Differential graded versus Simplicial categories

A differential graded (=dg) category is a category enriched in the category of complexes of modules over some commutative base ring k. Dg categories provide a framework for 'homological geometry' and for 'non commutative algebraic geometry' in the sense of Drinfeld and Kontsevich [4] [5] [14] [15] [16]. In [23] the homotopy theory of dg categories was constructed. This theory was allowed several developments such as: the creation by Toën of a derived Morita theory [24]; the construction of a category of non commutative motives [23]; the first conceptual characterization of Quillen-Waldhausen's K-theory [23]. . ..

On the other hand a simplicial category is a category enriched over the category of simplicial sets. Simplicial categories (and their close cousins: quasi-categories) provide a framework for ‘homotopy theories’ and for ‘higher category theory’ in the sense of Joyal, Lurie, Rezk, Toën . . . [11] [17][20] [25]. In [1] Bergner constructed a homotopy theory of simplicial categories by fixing an error in a previous version of [6]. This theory can be considered as one of the four Quillen models for the theory of (∞, 1)-categories, see [2] for a survey.

We observe that the homotopy theories of differential graded and simplicial categories are formally similar and so a ‘bridge’ between the two should be developed. In this paper we establish the first connexion between these theories by constructing a zig-zag of Quillen adjunctions relating the two:

In first place, we construct a Quillen model structure on positive graded dg categories by ’truncating’ the model structure of [23], see theorem 4.7.

Secondly we generalize Shipley-Schwede’s work [21] on connective DG algebras by extending the Dold-Kan correspondence to a Quillen equivalence between categories enriched over positive graded chain complexes and simplicial k-modules, see theorem 5. 19.

Finally we extend the k-linearization functor to a Quillen adjunction between simplicial categories and simplicial k-linear categories.

As an application, the zig-zag of Quillen adjunctions obtained allow us to give a conceptual explanation of Simpson’s homotopy fiber construction [22] used in his nonabelian mixed Hodge theory.

I am deeply grateful to Gustavo Granja for several useful discussions and for his kindness.

In what follows, k will denote a commutative ring with unit. The tensor product ⊗ will denote the tensor product over k. Let Ch denote the category of complexes over k and Ch ≥0 the full subcategory of positive graded complexes. Throughout this article we consider homological notation (the differential decreases the degree).

Observe that Ch ≥0 is a full symmetric monoidal subcategory of Ch and that the inclusion Ch ≥0 ֒→ Ch commutes with limits and colimits. We denote by Ch ≥0 (-, -) the internal Hom-functor in Ch ≥0 with respect to ⊗. By a dg category, resp. positive graded dg category, we mean a category enriched over the symmetric monoidal category Ch, resp. Ch ≥0 , see [4] [12] [13] [23]. We denote by dgcat, resp. dgcat ≥0 , the category of small dg categories, resp. small positive graded dg categories.

Notice that dgcat ≥0 is a full subcategory of dgcat and the inclusion dgcat ≥0 ֒→ dgcat commutes with limits and colimits.

Let sSet be the symmetric monoidal category of simplicial sets and sMod the category of simplicial k-modules. We denote by ∧ the levelwise tensor product of simplicial k-modules. The category (sMod, -∧ -) is a closed symmetric monoidal category. We denote by sMod(-, -) its internal Hom-functor.

By a simplicial category, resp. simplicial k-linear category, we mean a category enriched over sSet, resp. sMod, see [1].

We denote by sSet-Cat, resp. sMod-Cat, the category of small simplicial categories, resp. simplicial k-linear categories.

Let (C, -⊗ -, I C ) and (D, -∧ -, I D ) be two symmetric monoidal categories. A lax monoidal functor is a functor F : C → D equipped with:

-a morphism η :

which are coherently associative and unital (see diagrams 6.27 and 6.28 in [3]). A lax monoidal functor is strong monoidal if the morphisms η and ψ X,Y are isomorphisms.

Throughout this article the adjunctions are displayed vertically with the left, resp. right, adjoint on the left side, resp. right side.

In this section we will construct a Quillen model structure on dgcat ≥0 . For this we will adapt to our situation the Quillen model structure on dgcat constructed in chapter 1 of [23].

