Here are abstracts of a few papers of mine that connect to higher category theory, with links to the full texts. I will be glad to discuss this material.
Let X be a smooth algebraic variety over a field K containing the real numbers. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf of X. These are stack-like versions of usual deformations. We prove that there is a twisted quantization map from twisted Poisson deformations to twisted associative deformations, which is canonical and bijective on gauge equivalence classes. This result extends work of Kontsevich, and our own earlier work, on deformation quantization of algebraic varieties.
We construct a 2-dimensional twisted nonabelian multiplicative integral. This is done in the context of a Lie crossed module (an object composed of two Lie groups interacting), and a pointed manifold. The integrand is a connection-curvature pair, that consists of a Lie algebra valued 1-form and a Lie algebra valued 2-form, satisfying a certain differential equation. The geometric cycle of the integration is a kite in the pointed manifold. A kite is made up of a 2-dimensional simplex in the manifold, together with a path connecting this simplex to the base point of the manifold. The multiplicative integral is an element of the second Lie group in the crossed module.
We prove several properties of the multiplicative integral. Among them is the 2-dimensional nonabelian Stokes Theorem, which is a generalization of Schlesinger’s Theorem. Our main result is the 3-dimensional nonabelian StokesTheorem. This is a totally new result.
The methods we used are: the CBH Theorem for the nonabelian exponential map; piecewise smooth geometry of polyhedra; and some basic algebraic topology.
The motivation for this work comes from twisted deformation quantization. In the paper (no. 1 above) we encountered a problem of gluing nonabelian gerbes, where the input was certain data in differential graded algebras. The 2-dimensional multiplicative integral gives rise, in that situation, to a nonabelian 2-cochain; and the 3-dimensional Stokes Theorem shows that this cochain is a twisted 2-cocycle. (This was superseded by a simpler approach; see no. 3 below.)
We consider descent data in cosimplicial crossed groupoids. This is a combinatorial abstraction of the descent data for gerbes in algebraic geometry. The main result is this: a weak equivalence between cosimplicial crossed groupoids induces a bijection on gauge equivalence classes of descent data. This result is used to construct the twisted quantization in paper no. 1 above (replacing the earlier approach with surface integration).
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