Hochschild cohomology and “smoothness” in monoidal categories

We introduce and investigate the properties of Hochschild cohomology of algebras in an abelian monoidal category MM. We show that the second Hochschild cohomology group of an algebra in MM classifies extensions of AA up to an equivalence. We characterize algebras of Hochschild dimension 0 (separable algebras), and of Hochschild dimension ≤1≤1 (formally smooth algebras). Several particular cases and applications are included in the last section of the paper.