Remark 4.1. Chapter 1 of [23] (and the whole thesis) is written using cohomological notation. Throughout this article we are always using homological notation.

We now define the weak equivalences in dgcat ≥0 . Remark 4.4. Notice that the class Q qe consist exactly of those quasi-equivalences in dgcat, see [23, 1.6], which belong to dgcat ≥0 .

In order to build a Quillen model structure on dgcat ≥0 we consider the generating (trivial) cofibrations in dgcat which belong to dgcat ≥0 and introduce a new generating cofibration. Let us now recall these constructions, see section 1.3 in [23]. Definition 4.5. Following Drinfeld [4, 3.7.1] we define K to be the dg category that has two objects 1, 2 and whose morphisms are generated by

Let k be the dg category with one object 3, such that k(3, 3) = k. Let F be the dg functor from k to K that sends 3 to 1. Let B be the dg category with two objects 4 and 5 such that B(4, 4) = k , B(5, 5) = k , B(4, 5) = 0 and B(5, 4) = 0. Let n ≥ 1, S n-1 the complex k[n -1] and let D n be the mapping cone on the identity of S n-1 . We denote by P(n) the dg category with two objects 6 and 7 such that P(n)(6, 6) = k , P(n)(7, 7) = k , P(n)(7, 6) = 0 , P(n)(6, 7) = D n and whose composition given by multiplication. Let R(n) be the dg functor from B to P(n) that sends 4 to 6 and 5 to 7. Let C(n) be the dg category with two objects 8 et 9

and whose composition given by multiplication. Let S(n) be the dg functor from C(n) to P(n) that sends 8 to 6, 9 to 7 and S n-1 to D n by the identity on k in degree n -1.

Let Q be the dg functor from the empty dg category ∅, which is the initial object in dgcat ≥0 , to k. Finally let N be the dg functor from B to C(1) that sends 4 to 8 and 5 to 9.

Let us now recall the following standard recognition theorem:

Theorem 4.6. [8, 2.1.19] Let M be a complete and cocomplete category, W a class of maps in M and I and J sets of maps in M such that:

1. The class W satisfies the two out of three axiom and is stable under retracts.

2. The domains of the elements of I are small relative to I-cell.

3. The domains of the elements of J are small relative to J-cell.

Then there is a cofibrantly generated model category structure on M in which W is the class of weak equivalences, I is a set of generating cofibrations, and J is a set of generating trivial cofibrations. Theorem 4.7. If we let M be the category dgcat ≥0 , W be the class Q qe , J be the set of dg functors F and R(n), n ≥ 1, and I the set of dg functors Q, N and S(n), n ≥ 1, then the conditions of the recognition theorem 4.6 are satisfied. Thus, the category dgcat ≥0 admits a cofibrantly generated Quillen model structure whose weak equivalences are the quasi-equivalences.

Proof of Theorem 4.7. We start by observing that the category dgcat ≥0 is complete and cocomplete and that the class Q qe satisfies the two out of three axiom and that it is stable under retracts. We observe also that the domains and codomains of the morphisms in I and J are small in the category dgcat ≥0 . This implies that the first three conditions of the recognition theorem 4.6 are verified.

Proof. Since the inclusion dgcat ≥0 ֒→ dgcat preserves colimits and the class Q qe consist exactly of those quasi-equivalences in dgcat which belong to dgcat ≥0 , the proof follows from lemma 1.10 in [23].

√

We now prove that Jinj ∩ Q qe = Iinj. For this we introduce the following auxiliary class of dg functors: Definition 4.9. Let Surj ≥0 be the class of dg functors G : H → I in dgcat ≥0 such that:

-G(x, y) : H(x, y) → I(Gx, Gy) is a surjective quasi-isomorphism for all objects x, y ∈ H and -G induces a surjective map on objects. Remark 4.10. Notice that the class Surj ≥0 consist exactly of those dg functors in Surj, see section 1.3.1 in [23], which belong to dgcat ≥0 . Lemma 4.11. Iinj = Surj ≥0 .

