The de Rham homotopy theory and differential graded category

Homotopy Coherent Category Theory

Coalgebraic Approach to the Loday Infinity Category, Stem Differential for 2n-ary Graded and Homotopy Algebras

We define a graded twisted-coassociative coproduct on the tensor algebra TW of any Zn-graded vector space W . If W is the desuspension space ↓V of a graded vector space V , the coderivations (resp. quadratic “degree 1” codifferentials, arbitrary odd codifferentials) of this coalgebra are 1to-1 with sequences πs, s ≥ 1, of s-linear maps on V (resp. Zn-graded Loday structures on V , sequences tha...

Semi-Infinite de Rham Theory

I shall start by explaining just what “semi-infinite” means and how it gives rise to interesting structures in geometry. These structures are studied as part of String Theory and are closely related to Floer Theory. Manifolds which carry a “semi-infinite” structure include loop spaces (C∞(S1, M) for M a closed manifold) and the theory seems to have particular simplicities when the original mani...

On the de Rham Cohomology of Differential and Algebraic Stacks

We introduce the notion of cofoliation on a stack. A cofoliation is a change of the differentiable structure which amounts to giving a full representable smooth epimorphism. Cofoliations are uniquely determined by their associated Lie algebroids. Cofoliations on stacks arise from flat connections on groupoids. Connections on groupoids generalize connections on gerbes and bundles in a natural wa...

Lecture Notes on De Rham–Hodge Theory

These are lecture notes for a 1–semester undergraduate course (in mathematics, physics, engineering, chemistry or biology) on the De Rham–Hodge theory. This landmark theory of the 20th Century mathematics gives a rigorous foundation to modern field and gauge theories in physics, engineering and physiology. The only necessary background for comprehensive reading of these notes is Green’s theorem...