Related papers: Homological Algebra for Diffeological Vector Space…
Thin homotopies, introduced by Caetano-Picken, serve to axiomatize the holonomy of connections on principal bundles. This approach has been generalized to higher non-abelian bundles with connection through transport functors and higher…
We construct a diffeomorphism invariant (Colombeau-type) differential algebra canonically containing the space of distributions in the sense of L. Schwartz. Employing differential calculus in infinite dimensional (convenient) vector spaces,…
We introduce a cohomology theory that classifies differential objects that arise from Picard-Vessiot theory, using the differential Hopf-Galois descent. To do this, we provide an explicit description of Picard-Vessiot theory in terms of…
A projection space is a collection of spaces interrelated by the combinatorics of projection onto tensor factors in a symmetric monoidal background category. Examples include classical configuration spaces, orbit configuration spaces, the…
We introduce full diffeomorphism-invariant Colombeau algebras with added $\varepsilone$-dependence in the basic space. This unites the full and special settings of the theory into one single framework. Using locality conditions we find the…
This is a paper in a series that studies smooth relative Lie algebra homologies and cohomologies based on the theory of formal manifolds and formal Lie groups. In a previous paper, we introduce the notion of formal manifolds and develop the…
Starting from the Colombeau's full generalized functions, the sharp topologies and the notion of generalized points, we introduce a new kind differential calculus (for functions between totally disconnected spaces). We study generalized…
It is often noted that many of the basic concepts of differential geometry, such as the definition of connection, are purely algebraic in nature. Here, we review and extend existing work on fully algebraic formulations of differential…
We provide base change theorems, projection formulae and Verdier duality for both cohomology and homology in the context of finite topological spaces
The notion of a $F$-manifold algebras is an algebraic description of a $F$-manifold. In this paper, we introduce the notion of Hom-$F$-manifold algebras which is generalisation of $F$-manifold algebras and Hom-Poisson algebras. We develop…
This is the first paper in a series that studies smooth relative Lie algebra homologies and cohomologies based on the theory of formal manifolds and formal Lie groups. In this paper, we lay the foundations for this study by introducing the…
In the present paper we determine for each parallelizable smooth compact manifold $M$ the cohomology spaces $H^2(V_M,\bar\Omega^p_M)$ of the Lie algebra $V_M$ of smooth vector fields on $M$ with values in the module $\bar\Omega^p_M =…
We first introduce global arithmetic cohomology groups for quasi-coherent sheaves on arithmetic varieties, adopting an adelic approach. Then, we establish fundamental properties, such as topological duality and inductive long exact…
This contribution is the first in a series of three: it reports on the construction of (a fine sheaf of) diffeomorphism invariant Colombeau algebras on open sets of Eucildean space, which completes earlier approaches. Part II and III will…
D. Davis introduced projective product spaces in 2010 as a generalization of real projective spaces and discussed some of their topological properties. On the other hand, Dold manifolds were introduced by A. Dold in 1956 to study the…
Among plenty of applications, low-dimensional homogeneous spaces appear in cosmological models as both, classical factor spaces of multidimensional geometry and minisuperspaces in canonical quantization. Here a new tool to restrict their…
As an algebraic study of differential equations, differential algebras have been studied for a century and and become an important area of mathematics. In recent years the area has been expended to the noncommutative associative and Lie…
We decategorify the Heisenberg 2-category of Gyenge-Koppensteiner-Logvinenko using Hochschild homology. We use this to generalise the Heisenberg algebra action of Grojnowski and Nakajima to all smooth and proper noncommutative varieties in…
We introduce the \verb|Macaulay2| package \verb|RepHomology| for the computations of representation homology of certain spaces. The main methods implement computing the representation homology of surfaces (with group coefficients, and…
Over a non-closed field, it is a common strategy to use separable algebras as invariants to distinguish algebraic and geometric objects. The most famous example is the deep connection between Severi-Brauer varieties and central simple…