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Characteristic classes of transitive Lie algebroids. Categorical point of view

The talk was done at the International Conference "Analysis, Topology and Applications", Harbin, China, 23.08.2011. Transitive Lie algebroids have specific properties that allow to look at the transitive Lie algebroid as an element of the object of a homotopy functor. Roughly speaking each transitive Lie algebroids can be described as a vector bundle over the tangent bundle of the manifold which is endowed with additional structures. Therefore transitive Lie algebroids admits a construction of inverse image generated by a smooth mapping of smooth manifolds. The intention of my talk is to use a homotopy classification of transitive Lie algebroids due to K.Mackenzie (2005) and on this basis to construct a classifying space. The realization of the intention allows to describe characteristic classes of transitive Lie algebroids form the point of view a natural transformation of functors similar to the classical abstract characteristic classes for vector bundles and to compare them with that derived from the Chern-Weil homomorphism by J.Kubarski (1991).