Vladimir Hinich

Homological algebra of homotopy algebras

We define closed model category structures on different categories connected to the world of operad algebras over the category C(k) of (unbounded) complexes of k-modules: on the category of operads, on the category of algebras over a fixed operad, on the category of modules over a fixed operad algebra. In Sections 2 - 6 we define the necessary structures and provide some standard comparison results. In Section 7 we define cotangent complex. In Section 8 we define a canonical structure of homotopy Lie algebra on the tangent complex.

DG coalgebras as formal stacks

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 the most 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. As an example, we calculate the coalgebra of formal deformations of a principal G-bundle on a scheme X.

Deformations of Homotopy Algebras

Abstract Let k be a field of characteristic zero, 𝒪 be a dg operad over k and let A be an 𝒪-algebra. In this note we suggest a definition of a formal deformation functor of A from the category of artinian local dg algebras to the category of simplicial sets. This functor generalizes the classical deformation functor for an algebra over a linear operad. In the case 𝒪 and A are non-positively graded, we prove that Def A is governed by the tangent Lie algebra T A which can be calculated as the Lie algebra of derivations of a cofibrant resolution of A. An example shows that the result does not necessarily hold without the non-positivity condition.

Virtual operad algebras and realization of homotopy types

Abstract We prove that the category of algebras over a cofibrant operad admits a closed model category structure. This leads to the notion of “virtual operad algebra” – the algebra over a cofibrant resolution of the given operad. In particular, virtual commutative algebras can serve to an algebraic description of homotopy p-types as in the recent preprint of Mandell (M. Mandell, E ∞ -algebras and p -adic homotopy theory, Hopf preprint server, October, 1998). Our main result allows one to simplify the proof of Mandells theorem.

Noncommutative unfolding of hypersurface singularity

A version of Kontsevich Formality theorem is proven for smooth DG algebras. As an application of this, it is proven that any quasiclassical datum of noncommutative unfolding of an isolated surface singularity can be quantized.