We discuss two aspects of the presentation of the theory of principal ∞-bundles in an ∞-topos, introduced in [NSSa], in terms of categories of simplicial (pre)sheaves. First we show that over a cohesive site C and for G a presheaf of simplicial groups which is C-acyclic, G-principal ∞-bundles over any object in the ∞-topos over C are classified by hyper-Čech-cohomology with coefficients in G. T...

This paper solves the problem of computing conformal structures of general 2manifolds represented as triangular meshes. We approximate the De Rham cohomology by simplicial cohomology and represent the Laplace-Beltrami operator, the Hodge star operator by linear systems. A basis of holomorphic one-forms is constructed explicitly. We then obtain a period matrix by integrating holomorphic differen...

The classical HKR-theorem gives an isomorphism of the n-th Hochschild cohomology of a smooth algebra and the n-th exterior power of its module of Kähler differentials. Here we generalize it for simplicial, graded and anticommutative objects in “good pairs of categories”. We apply this generalization to complex spaces and noetherian schemes and deduce two decomposition theorems for their (relati...

This is a further investigation of our approach to group actions in homological algebra the settings homology $$\Gamma $$ -simplicial groups, particularly -equivariant and cohomology -groups. new direction called -homological algebra. The abstract kernel non-abelian extensions its relationship with obstruction theory, second are extended case -extensions We compute rational (co)homology groups ...

We show that Sullivan's model of rational differential forms on a simplicial set X may be interpreted as (kind of) 0 | 1 -dimensional supersymmetric quantum field theory over , and, consequence, concordance classes such theories represent the cohomology . introduce notion superalgebraic cartesian sets concept space which should roughly thought blend and supermanifolds, but valid an arbitrary ba...

The theory of abelian categories proved very useful, providing an ax-iomatic framework for homology and cohomology of modules over a ring (in particular, abelian groups) [5]. A similar framework has been lacking for non-abelian (co)homology, the subject of which includes the categories of groups and Lie algebras etc. The point of my thesis is that semi-abelian categories (in the sense of Janeli...

The counting function on the natural numbers defines a discrete Morse-Smale complex with a cohomology for which topological quantities like Morse indices, Betti numbers or counting functions for critical points of Morse index are explicitly given in number theoretical terms. The Euler characteristic of the Morse filtration is related to the Mertens function, the Poincaré-Hopf indices at critica...