[Paper Review] Differential graded versus Simplicial categories

Gonçalo Tabuada|ArXiv.org|Nov 24, 2007

Homotopy and Cohomology in Algebraic Topology2 references3 citations

TL;DR

This paper establishes a zig-zag of Quillen adjunctions between the homotopy theories of differential graded (dg) categories and simplicial categories by generalizing the Dold-Kan correspondence to a Quillen equivalence between categories enriched over positive graded chain complexes and simplicial $k$-modules. The key contribution is a conceptual explanation of Simpson’s homotopy fiber construction via this bridge between dg and simplicial homotopy theories.

ABSTRACT

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.

Motivation & Objective

• To construct a bridge between the homotopy theories of differential graded and simplicial categories.
• To generalize Shipley-Schwede’s work on connective DG algebras to enriched categories over positive graded chain complexes.
• To extend the Dold-Kan correspondence to a Quillen equivalence between positive graded dg categories and simplicial $k$-modules.
• To provide a conceptual explanation of Simpson’s homotopy fiber construction in non-abelian Hodge theory.
• To establish a zig-zag of Quillen adjunctions linking the homotopy theories of dg and simplicial categories.

Proposed method

• Construct a Quillen model structure on the category of positive graded dg categories by truncating the model structure from [23].
• Generalize the Dold-Kan correspondence to a Quillen equivalence between categories enriched over positive graded chain complexes and simplicial $k$-modules.
• Define a $k$-linearization functor from simplicial categories to simplicial $k$-linear categories, and show it forms a Quillen adjunction.
• Use the Dold-Kan correspondence to relate the homotopy theory of positive graded dg categories to that of simplicial $k$-linear categories.
• Construct a zig-zag of Quillen adjunctions: $s\mathbf{Set}$-Cat $\xrightleftarrows[k(-)]{U}$ $s\mathbf{Mod}$-Cat $\xrightleftarrows[N]{L}$ $\mathsf{dgcat}_{\geq 0}$ $\xrightleftarrows[\tau_{\geq 0}]{L}$ $\mathsf{dgcat}$.
• Leverage the Quillen adjunctions to relate homotopy limits and colimits, particularly for the derived functors of the composition.

Experimental results

Research questions

• RQ1 How can the homotopy theories of differential graded and simplicial categories be connected via Quillen adjunctions?
• RQ2 Can the Dold-Kan correspondence be extended to a Quillen equivalence between categories enriched over positive graded chain complexes and simplicial $k$-modules?
• RQ3 What is the conceptual role of the $k$-linearization functor in relating simplicial and dg categories?
• RQ4 How does the derived functor of the composition of adjunctions relate to Simpson’s homotopy fiber construction?
• RQ5 Can the homotopy fiber of a dg functor be understood as the homotopy fiber of its image under the composed derived functor?

Key findings

• A Quillen model structure is constructed on the category of positive graded dg categories by truncating the model structure from [23].
• The Dold-Kan correspondence is generalized to a Quillen equivalence between categories enriched over positive graded chain complexes and simplicial $k$-modules.
• The $k$-linearization functor $k(-)$ and its right adjoint $U$ form a Quillen adjunction between simplicial $k$-linear categories and simplicial categories.
• The zig-zag of Quillen adjunctions establishes a derived equivalence between the homotopy theories of dg and simplicial categories.
• The derived functor $\mathbb{R}(U \circ L \circ \tau_{\geq 0})$ preserves homotopy limits, and the homotopy fiber of a dg functor is equivalent to the homotopy fiber of its image under this composition.
• The construction provides a conceptual explanation of Simpson’s homotopy fiber construction in non-abelian mixed Hodge theory.