We establish a functor Kan from local Kan simplicial manifolds to weak Kan simplicial manifolds. It gives a solution to the problem of extending local Lie groupoids to Lie 2-groupoids.

This is an introductory survey of the passage from groups to groupoids and their higher dimensional versions, with most emphasis on calculations with crossed modules and the construction and use of homotopy double groupoids.

We fill a lacuna in the literature by giving a version in dimension 1 of the Relative Hurewicz Theorem, and relate this to abelianisations of groupoids, covering spaces, covering morphisms of groupoids, and Crowell’s notion of derived modules.

This paper illustrates the themes of the title in terms of: van Kampen type theorems for the fundamental groupoid; holonomy and monodromy groupoids; and higher homotopy groupoids. Interaction with work of the writer is explored.

In this paper we study the differential systems on Leibniz algebroids. We introduce a class of almost metriplectic manifolds as a special case of Leibniz manifolds. Also, the notion of almost metriplectic algebroid is introduced. These types of algebroids are used in the presentation of associated differential systems. We give some interesting examples of differential systems on algebroids and ...

This paper illustrates the themes of the title in terms of: van Kampen type theorems for the fundamental groupoid; holonomy and monodromy groupoids; and higher homotopy groupoids. Interaction with work of the writer is explored. 1

A Quillen model structure on the category of orbifold groupoids is constructed. The fibrant objects for this category are the stacks groupoids and the homotopy category is the category of orbifolds.

As shown by Bursztyn et al. (2019), germs of tubular neighborhood embeddings for submanifolds N?M are in one–one correspondence with Euler-like vector fields near N. In many contexts, this reduces the proof ‘normal forms results’ geometric structures to construction an field compatible given structure. We illustrate principle a variety examples, including Morse–Bott lemma, Weinstein’s Lagrangia...

A theorem of Muhly–Renault–Williams states that if two locally compact groupoids with Haar system are Morita equivalent, then their associated convolution C-algebras are strongly Morita equivalent. We give a new proof of this theorem for Lie groupoids. Subsequently, we prove a counterpart of this theorem in Poisson geometry: If two Morita equivalent Lie groupoids are s-connected and s-simply co...

We discuss the integrability of Jacobi manifolds by contact groupoids, and then look at what the Jacobi point of view brings new into Poisson geometry. In particular, using contact groupoids, we prove a Kostant-type theorem on the prequantization of symplectic groupoids, which answers a question posed by Weinstein and Xu [20]. The methods used are those of CrainicFernandes on A-paths and monodr...