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A new model of dg-categories

Elena Dimitriadis Bermejo

Abstract

In this article, we develop a new model for the category of dg-categories. Following Rezk's example in the case of classic Segal spaces, we define dg-Segal spaces: functors between free dg-categories of finite type and simplicial spaces to which we add certain properties. We define also complete dg-Segal spaces, and make their relationship to classic Segal spaces explicit. With the help of two new hypercover constructions, and up to a certain hypothesis, we prove that there exists an equivalence between the homotopy category of dg-categories and the homotopy category of functors defined above with a model structure making the complete dg-Segal spaces into its fibrant objects.

A new model of dg-categories

Abstract

In this article, we develop a new model for the category of dg-categories. Following Rezk's example in the case of classic Segal spaces, we define dg-Segal spaces: functors between free dg-categories of finite type and simplicial spaces to which we add certain properties. We define also complete dg-Segal spaces, and make their relationship to classic Segal spaces explicit. With the help of two new hypercover constructions, and up to a certain hypothesis, we prove that there exists an equivalence between the homotopy category of dg-categories and the homotopy category of functors defined above with a model structure making the complete dg-Segal spaces into its fibrant objects.
Paper Structure (23 sections, 46 theorems, 99 equations)

This paper contains 23 sections, 46 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. A historical perspective
  3. Motivation
  4. Main results and definitions
  5. Plan
  6. Acknowledgments
  7. Notation
  8. Background notions
  9. Differential graded categories
  10. Simplicial localizations
  11. The universal model category
  12. The Sing functor and dg-Segal spaces
  13. Constructing the adjunction
  14. dg-Segal spaces and the delinearization functor
  15. Complete dg-Segal spaces
  16. ...and 8 more sections

Key Result

Proposition 1.3

There exists a morphism, called the linearisation of $\Delta$, between the categories $\Delta$ and $c\mathcal{L}_\mathbb{S}$, and it defines a Quillen adjunction between the categories $\mathop{\mathrm{Fun}}\nolimits^\mathbb{S}(c\mathcal{L}_\mathbb{S}^{op},\mathbf{sSet})$ and $\mathop{\mathrm{Fun}}\

Theorems & Definitions (138)

  • Definition 1.1: \ref{['Def: dg-Segal spaces']}
  • Definition 1.2: \ref{['Def: G alpha']}
  • Proposition 1.3: \ref{['Prop: delinearisation']}
  • Definition 1.4: \ref{['Def: complete dg-Segal space']}
  • Theorem 1.5: \ref{['Th: complee dg-Segal model structure']}
  • Theorem 1.7: \ref{['Th: fully faithfulness']}, \ref{['Th: equivalence']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.4
  • Definition 2.5
  • ...and 128 more