13,753 research outputs found
From double Lie groupoids to local Lie 2-groupoids
We apply the bar construction to the nerve of a double Lie groupoid to obtain
a local Lie 2-groupoid. As an application, we recover Haefliger's fundamental
groupoid from the fundamental double groupoid of a Lie groupoid. In the case of
a symplectic double groupoid, we study the induced closed 2-form on the
associated local Lie 2-groupoid, which leads us to propose a definition of a
symplectic 2-groupoid.Comment: 23 pages, a few minor changes, including a correction to Lemma 6.
Differentiable stratified groupoids and a de Rham theorem for inertia spaces
We introduce the notions of a differentiable groupoid and a differentiable
stratified groupoid, generalizations of Lie groupoids in which the spaces of
objects and arrows have the structures of differentiable spaces, respectively
differentiable stratified spaces, compatible with the groupoid structure. After
studying basic properties of these groupoids including Morita equivalence, we
prove a de Rham theorem for locally contractible differentiable stratified
groupoids. We then focus on the study of the inertia groupoid associated to a
proper Lie groupoid. We show that the loop and the inertia space of a proper
Lie groupoid can be endowed with a natural Whitney B stratification, which we
call the orbit Cartan type stratification. Endowed with this stratification,
the inertia groupoid of a proper Lie groupoid becomes a locally contractible
differentiable stratified groupoid
The H-Covariant Strong Picard Groupoid
The notion of H-covariant strong Morita equivalence is introduced for
*-algebras over C = R(i) with an ordered ring R which are equipped with a
*-action of a Hopf *-algebra H. This defines a corresponding H-covariant strong
Picard groupoid which encodes the entire Morita theory. Dropping the positivity
conditions one obtains H-covariant *-Morita equivalence with its H-covariant
*-Picard groupoid. We discuss various groupoid morphisms between the
corresponding notions of the Picard groupoids. Moreover, we realize several
Morita invariants in this context as arising from actions of the H-covariant
strong Picard groupoid. Crossed products and their Morita theory are
investigated using a groupoid morphism from the H-covariant strong Picard
groupoid into the strong Picard groupoid of the crossed products.Comment: LaTeX 2e, 50 pages. Revised version with additional examples and
references. To appear in JPA
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