[Paper Review] DG quotients of DG categories

Vladimir Drinfeld|ArXiv.org|Oct 8, 2002

Homotopy and Cohomology in Algebraic Topology27 references4 citations

TL;DR

This paper develops a theory of derived quotients (DG quotients) for differential graded categories, generalizing Verdier's triangulated quotient to the DG setting. It provides two constructions—one via resolution under flatness assumptions, and another via a new localization-like method—proving the DG quotient is unique up to quasi-equivalence and characterizing its Hom complexes via cone constructions in derived categories.

ABSTRACT

Keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with Verdier's notion of quotient of a triangulated category modulo a triangulated subcategory. This work is an attempt to further develop his theory. More than a half of the text is devoted to an overview of "well known" definitions and results. As a result, the e-print is essentially self-contained.

Motivation & Objective

• To extend Keller's DG quotient theory to a more systematic and self-contained framework.
• To resolve the limitations of classical triangulated quotients by working in the DG category setting, preserving more homotopical information.
• To provide a canonical construction of the DG quotient of a DG category modulo a full DG subcategory.
• To characterize the Hom complexes of the DG quotient using derived tensor products and cones.
• To establish uniqueness of the DG quotient up to quasi-equivalence, enabling consistent use in derived algebraic geometry and homological algebra.

Proposed method

• Introduces a new construction of the DG quotient using a resolution of the DG category when the flatness condition fails.
• Employs a localization-like approach inspired by Dwyer-Kan, simplifying the construction compared to Keller's original method.
• Uses the ind-version of the DG category to define the orthogonal complement, ensuring the quotient is well-behaved.
• Characterizes the Hom complexes of the DG quotient as cones: $\operatorname{Cone}(h_Y \buildrel L\over{\otimes}_{{\mathcal{B}}} \tilde{h}_X \to \operatorname{Hom}(X,Y))$.
• Establishes uniqueness of the DG quotient via homotopical equivalence and quasi-equivalence, allowing the use of 'the' DG quotient.
• Relies on the triangulated category associated to a DG category ($\mathcal{A}^{\operatorname{tr}}$) to relate DG quotients to classical Verdier quotients.

Experimental results

Research questions

• RQ1 How can one systematically construct a derived quotient of a DG category modulo a full DG subcategory?
• RQ2 What is the homological structure of the Hom complexes in such a DG quotient?
• RQ3 In what sense is the DG quotient unique, and how does it relate to classical triangulated quotients?
• RQ4 How does the DG quotient construction behave under flatness assumptions or when resolving non-flat categories?
• RQ5 Can the DG quotient be characterized in terms of derived functors and $A_\infty$-structures?

Key findings

• A DG quotient of a small DG category $\mathcal{A}$ modulo a full DG subcategory $\mathcal{B}$ always exists and is unique up to quasi-equivalence.
• Under a flatness assumption, the DG quotient can be constructed by simply killing the objects of $\mathcal{B}$, simplifying the construction.
• The Hom complexes in the DG quotient are canonically isomorphic to the cone of the map $h_Y \buildrel L\over{\otimes}_{{\mathcal{B}}} \tilde{h}_X \to \operatorname{Hom}(X,Y)$, where $h_Y$ and $\tilde{h}_X$ are associated DG modules.
• The DG quotient construction induces an equivalence $\mathcal{A}^{\operatorname{tr}} / \mathcal{B}^{\operatorname{tr}} \to \mathcal{C}^{\operatorname{tr}}$, linking it to classical Verdier quotients.
• The theory allows for a well-defined, canonical DG quotient, enabling consistent use in derived algebraic geometry and $A_\infty$-category theory.
• The DG quotient is compatible with derived functors and provides a framework for defining $T(\mathcal{A}_1, \mathcal{A}_2)$ via Keller's model or Kontsevich's model.