We propose a notion of 1-homotopy for generalized maps. This notion generalizes those of
natural transformation and ordinary homotopy for functors. The 1-homotopy type of a Lie …

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We give an explicit description of the free path and loop groupoids in the Morita bicategory
of translation topological groupoids. We prove that the free path groupoid of a discrete group …

We introduce the notion of Lusternik-Schnirelmann category for differentiable stacks and
establish its relation with the groupoid Lusternik-Schnirelmann category for Lie groupoids …

We model systems as objects in a certain ambient Grothendieck site with additional
structure. We introduce generalized sheaves, called virtual manifolds. These sheaves …

The concept of a groupoid was first introduced in 1926 by H. Brandt in his fundamental
paper [7]. The idea behind it is a small category in which every arrow is invertible. This …