Proof. We prove first the inclusion ⊇. Let G : H → I be a dg functor in Surj ≥0 . By remark 4.10, G belongs to Surj and so lemma 1.11 in [23] implies that G has the right lifting property with respect to the dg functors Q and S(n), n ≥ 1. Since the morphism of complexes G(x, y) : H(x, y) → I(Gx, Gy), x, y ∈ H is surjective on the degree zero component, the dg functor G also has the right lifting property with respect to N . This proves the inclusion ⊇.

We now prove the inclusion ⊆. Let R : C → D be a dg functor in Iinj. Lemma 1.11 in [23] implies that:

-R induces a surjective map on objects and -for all objects x, y ∈ C: -R(x, y) : C(x, y) → D(Rx, Ry) is a surjective quasi-isomorphism for n ≥ 1 and -H 0 R(x, y) : H 0 C(x, y) → H 0 D(Rx, Ry) is an injective map. Since R belongs to Iinj it has the right lifting property with respect to N and so the morphism of complexes R(x, y) is also surjective on the degree zero component. This clearly implies that R belongs to Surj ≥0 and proves the inclusion ⊆. √

We now consider the following ‘diagram chasing’ lemma:

Lemma 4.12. Let f : M • → N • be a morphism in Ch ≥0 such that:

Proof. The inclusion ⊇ follows from remark 4.10 and from the inclusion ⊇ in lemma 1.12 of [23]. We now prove the inclusion ⊆. Let R : C → D be a dg functor in Jinj ∩ Q qe . Since R belongs to Q qe and it has the right lifting property with respect to the dg functors R(n), n ≥ 1 the morphism of complexes

satisfies the conditions of lemma 4.12 and so R(x, y) is a surjective quasi-isomorphism.

Finally the fact that R induces a surjective map on objects follows from lemma 1.12 in [23]. This proves the lemma. √ Lemma 4.14. Jcell ⊆ Icof.

Proof. Observe that the morphisms in Jcell have the left lifting property with respect to the class Jinj. By lemmas 4.11 and 4.13 Iinj = Jinj ∩ Q qe and so the morphisms in Jcell have also the left lifting property with respect to the class Iinj, i.e. Jcell ⊆ Icof. √

We have shown that Jcell ⊆ Q qe ∩ Icof (lemmas 4.8 and 4.14) and that Iinj = Jinj ∩ Q qe (lemmas 4.11 and 4.13). This implies that the last three conditions of the recognition theorem 4.6 are satisfied. This finishes the proof of theorem 4.7.

Remark 4.15. Since every object in dgcat is fibrant, see remark 1.14 in [23], and the set J of generating trivial cofibrations in dgcat ≥0 is a subset of the generating trivial cofibrations in dgcat we conclude that every object in dgcat ≥0 is also fibrant.

In this subsection we construct a functorial path object in the Quillen model category dgcat ≥0 .

Consider the following adjunction:

where τ ≥0 denotes the ‘intelligent’ truncation functor: to a complex

The truncation functor τ ≥0 is a lax monoidal functor. In particular we have natural morphisms

which satisfy the associativity conditions. Observe that the truncation functor τ ≥0 preserve the unit

Definition 4.16. Let A be a small dg category. The truncation τ ≥0 A of A is the positive graded dg category with the same objects as A and whose complexes of morphisms are defined as

For x, y and z objects in τ ≥0 A the composition is defined as

where c denotes the composition operation in A. The units in τ ≥0 A are the same as those of A.

Observe that we have a natural adjunction

Remark 4.17. Notice that both functors i and τ ≥0 preserve quasi-equivalences.

Proposition 4.18. The adjunction (i, τ ≥0 ) is a Quillen adjunction.

Proof. Clearly, by remark 4.4 the functor i preserves weak equivalences. We now show that it also preserves cofibrations. The Quillen model structure of theorem 4.7 is cofibrantly generated and so by proposition 11.2.1 in [7] the class of cofibrations equals the class of retracts of relative I-cell complexes. Since the functor i preserves colimits it is then enough to prove that it sends the generating cofibrations in dgcat ≥0 to cofibrations in dgcat. This is clear, by definition, for the generating cofibrations Q and S(n), n ≥ 1. We now observe that i(N ) = N is also a cofibration in dgcat. In fact N can be obtained by the following push-out where I is a quasi-equivalence and P a fibration. By remark 4.17 and lemma 4.18 the functor τ ≥0 preserves quasi-equivalences and fibrations. Since the functor τ ≥0 also preserves limits we obtain the following factorization

This proves the lemma. √

In this section we will first construct a Quillen model structure on sMod-Cat and then show that it is Quillen equivalent to the model structure on dgcat ≥0 of theorem 4.7.

