m at h . D G ] 1 2 Fe b 20 09 On Morphic Actions and Integrability of LA - Groupoids

ar X iv : 0 90 2 . 29 25 v 1 [ m at h . G T ] 1 7 Fe b 20 09 COVERINGS AND ACTIONS OF STRUCTURED LIE GROUPOIDS

In this work we deal with coverings and actions of Lie groupgroupoids being a sort of the structured Lie groupoids. Firstly, we define an action of a Lie group-groupoid on some Lie group and the smooth coverings of Lie group-groupoids. Later, we show the equivalence of the category of smooth actions of Lie group-groupoids on Lie groups and the category of smooth coverings of Lie group-groupoids.

49 9 v 3 [ m at h . D G ] 2 6 Fe b 20 06 Three themes in the work of Charles Ehresmann : Local - to - global ; Groupoids ; Higher dimensions . ∗

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.

ar X iv : 0 90 2 . 19 17 v 1 [ m at h . D S ] 1 1 Fe b 20 09 AVERAGES ON ANNULI OF EULIDEAN SPACE

We study the range of validity of differentiation theorems and ergodic theorems for R actions, for averages on “thick spheres” of Euclidean space.

ar X iv : 0 90 2 . 32 39 v 1 [ m at h . D G ] 1 8 Fe b 20 09 Gauge theory in higher dimensions , II

A History of Selected Topics in Categorical Algebra I: From Galois Theory to Abstract Commutators and Internal Groupoids

This paper is a chronological survey, with no proofs, of a direction in categorical algebra, which is based on categorical Galois theory and involves generalized central extensions, commutators, and internal groupoids in Barr exact Mal’tsev and more general categories. Galois theory proposes a notion of central extension, and motivates the study of internal groupoids, which is then used as an a...