# Categorical Perspective on Quantization of Poisson Algebra

## TL;DR

This paper introduces a categorical framework for quantization of Poisson algebras, unifying various quantization methods and analyzing their categorical relationships.

## Contribution

It generalizes quantization as a categorical concept, defining and comparing multiple quantization categories within a unified framework.

## Key findings

- Strict deformation, prequantization, and matrix regularization categories are equivalent under certain conditions.
- Poisson enveloping algebra category is not equivalent to the other quantization categories.
- Provides a categorical perspective that unifies and distinguishes different quantization approaches.

## Abstract

We propose a generalization of quantization as a categorical way. For a fixed Poisson algebra quantization categories are defined as subcategories of R-module category with the structure of classical limits. We construct the generalized quantization categories including matrix regularization, strict deformation quantization, prequantization, and Poisson enveloping algebra, respectively. It is shown that the categories of strict deformation quantization, prequantization, and matrix regularization with some conditions are categorical equivalence. On the other hand, the categories of Poisson enveloping algebra is not equivalent to the other categories.