On the loop space of a 2-category

Every small category $C$ has a classifying space $BC$ associated in a natural way. This construction can be extended to other contexts and set up a fruitful interaction between categorical structures and homotopy types. In this paper we study the classifying space $B_2C$ of a 2-category $C$ and prove that, under certain conditions, the loop space $\Omega_c B_2C$ can be recovered up to homotopy from the endomorphisms of a given object. We also present several subsidiary results that we develop to prove our main theorem.Comment: 21 pages, final version. Section 8 concerning the main theorem was rewritten. In particular, a partial converse for the main theorem was adde

Riemannian metrics on Lie groupoids

We introduce a notion of metric on a Lie groupoid, compatible with multiplication, and we study its properties. We show that many families of Lie groupoids admit such metrics, including the important class of proper Lie groupoids. The exponential map of these metrics allow us to establish a Linearization Theorem for Riemannian groupoids, obtaining both a simpler proof and a stronger version of the Weinstein-Zung Linearization Theorem for proper Lie groupoids. This new notion of metric has a simplicial nature which will be explored in future papers of this series.Comment: 29 pages; Final version accepted for publication in Journal f\"ur die reine und angewandte Mathematik (Crelle

Lie Groupoids

A Lie groupoid can be thought of as a generalization of a Lie group in which the multiplication is only defined for certain pairs of elements. From another perspective, Lie groupoids can be regarded as manifolds endowed with a type of action codifying internal and external symmetries. The vigorous development of their theory in the last few decades has been largely stimulated by their connections with such areas as Poisson geometry and non-commutative geometry, as well as several topics in mathematical physics, including classical mechanics, quantization and topological field theories. This article is an overview on Lie groupoids, including basic definitions, key examples and constructions, and topics such as actions and representations, local models, Morita equivalence and cohomology.Comment: Contribution to Encyclopedia of Mathematical Physic