The Category Game is a multi-agent model that accounts for the emergence of shared categorization patterns in a population of interacting individuals. In the framework of the model, linguistic categories appear as long lived consensus states that are constantly reshaped and re-negotiated by the communicating individuals. It is therefore crucial to investigate the long time behavior to gain a clear understanding of the dynamics. However, it turns out that the evolution of the emerging category system is so slow, already for small populations, that such an analysis has remained so far impossible. Here, we introduce a fast no-rejection algorithm for the Category Game that disentangles the physical simulation time from the CPU time, thus opening the way for thorough analysis of the model. We verify that the new algorithm is equivalent to the old one in terms of the emerging phenomenology and we quantify the CPU performances of the two algorithms, pointing out the neat advantages offered by the no-rejection one. This technical advance has already opened the way to new investigations of the model, thus helping to shed light on the fundamental issue of categorization.