On the Quantization-Dequantization Correspondence for (co)Poisson Hopf Algebras

In this paper, we construct a functorial quantization of (co)Poisson Hopf algebras within a broad categorical framework. We further introduce categories naturally associated with (co)Poisson Hopf algebras, namely Drinfeld-Yetter modules. These categories provide a canonical setting in which we define explicit dequantization functors that are inverse to the quantization functors. Using this framework, we also establish functorial (de)quantization results for the corresponding module categories. Finally, we recover the classical results of Etingof and Kazhdan as special cases of our construction and discuss applications to deformation quantization \`a la Tamarkin.