in DGcoAlg, that is We are not able to show that the Hovey criterion holds for an arbitrary fibrant differential graded coalgebra, but we investigate the Hovey criterion for some classes of examples of differential graded coalgebras.
Proof. As the functor R com is a right adjoint, it carries the terminal object K[0] to the terminal object of ScoAlg which is I(K), the simplicial coalgebra with the one dimensional coalgebra K in each degree. Therefore, we obtain hence K[0] satisfies the Hovey criterion.
Recall that every object is fibrant in the model structure on the category DGVct. Since the functor S d preserves fibrations, it follows that every cofree coalgebra is fibrant. In particular S d (S 0 ) is fibrant. Now the definition of the functor R com on cofree coalgebras and the Dold-Kan correspondence yields Since the face maps in the simplicial coalgebra S s I(K) are all identities on S(K), their alternating sums are either 0 or id S(K) and the associated complex is: Finally the Hovey criterion for the differential graded coalgebra S d (S 0 ) follows from the fact that S d (S 0 ) is concentrated in degree zero and from the identification of S d (S 0 ) with S(K). Now let V be an acyclic differential graded vector space. Since the functors S d , Γ and S s preserve weak equivalences, we deduce on one hand that S d (V ) is weakly equivalent to S d (0) = K[0] and on the other hand that Hence, we obtain the required result for acyclic differential graded vector spaces.
Notice in particular that the Hovey criterion holds for the cofree coalgebra on every n-disk D n .

Categories of connected coalgebras
4.1. Connected differential graded coalgebras. In this section we restrict to connected differential graded objects. An object V in DGVct is connected if V 0 = 0 and we denote by DGVct c the category of connected differential vector spaces. With mild changes, the category DGVct c inherits from DGVct a model category structure. Indeed, DGVct c has limits and colimits constructed degreewise as in DGVct. Moreover, the model category factorization axioms may be performed as in [GS06, Section 1.3], but discarding this time objects such as the 0-sphere S 0 and the 1-disk D 1 / ∈ DGVct c . In this way, a map f in the category DGVct c is a weak equivalence if H * f is an isomorphism, a cofibration if for each n ≥ 1, f n is injective, and a fibration if for each n ≥ 2, f n is surjective. An object C in DGcoAlg is connected if C 0 = K. We denote by DGcoAlg c the category of connected differential graded coalgebras and in the rest of this section we discuss its model category structure. Now, let V be an object in the category DGVct c . Then, the tensor coalgebra on V is defined by We mention that the structure maps on T ′ d (V ) are given by [HMS74, Section II.2]. Let C be a connected differential graded coalgebra. Then, the functor I ′ d : DGcoAlg c → DGVct c is defined by . In other words, the differential graded vector space I ′ d (C) n is C n for n ≥ 1 and 0 otherwise. Proof. We start with limits. A terminal object in DGcoAlg c is given by K[0]. For other limits, we recall from [GG99, Proposition 1.7] that there is an anti-equivalence between the category of differential graded coalgebras and the category of profinite differential graded algebras. Since by the appendix in Section 5, the category of connected differential graded algebras is complete and cocomplete, we derive limits for DGcoAlg c by applying the steps given in [GG99, Proof of Proposition 1.8]. We mention that the usual tensor product ⊗ of differential graded coalgebras is not the categorical product in DGcoAlg c since we do not assume cocommutativity.
Next, we investigate colimits. We refer to [Nei78, Section 1] where similar constructions appear for cocommutative coalgebras. Since, as a left adjoint, the functor I ′ d must preserve initial objects, we deduce that K[0] is initial in DGcoAlg c . Let f, g : C → D be two maps in DGcoAlg c . Then, their coequalizer is given by D/im(f − g). This quotient is constructed degreewise and each of its homogeneous parts is in fact a coalgebra by [Swe69, Proposition 1.4.8]. Finally, if C and D are two objects in DGcoAlg c , we may form the maps Then the coproduct of C and D in DGcoAlg c is given by Notice that the direct sum is the coproduct of the underlying differential graded vector spaces and that the quotient guarantees the required connectedness condition.
After these categorical facts, we endow the category of connected differential graded coalgebras with a model structure which is defined exactly as in Section 3.2.2. We essentially concentrate on the model category factorization axioms.
Lemma 4.4. Let V be an acyclic (H * V = 0) connected differential graded vector space and C, a connected differential graded coalgebra. Then the projection is an acyclic fibration.