Recall from [9, III-2.3] the Dold-Kan equivalence between simplicial k-modules and positive graded complexes sMod

where N is the normalization functor and Γ its inverse. The normalization functor N is a lax monoidal functor, see [21, 2.3], via the Eilenberg-MacLane shuffle map, see [18,]

Observe that the normalization functor N preserves the unit of the two symmetric monoidal structures.

As it is shown in [21, 2.3] the lax monoidal structure on N , given by the shuffle map ∇, induces a lax comonoidal structure on Γ:

Now, let I be a set. Notation 5.1. We denote by Ch I ≥0 -Gr, resp. by Ch I ≥0 -Cat, the category of Ch ≥0graphs with a fixed set of objects I, resp. the category of categories enriched over Ch ≥0 which have a fixed set of objects I. The morphisms in Ch I ≥0 -Gr and Ch I ≥0 -Cat induce the identity map on the objects.

We have a natural adjunction

where U is the forgetful functor and T I is defined as

Composition is given by concatenation and the unit corresponds to 1 ∈ k.

Remark 5.2.

-Notice that the categories Ch I ≥0 -Gr and Ch I ≥0 -Cat admit standard Quillen model structures whose weak equivalences (resp. fibrations) are the morphisms F : A → B such that F (x, y) : A(x, y) -→ B(x, y), x, y ∈ I is a weak equivalence (resp. fibration) in Ch ≥0 . In fact the projective Quillen model structure on Ch ≥0 , see [9,, naturally induces a model structure on Ch I ≥0 -Gr which can be lifted along the functor T I using theorem 11.3.2 in [7].

-If the set I has a unique element, then the previous adjunction corresponds to the (Quillen) adjunction between connective dg algebras and positive graded complexes, see [10].

Notation 5.3. We denote by sMod I -Gr, resp. by sMod I -Cat, the category of sMod-graphs with a fixed set of objects I, resp. the category of categories enriched over sMod which have a fixed set of objects I. The morphisms in sMod I -Gr and sMod I -Cat induce the identity map on the objects.

In an analogous way we have an adjunction

where U is the forgetful functor and T I is defined as

Composition is given by concatenation and the unit corresponds to 1 ∈ k∆ 0 . In this subsection we will construct the left adjoint of N .

Let A ∈ dgcat ≥0 and denote by I its set of objects. Define L(A) as the simplicial k-linear category L I (A). Now, let F : A → A ′ be a dg functor. We denote by I ′ the set of objects of A ′ . The dg functor F induces the following diagram in sMod-Cat:

Notice that the square whose horizontal arrows are ψ 1 (resp. Notice that G induces a bijection on objects and so it belongs to sMod I -Cat. Finally define φ(G) as the following composition

where G ♯ denotes the morphism in Ch I ≥0 -Cat which corresponds to G under the adjunction (L I , N I ).

We now construct in a similar way the map η. Let F : A → N B be a dg functor and (N B) ′ be the full subcategory of N B whose objects are those which belong to the image of F . We have a natural factorization

Now, let N B be the positive graded dg category whose set of objects is

and whose positive graded complex of morphisms is defined as

The composition is given by the composition in (N B) ′ . Consider the dg functor

which maps a to (a, F ′ (a)) and the dg functor Notice that F induces a bijection on objects and so belongs to Ch I ≥0 -Cat. Since the normalization functor N preserve the set of objects, the above construction

can be naturally lifted to the category sMod-Cat. We have the folowing diagram sMod-Cat

We can now define η(F ) as the following composition

where F ♮ denotes the morphism in sMod I -Cat, which corresponds to F under the adjunction (L I , N I ). The maps η and φ are clearly inverse of each other and so the proposition is proven. √ -

-a cofibration if it has the left lifting property with respect to all trivial fibrations in sMod-Cat.

