We construct a zig-zag of Quillen adjunctions between the homotopy theories of differential graded and simplicial categories. In an intermediate step we generalize Shipley-Schwede's work 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. As an application we obtain a conceptual explanation of Simpson's homotopy fiber construction.
Deep Dive into Differential graded versus Simplicial categories.
We construct a zig-zag of Quillen adjunctions between the homotopy theories of differential graded and simplicial categories. In an intermediate step we generalize Shipley-Schwede’s work 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. As an application we obtain a conceptual explanation of Simpson’s homotopy fiber construction.
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 the
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