Proof. By using [GG99, Proof of Lemma 1.12], we obtain the isomorphisms where T C * denotes the tensor C * -algebra functor. The cohomological Künneth formula yields The required homology isomorphism H * (C ⊓ T ′ d (V )) ∼ = H * C comes from applying the cohomological universal coefficient theorem. It remains to prove that each projection is a fibration. To see this, we consider the diagram / / C where i is a cofibration in DGcoAlg c . By adjointness, this amounts to considering the diagram  Proof. Let us consider the object X = D ⊓ T ′ d cone I ′ d (C) ∈ DGcoAlg c . Since cone I ′ d (C) ∈ DGVct c is acyclic, Lemma 4.4 implies that the projection p : X → D is an acyclic fibration. Next, the following diagram shows how to obtain the coalgebra map i Indeed, j comes from the composition C −→ I ′ d (C) −→ cone (I ′ d (C)) and the universal property of the tensor coalgebra functor T ′ d . Finally, the map i is a cofibration since is a canonical injection map.
Proposition 4.6. Any morphism f : C → D in DGcoAlg c can be factored as with i an acyclic cofibration and p a fibration.
Sketch of proof. We exploit arguments in [Smi11, Definition 4.14, Lemma 4.16] for our purpose. First, factor f as C −→ C where the map i 0 is a canonical injection, hence a cofibration in DGcoAlg c and p 0 a fibration. Then, construct the maps (i n ) n≥1 and (p n ) n≥1 as displayed in the diagram in which (1) for n ≥ 0, the objects G(n + 1) are pullbacks of Lemma 4.7. Let C be a fibrant connected differential graded coalgebra and be the factorization of ǫ C : C → K[0] as in the previous lemma. Then, C is a retract of G. Moreover, G is a cofree connected differential graded coalgebra.
Proof. Since C is fibrant, the counit ǫ C is a fibration. It follows that a lift exists in the diagram C and therefore that C is a retract of G. For the second statement, note first that G(0) is cofree in DGcoAlg c since . As a right adjoint functor, T ′ d commutes with inverse limits. Hence, we may recursively deduce that for n ≥ 1 the objects G(n) are cofree in DGcoAlg c and finally that G = lim n G(n) is cofree as required.
4.2. Connected simplicial coalgebras. This section is the simplicial counterpart of the previous one. An object V in SVct is connected if V 0 = 0. A connected simplicial coalgebra C is an object in ScoAlg with C 0 = K. We denote by SVct c and ScoAlg c the categories of connected simplicial vector spaces and coalgebras.
Lemma 4.8. Let C be a connected simplicial coalgebra. Then, the constant simplicial object I(K) splits off the object C.
Proof. In the category ScoAlg c , the object I(K) is both initial and terminal as proven in Proposition 4.11 below. Consequently, the canonical map i : I(K) → C is an injection and therefore I(K) splits off C.
We next consider T ′ : Vct → coAlg the tensor coalgebra functor constructed in [Swe69, Theorem 6.4.1]. The simplicial tensor coalgebra T ′ s : SVct c → ScoAlg c is defined as a degreewise prolongation of the tensor coalgebra functor T ′ .
Definition 4.9. Let C be a connected simplicial coalgebra. Then, the functor I ′ s : ScoAlg c → SVct c is defined by I ′ s (C) = C/I(K). Notice that this definition makes sense because of Lemma 4.8.
Proposition 4.10. Let V be a connected simplicial vector space and 1 K denotes a generator of I(K). If C is a connected simplicial coalgebra, then there is a bijection from the set of simplicial coalgebras maps C → T ′ s (V ) to the set of connected simplicial vector spaces maps I ′ s (C) → V such that u(1 K ) = 0. Proof. In the following, = ScoAlg c (I ′ s (C), V ) the first bijection comes from the fact that T ′ s is right adjoint to the forgetful functor from simplicial coalgebras to simplicial vector spaces. The second bijection follows from the assumption u(1 K ) = 0 which implies that I(K) ⊆ ker u.
Proposition 4.11. The category of connected simplicial coalgebras is complete and cocomplete.
