Comparison of categorical characteristic classes of transitive Lie algebroid with Chern-Weil homomorphism

Abstract

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 $\mathcal{TLA}_{\rg}$ from category of smooth manifolds to the transitive Lie algebroids. The functor $\mathcal{TLA}_{\rg}$ associates with each smooth manifold $M$ the set $\mathcal{TLA}_{\rg}(M)$ 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 $[M,\cB_{\rg}]$: $\mathcal{TLA}_{\rg}(M)\approx [M,\cB_{\rg}].$ It allows to describe characteristic classes of transitive Lie algebroids from 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. As a matter of fact we show that the Chern-Weil homomorphism does not cover all characteristic classes from categorical point of view.