Motivated by properties of higher tangent lifts geometric structures, we introduce concepts weighted structures for various objects on a manifold F equipped with homogeneity structure. The latter is smooth action the monoid $$(\mathbb {R},\cdot )$$ multiplicative reals. Vector bundles are particular cases and them call $$\mathrm{V\!B}$$ -structures. In case Lie algebroids groupoids, include -al...

We prove lower semicontinuity of the Galois groupoid a vector field depending on parameters. Apply to Painlevé equations, this result can be used compute their groupoids for general values

The aim of this paper is to introduce the notion of cat$^{bf {1}}-$groupoids which are the groupoid version of cat$^{bf {1}}-$groups and to prove the categorical equivalence between crossed modules over groupoids and cat$^{bf {1}}-$groupoids. In section 4 we introduce the notions of crossed squares over groupoids and of cat$^{bf {2}}-$groupoids, and then we show their categories are equivalent....

Fundamental representations of real simple Poisson Lie groups are Poisson actions with a suitable choice of the Poisson structure on the underlying (real) vector space. We study these (mostly quadratic) Poisson structures and corresponding phase spaces (symplectic groupoids).

Supergroupoids, double structures, and equivariant cohomology by Rajan Amit Mehta Doctor of Philosophy in Mathematics University of California, Berkeley Professor Alan Weinstein, Chair Q-groupoids and Q-algebroids are, respectively, supergroupoids and superalgebroids that are equipped with compatible homological vector fields. These new objects are closely related to the double structures of Ma...

Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of ‘degroupoidification’: a systematic process that turns groupoids and spans into vector spaces and linear operators. Then we present two applications of groupoidification. The first is to Fe...

In this paper we show that Cartan geometries can be studied via transitive Lie groupoids endowed with a special kind of vector-valued multiplicative 1-forms. This viewpoint leads us to more general notion, bundle, which encompasses both and G-structures.

Groupoidification is a form of categorification in which vector spaces are replaced by groupoids and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of ‘degroupoidification’: a systematic process that turns groupoids and spans into vector spaces and linear operators. Then we present three applications of groupoidification. The first is to F...

We interpret certain equivariant Kasparov groups as equivariant representable K-theory groups. We compute these groups via a classifying space and as K-theory groups of suitable σ-C-algebras. We also relate equivariant vector bundles to these σ-C-algebras and provide sufficient conditions for equivariant vector bundles to generate representable K-theory. Mostly we work in the generality of loca...