Deformations of Lie brackets: cohomological aspects

We introduce a new cohomology for Lie algebroids, and prove that it provides a differential graded Lie algebra which ``controls'' deformations of the structure bracket of the algebroid. We also have a closer look at various special cases such as Lie algebras, Poisson manifolds, foliations, Lie algebra actions on manifolds.Comment: 17 pages, Revised version: small corrections, more references adde

On the developability of subalgebroids

In this paper, the Almeida-Molino obstruction to developability of transversely complete foliations is extended to Lie groupoids

On the integrability of subalgebroids

Let G be a Lie groupoid with Lie algebroid g. It is known that, unlike in the case of Lie groups, not every subalgebroid of g can be integrated by a subgroupoid of G. In this paper we study conditions on the invariant foliation defined by a given subalgebroid under which such an integration is possible. We also consider the problem of integrability by closed subgroupoids, and we give conditions under which the closure of a subgroupoid is again a subgroupoid

Representing topoi by topological groupoids

It is shown that every topos with enough points is equivalent to the classifying topos of a topological groupoid

General static spherically symmetric solutions in Horava gravity

We derive general static spherically symmetric solutions in the Horava theory of gravity with nonzero shift field. These represent "hedgehog" versions of black holes with radial "hair" arising from the shift field. For the case of the standard de Witt kinetic term (lambda =1) there is an infinity of solutions that exhibit a deformed version of reparametrization invariance away from the general relativistic limit. Special solutions also arise in the anisotropic conformal point lambda = 1/3.Comment: References adde

Relative compactness conditions for topos

In this paper a systematic study is made of various notions of proper map in the context of toposes Modulo some separation conditions a proper map Y X of spaces is generally understood to be a continuous function which preserves compactness of subspaces under inverse image and which therefore in particular has compact bers In this spirit a rst denition of proper map between toposes was put forward by Johnstone in There a map of toposes fF E was called proper if fF is a compact lattice object in the topos E This is probably the most direct way of expressing that F is compact when viewed as a topos over the base E In fact Johnstone used the term perfect rather than proper and developed the theory mostly in the context of localic maps between toposes se

Minimal fibrations of dendroidal sets

We prove the existence of minimal models for fibrations between dendroidal sets in the model structure for ∞–operads, as well as in the covariant model structure for algebras and in the stable one for connective spectra. We also explain how our arguments can be used to extend the results of Cisinski (2014) and give the existence of minimal fibrations in model categories of presheaves over generalized Reedy categories of a rather common type. Besides some applications to the theory of algebras over ∞–operads, we also prove a gluing result for parametrized connective spectra (or Γ–spaces)

On the equivalence between Lurie's model and the dendroidal model for infinity-operads

© 2016 Elsevier Inc.We compare two approaches to the homotopy theory of ∞-operads. One of them, the theory of dendroidal sets, is based on an extension of the theory of simplicial sets and ∞-categories which replaces simplices by trees. The other is based on a certain homotopy theory of marked simplicial sets over the nerve of Segal's category Γ. In this paper we prove that for operads without constants these two theories are equivalent, in the precise sense of the existence of a zig-zag of Quillen equivalences between the respective model categories

Algebroid Yang-Mills Theories

A framework for constructing new kinds of gauge theories is suggested. Essentially it consists in replacing Lie algebras by Lie or Courant algebroids. Besides presenting novel topological theories defined in arbitrary spacetime dimensions, we show that equipping Lie algebroids E with a fiber metric having sufficiently many E-Killing vectors leads to an astonishingly mild deformation of ordinary Yang-Mills theories: Additional fields turn out to carry no propagating modes. Instead they serve as moduli parameters gluing together in part different Yang-Mills theories. This leads to a symmetry enhancement at critical points of these fields, as is also typical for String effective field theories.Comment: 4 pages; v3: Minor rewording of v1, version to appear in Phys. Rev. Let