Definition 5.7. Let A be a small simplicial k-linear category. The homotopy category π 0 (A) of A is the category which has the same objects as A and whose morphisms are defined as

Proof. We start by observing that if we restrict ourselves to the 0-simplex morphisms in A and to the degree zero morphisms in N A we have the same category. In fact the degree zero component of the shuffle map ∇, used in the definition of N A, is the identity map, see [18,. Now suppose that π 0 (f ) is invertible. Then there exists a 0-simplex morphism g : y → x and 1-simplex morphisms h 1 ∈ A(x, x) and h 2 ∈ A(y, y) such that

Observe that the image of h 1 , resp. h 2 , by the normalization functor N is a degree 1 morphism in N A(x, x), resp. in N A(y, y), whose differential is gf -1 X (resp. f g -1 Y ). This implies that H 0 (N f ) is also invertible in H 0 (N A).

To prove the converse we consider an analogous argument. √ Proposition 5.9. A simplicial k-linear functor G : A → B is a weak equivalence iff:

(1) G(x, y) : A(x, y) → B(Gx, Gy) induces an isomorphism on π i for all i ≥ 0 and for all objects x, y ∈ A, and (2) π 0 (G) : π 0 (A) → π 0 (B) is essentially surjective.

Proof. We show that condition (1), resp. condition (2), is equivalent to condition (i), resp. condition (ii), of definition 4.2. By the Dold-Kan equivalence, we have the following commutative diagram

where the vertical arrows are isomorphisms. This implies that condition (1) is equivalent to condition (i) of definition 4.2.

Concerning condition (2), we start by supposing that π 0 (G) is essentially surjective. Consider the functor

and let z be an object in H 0 (N B). Since π 0 (B) and H 0 (N B) have the same objects we can consider z as an object in π 0 (B). By hypothesis, π 0 (G) is essentially surjective and so there exists an object w ∈ π 0 (A) and a 0-simplex morphism Gw f → z which becomes invertible in π 0 (B). Now lemma 5.8 implies that N f is invertible in H 0 (N B) and so we conclude that the functor H 0 (N G) is essentially surjective. This shows that condition (2) implies condition (ii) of definition 4.2. To prove the converse we consider an analogous argument. √ Theorem 5.10. The category sMod-Cat when endowed with the notions of weak equivalence, fibration and cofibration as in definition 5.6, becomes a cofibrantly generated Quillen model category and the adjunction (L, N ) becomes a Quillen adjunction.

The proof will consist on verifying the conditions of theorem 5.12 and proposition 5.13 in [23]. Since the Quillen model structure on dgcat ≥0 is cofibrantly generated, see theorem 4.7; every object in dgcat ≥0 is fibrant, see remark 4.15; and the functor N clearly commutes with filtered colimits it is enough to show that:

-for each simplicial k-linear category A, we have a factorization

with I A is a weak equivalence and P 0 × P 1 is a fibration in sMod-Cat. For this we need a few lemmas. We start with the following definition. Definition 5.11. Let us define P (A) as the simplicial k-linear category whose objects are the 0-simplex morphisms f : x → y in A which become invertible in π 0 (A). We define the simplicial k-module of morphisms

as the homotopy pull-back in sMod of the diagram

by which we mean the simplicial k-module

We denote the simplexes in A(x, x ′ ) and A(y, y ′ ) lateral morphisms and the simplexes in sMod(k∆ [1], A(x, y ′ )) homotopies. The composition operation

decomposes on:

-a composition of lateral morphisms, which is induced by the composition on A and -a composition of homotopies, which is given by the map

where the last map is induced by the diagonal map in k∆ [1].

Remark 5.12. Notice that a 0-simplex morphism α :

We have a natural commutative diagram in sMod-Cat

where I A is the simplicial k-linear functor that associates to an object x ∈ A the 0-simplex morphism x Id → x and P 0 , resp. P 1 , is the simplicial k-linear functor that sends a morphism x f → y in P (A) to x, resp. y. Notice that by applying the normalization functor N to the above diagram and lemma 4.20 to the dg category N A we obtain two factorizations

H H of the diagonal dg functor. By lemma 4.20 τ ≥0 P (N A) is a path object of N A in dgcat ≥0 . We will show in proposition 5.17 that N P (A) is also a path object of N A. Lemma 5.13. Let A, B ∈ sMod. The shuffle map ∇ induces a natural surjective chain homotopy equivalence

which has a natural section induced by the Alexander-Whitney map.

