ИСТИНА

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. Due to to K.Mackenzie (2005) the construction can be managed as a homotopy functor $TLA_{\rg}$ from category of smooth manifolds to the transitive Lie algebroids. The functor $TLA_{\rg}$ associates with each smooth manifold $M$ the set $TLA_{\rg}(M)$ of all transitive algebroids with fixed structural finite dimensional Lie algebra $\rg$. Hence one can construct a classifying space $\cB_{\rg}$ such that the family of all transitive Lie algebroids with fixed Lie algebra $\rg$ over the manifold $M$ has one-to-one correspondence with the family of homotopy classes of continuous maps $[M,\cB_{\rg}]$: $ \cA(M)\approx [M,\cB_{\rg}]. $ In spite of the evident categorical point of view we faced the challenge of geometrical construction of the classifying space, in particular generalization of the Eilenberg-MacLane spaces, realization of the cohomological obstructions for equivariant mapping and others.