Proof. For limits in ScoAlg c , it suffices to extend degreewise the construction by [Ago11, Theorem 1.1] for the category of coalgebras over fields. Hence, a terminal object in ScoAlg c is given by I(K), the constant simplicial coalgebra. Notice that, since the field K is a terminal object in coAlg, the product K ⊓ K is isomorphic to K. This ensures that the products C ⊓ D of objects in ScoAlg c is again connected Colimits in ScoAlg c are formed in the same way as for DGcoAlg c . In this way, an initial object is given by I(K). If f, g : C → D are two maps in ScoAlg c , their coequalizer is given by D/im(f − g). Finally, If C and D are two objects in ScoAlg c , we may form the maps Then the coproduct of C and D in ScoAlg c is given by Notice that the direct sum is taken degreewise and that the quotient guarantees the connectedness condition.
With the above definition, the category ScoAlg c is endowed with a model category structure exactly as in [Goe95, Section 3].
4.3. A Quillen equivalence for connected coalgebras. In this section, we improve the Quillen adjunction ( N , R com ) to a Quillen equivalence. Recall that we were not able to check the Hovey's criterion for arbitrary fibrant differential graded coalgebras. However, connectedness is a condition that guarantees such a criterion which yields a Quillen equivalence.
Lemma 4.12. There is an equivalence between the category of connected simplicial vector spaces and the category of connected differential graded vector spaces.
Proof. We notice that the restriction of the normalization functor N : SVct c → DGVct c is full and faithful since it is induced by the Dold-Kan equivalence. Moreover, if V an object in DGVct c , we may find an object W in SVct c such that N W ∼ = V . Indeed, since Γ(V ) 0 = V 0 = 0, it follows that Γ(V ) ∈ SVct c and setting W = Γ(V ) satisfies the required condition. Therefore, by [Par70, Section 2.1, Proposition 3], we deduce that the restriction of the normalization functor N induces an equivalence between connected vector spaces categories with an inverse given by the restriction of Γ.
The following result is dual to [Qui69, Part I, Proposition 4.5].
Lemma 4.13. Let V be a differential graded vector space. Then the following maps Proof. These maps are obtained by using the universal properties of the respective tensor coalgebras. Then, as vector spaces, the tensor coalgebra is the same as the tensor algebra, hence we obtain the isomorphisms by applying Künneth and Eilenberg-Zilber theorems.
Theorem 4.15. If C is a fibrant connected differential graded coalgebra, then the map N R com C −→ C is a weak equivalence. Hence, there is a Quillen equivalence between the category of connected differential graded coalgebras and the category of connected simplicial coalgebras.

Appendix on connected differential graded algebras
In this appendix, we consider the category of connected differential graded algebras. This category is somehow dual to the category of connected differential graded coalgebra. We refer to [GG99, Proposition 1.7] for the precise statement. Its main interest here, is that a particular dual of its limits is used to construct colimits for the category DGcoAlg c .
Definition 5.1. A connected differential graded algebra A is a differential graded vector space equipped with an associative, graded multiplication M : A ⊗ A → A and a unit µ : K[0] → A with A 0 = K. We denote by DGAlg c the category of connected differential graded algebras. Proposition 5.4. The category of connected differential graded algebras is complete and cocomplete.
Proof. Constructions of limits are well-known in the category of differential graded algebras. Because of connectedness, some refinements have to be performed. A terminal object in DGAlg c is given by K[0]. If f, g : A → B are two maps in DGAlg c , their equalizer is given by ker (f − g). Now let A and B be objects in DGAlg c . We may form the maps Then the product of A and B in DGAlg c is given by For colimits, we first notice that an initial object in DGAlg c is given by K[0]. Then the coequalizer of two maps f, g : A → B is given by B/ f (a) − g(a), a ∈ A where f (a) − g(a), a ∈ A denotes the ideal generated by f (a) − g(a) for a ∈ A. In [Jar97], the coproduct of two differential graded non-commutative algebras A and B is given by factoring out from the tensor algebra T d (A ⊗ B) the ideal I which is generated by elements of the form (a 1 ⊗ b 1 ) ⊗ (1 ⊗ b 2 ) − a 1 ⊗ b 1 b 2 , (a 1 ⊗ 1) ⊗ (a 2 ⊗ b 2 ) − a 1 a 2 ⊗ b 2 .
However, the resulting object need not to be connected even if A and B are connected. To get around this problem, we define the coproduct of two objects in DGAlg c as A ⊔ B = T d (ker γ A ⊗ ker γ B ) /I. Since ker γ A and ker γ A are connected differential graded vector spaces, they will not contribute to the degree 0 part of the tensor algebra T d (ker γ A ⊗ ker γ B ). In this way, we will have (A ⊔ B) 0 = K and therefore A ⊔ B ∈ DGAlg c .