Proof. First note that if (L, R) and (L ′ , R ′ ) are adjoint pairs of functors, a natural transformation ζ : L → L ′ induces a natural transformation ζ ♯ : R ′ → R which is a natural equivalence iff ζ is also.

Fixing a chain complex N A ∈ Ch ≥0 let L, L ′ : Ch ≥0 → Ch ≥0 be defined by

Using the Dold-Kan equivalence in the case of L ′ , we see that these functors have right adjoints

respectively.

The shuffle map determines a natural inclusion ∇ : L → L ′ which has a right inverse given by the Alexander-Whitney map AW , see [21, 2.7]. It follows that ∇ ♯ : R ′ → R is a natural surjection with a section given by AW ♯ .

The fact that ∇ ♯ is a natural transformation of bi-functors is clear. Since ∇ is a chain homotopy equivalence, in order to finish the proof it is now enough to show that the functors L, L ′ , R, R ′ send chain homotopic maps to chain homotopic maps (for (L, R) and (L ′ , R ′ ) will then induce adjunctions on the homotopy category Ho(Ch ≥0 ) and ∇ : L → L ′ will be a natural isomorphism between endo-functors of Ho(Ch ≥0 )).

The functors L and R clearly preserve the chain homotopy relation. For the same reason, L ′ and R ′ preserve the relation on Ch ≥0 (C, D) defined by the cylinder object

x x q q q q q q q q q q q q C . Since the Alexander-Whitney and shuffle maps give maps between this cylinder object and the usual one, we see that this relation is the usual chain homotopy relation. This concludes the proof.

√

We now define a map φ relating the Ch ≥0 -graphs associated with the dg categories N P (A) and τ ≥0 P (N A). Observe that:

-By lemma 4.20, N P (A) and τ ≥0 P (N A) have exactly the same objects and -For each pair of objects

, the map of lemma 5.13 (with

Notation 5.14. We denote by

the map of Ch ≥0 -graphs which is the identity on objects and φ f,f ′ on the complexes of morphisms.

Remark 5.15. Notice that by definition of P (A) and remark 5.12 the map φ preserve identities and the composition of degree zero morphisms.

We now establish a ‘homotopy equivalence lifting property’ of φ.

Proposition 5.16. Let α be a degree zero morphism in τ ≥0 P (N A) that becomes invertible in H 0 (τ ≥0 P (N A)). Then there exists a degree zero morphism α in N P (A) which becomes invertible in H 0 (N P (A)) and φ(α) = α.

Proof. Let

be a degree zero morphism in τ ≥0 P (N A). Notice that α is of the form (m x , h, m y ) with m x : x → x ′ and m y : y → y ′ degree zero morphisms in N A and h : x → y a degree 1 morphism in N A. Now, by definition of P (A) we can choose a representative h ∈ A(x, y) 1 of h and so we obtain a degree zero morphism α = (m x , h, m y ) in N P (A) such that

Now suppose that α is invertible in H 0 (τ ≥0 P (N A)). Then there exist morphisms β of degree 0 and r 1 and r 2 of degree 1 such that d(r 1 ) = βα -1 and d(r 2 ) = αβ -1.

As above, we can lift β to a morphism β in N P (A). Since the map φ preserve the identities and the composition of degree zero morphisms it maps αβ to αβ and βα to βα. Finally since the maps φ f,f ′ are surjective quasi-isomorphisms we can lift r 1 to r 1 , resp. r 2 to r 2 , in N P (A) by applying the lemma [8, 2.3.5] to the couple (r 1 , 1), resp. (r 2 , 1). This implies that α is also invertible in H 0 (N P (A)). √ Proposition 5.17 8 8 q q q q q q q q q q q obtained by applying the normalization functor N to the diagram (1) in sMod-Cat, the dg functor N (I A ) is a quasi-equivalence and N (P 0 ) × N (P 1 ) is a fibration.

