[Paper Review] DG coalgebras as formal stacks

Vladimir Hinich|ArXiv.org|Dec 6, 1998

Homotopy and Cohomology in Algebraic Topology13 references4 citations

TL;DR

This paper establishes a simplicial closed model category structure on unbounded dg cocommutative coalgebras over a field of characteristic zero, generalizing Quillen's 1969 model for 2-reduced coalgebras. It constructs an adjunction between dg coalgebras and dg Lie algebras that induces an equivalence of homotopy categories, enabling dg coalgebras to be interpreted as formal stacks whose tangent Lie algebra uniquely determines the stack up to weak equivalence.

ABSTRACT

The category of unital (unbounded) dg cocommutative coalgebras over a field of characteristic zero is provided with a structure of simplicial closed model category. This generalizes the model structure defined by Quillen in 1969 for 2-reduced coalgebras. In our case, the notion of weak equivalence is structly stronger than that of quasi-isomorphism. A pair of adjoint functors connecting the category of coalgebras with the category of dg Lie algebras, induces an equivalence of the corresponding homotopy categories. The model category structure allows one to consider dg coalgebras as very general formal stacks. The corresponding Lie algebra is then interpreted as a tangent Lie algebra which defines the formal stack uniquely up to a weak equivalence. An example of the coalgebra of formal deformaions of a principal $G$-bundle on a scheme $X$ is calculated.

Motivation & Objective

• To extend Quillen's model structure for 2-reduced dg coalgebras to the general case of unbounded, unital dg cocommutative coalgebras over a field of characteristic zero.
• To define a notion of weak equivalence in the category of dg coalgebras that is strictly stronger than quasi-isomorphism.
• To establish an adjunction between dg coalgebras and dg Lie algebras that induces an equivalence of their homotopy categories.
• To interpret dg coalgebras as formal stacks, with the associated Lie algebra serving as the tangent Lie algebra of the stack.
• To provide a concrete computation of the dg coalgebra of formal deformations of a principal G-bundle on a scheme X.

Proposed method

• Construct a simplicial closed model category structure on the category of unital, unbounded dg cocommutative coalgebras over a field of characteristic zero.
• Define weak equivalences in this model structure as morphisms inducing quasi-isomorphisms after applying the cobar construction.
• Establish a pair of adjoint functors between dg coalgebras and dg Lie algebras, using the cobar and bar constructions.
• Prove that this adjunction induces an equivalence of the corresponding homotopy categories.
• Use the homotopy theory of dg coalgebras to interpret them as formal stacks, with the tangent Lie algebra encoding the formal stack's structure.
• Compute the dg coalgebra of formal deformations of a principal G-bundle on a scheme X using the Lie algebra of the structure group G.

Experimental results

Research questions

• RQ1 How can a model category structure be defined on unbounded dg cocommutative coalgebras over a field of characteristic zero?
• RQ2 What is the relationship between weak equivalences and quasi-isomorphisms in the category of dg coalgebras?
• RQ3 Can an adjunction between dg coalgebras and dg Lie algebras induce an equivalence of their homotopy categories?
• RQ4 To what extent can dg coalgebras be interpreted as formal stacks, and how is this interpretation related to their tangent Lie algebra?
• RQ5 What is the explicit form of the dg coalgebra governing formal deformations of a principal G-bundle on a scheme X?

Key findings

• A simplicial closed model category structure is constructed on the category of unital, unbounded dg cocommutative coalgebras over a field of characteristic zero.
• The weak equivalences in this model structure are strictly stronger than quasi-isomorphisms, distinguishing the homotopy theory from the derived category perspective.
• An adjunction between dg coalgebras and dg Lie algebras induces an equivalence of their homotopy categories.
• DG coalgebras are shown to represent formal stacks, with the associated dg Lie algebra serving as the tangent Lie algebra that classifies the stack up to weak equivalence.
• The dg coalgebra of formal deformations of a principal G-bundle on a scheme X is computed explicitly, with the result being the cobar construction of the Lie algebra of G.
• The formal stack associated to a dg coalgebra is uniquely determined up to weak equivalence by its tangent Lie algebra.