Let A be an associative and unital algebra over a commutative ring K, such that A is K-projective. The Hochschild cohomology ring HH*(A) of A is, as a graded algebra, isomorphic to the Ext-algebra of A in the category of A-bimodules. In 1963, M. Gerstenhaber established a graded Lie bracket on HH*(A) of degree -1 which he described in terms of the so called bar resolution. While the multiplication admits an intrinsic description (Yoneda product), this Lie bracket has resisted such an interpretation at first. A serious attempt to find a categorical description of the Lie bracket was given by S. Schwede in 1998. By using the monoidal structure on the category of A-bimodules, he was able to formulate Gerstenhaber's bracket in terms of self-extensions of A. The present thesis extends Schwede's construction to exact and monoidal categories. Therefore we will establish an explicit description of an isomorphism by A. Neeman and V. Retakh, linking Ext-groups with fundamental groups of categories of extensions.

Our main result shows that our construction behaves well with respect to structure preserving functors between strong exact monoidal categories. We use our main result to conclude, that the graded Lie bracket on HH*(A) is an invariant under Morita equivalence. For quasitriangular Hopf algebras over K, we further determine a significant part of the Lie bracket's kernel.