In this paper we study quotients of Lie algebroids and groupoids endowed with compatible differential forms. We identify theoretic conditions under which such forms become basic characterize the induced on quotients. apply these results to describe generalized quotient reduction processes for (twisted) Poisson Dirac structures, as well their integration by (twisted, pre-)symplectic groupoids. p...

Groupoids graded by the groupoid of bijections between finite sets admit generating functions which encode cardinalities their components. As suggested in work Baez and Dolan, we use analytic continuation such to define a complex-valued cardinality for groupoids whose usual diverges. The complex nature invariant is shown reflect recursion structure refer as ‘nested equivalence’.

For almost finite groupoids, we study how their homology groups reflect dynamical properties of their topological full groups. It is shown that two clopen subsets of the unit space has the same class in H0 if and only if there exists an element in the topological full group which maps one to the other. It is also shown that a natural homomorphism, called the index map, from the topological full...

This paper continues the study of generalized amalgamation properties begun in [1], [2], [3], [5], and [6]. Part of the paper provides a finer analysis of the groupoids that arise from failure of 3-uniqueness in a stable theory. We show that such groupoids must be abelian and we link the binding group of the groupoids to a certain automorphism group of the monster model, showing that the group ...

In this note a functorial approach to the integration problem of an LAgroupoid to a double Lie groupoid is discussed. To do that, we study the notions of fibered products in the categories of Lie groupoids and Lie algebroids, giving necessary and sufficient conditions for the existence of such. In particular, it turns out, that the fibered product of Lie algebroids along integrable morphisms is...

In this note a functorial approach to the integration problem of an LAgroupoid to a double Lie groupoid is discussed. To do that, we study the notions of fibred products in the categories of Lie groupoids and Lie algebroids, giving necessary and sufficient conditions for the existence of such. In particular, it turns out, that the fibred product of Lie algebroids along integrable morphisms is a...