On deformation theory of associative algebras in monoidal categories

We extend the classical concept of deformation of an associative algebra, as introduced by Gerstenhaber, by using monoidal linear categories and cocommutative coalgebras as foundational tools. To achieve this goal, we associate to each cocommutative coalgebra $C$ and each linear monoidal category $\M$, a $\Bbbk$-linear monoidal category $\M_{C}$. This construction is functorial: any coalgebra morphism $\iota:C\to\widetilde{C}$ induces a strict monoidal functor $\iota^{*}:\M_{\widetilde{C}}\to\M_{C}$. An $\iota$-deformation of an algebra $(A,m)$ is defined as an algebra $(A,\widetilde{m})$ in the fiber of $\iota^{*}$ over $(A,m)$. Within this framework, the deformations of $(A,m)$ are organized into a presheaf, which is shown to be representable. In other words, there exists a universal deformation satisfying a specific universal property. It is well established that classical deformation theory is deeply connected to Hochschild cohomology. We identify and analyze the cohomology theory that governs $\iota$-deformations in the second part of the paper. Additionally, several particular cases and applications of these results are examined in detail.