Proof. We first prove that N (I A ) is a quasi-equivalence. By definition of P (A) the dg functor I A clearly satisfies condition (1) of proposition 5.9. We now prove that N (I A ) satisfies condition (ii) of definition 4.2. Let f be an object in N P (A). The dg categories N P (A) and τ ≥0 P (N A) have the same objects and so we can consider f as an object in τ ≥0 P (N A). Since the dg functor

is a quasi-equivalence, see lemma 4.20, there exists an object x in N A and a homotopy equivalence α in τ ≥0 P (N A) between I(x) and f . By proposition 5.16 we can lift α to a homotopy equivalence α in N P (A) and so the dg functor

satisfies condition (ii) of definition 4.2. This proves that N (I) is a quasi-equivalence. We now prove that N (P 0 ) × N (P 1 ) is a fibration. By definition of P (A) the dg functor N (P 0 ) × N (P 1 ) is clearly surjective on the complexes of morphisms. We now prove that it has the right lifting property with respect to the generating trivial cofibration F , see definition 4.5. Let x f → y be an object in N P (A) and γ : (x, y) → (x ′ , y ′ ) a homotopy equivalence in N A × N A. Since the dg functor τ ≥0 (P ) :

is a fibration there exists a homotopy equivalence α : f → f ′ in τ ≥0 P (N A) such that τ ≥0 (P )(α) = γ. Now, by proposition 5.16 we can lift α to a homotopy equivalence

This proves the proposition. √

Notice that the previous proposition implies theorem 5.10.

Remark 5.18. Since every object in dgcat ≥0 is fibrant, see remark 4.15, all simplicial k-linear categories will be fibrant with respect to this Quillen model structure. Proof. Let A ∈ dgcat ≥0 be a cofibrant dg category and B a simplicial k-linear category. Recall from remark 5.18 that every object in sMod-Cat is fibrant. We need to show that a simplicial k-linear functor F : L(A) -→ B is a weak equivalence in sMod-Cat iff the corresponding dg functor

We have the folowing commutative diagram

where η is the counit of the adjunction (L, N ). Since, by definition, F is a weak equivalence in sMod-Cat iff N F is a quasi-equivalence it is enough to show that η is a quasi-equivalence. The dg functor η is the identity map on objects and so it is enough to show that η(x, y) : A(x, y) -→ N L(A)(x, y), x, y ∈ A is a quasi-isomorphism. Now, let I be the set of objects of A. Since A is cofibrant in dgcat ≥0 it clearly stays cofibrant when considered as an object of the Quillen model structure on Ch I ≥0 -Cat, see remark 5.2. By proposition 6.4 of [21] the adjunction morphism in Ch

induces an isomorphism in π i for i ≥ 0 and for all objects x, y ∈ ΓU (A). This implies by the Dold-Kan equivalence that -for any objects x and y in A, the map F (x, y) : A(x, y) -→ B(F x, F y) is a weak equivalence of simplicial sets and -the induced functor π 0 (F ) : π 0 (A) -→ π 0 (B) is essentially surjective. Proposition 6.2. The adjunction (k(-), U ) is a Quillen adjunction, when we consider on sMod-Cat the Quillen model structure of theorem 5.10.

Proof. We first observe that by proposition 5.9 the functor U : sMod-Cat -→ sSet-Cat preserves weak equivalences.

We now show that it also preserves fibrations. Let G : A → B be a simplicial k-linear functor such that N G : N A → N B is a fibration in dgcat ≥0 . We need to show that U G is a fibration in sSet-Cat. Recall from [1] that U G is a fibration iff: Concerning condition (F2), let x ∈ U A, y ∈ U B and f : Gx → y be a homotopy equivalence in U B. This means that f is invertible in π 0 (B) and so by lemma 5.8 N (f ) is also invertible in H 0 (N B). This data allow us to construct, using proposition 1.7 in [23], the following (solid) commutative square

Since N G is a fibration in dgcat ≥0 we can lift N f to a morphism h : x → z in N A which is invertible in H 0 (N A). Since the 0-simplex morphisms in A and the degree zero morphisms in N A are exactly the same, lemma 5.8 implies that h : x → z, when considered as a morphism in U A, satisfies condition (F2). This proves the proposition. √

We have obtained the following zig-zag of Quillen adjunctions relating the homotopy theories of differential graded and simplicial categories: Proof. It follows from remark 6.3. √

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