In [3,8], we proposed to reformulate Statistical Mechanics and Gibbs mea- sures introducing an appropriate category. Among others, in [4–7], we gave a characterization of independent variables in terms of projective objects in this category and an easy-to-verify condition that characterizes such objects. In [9], we proposed an Entropy functional for the categorical version of statistical systems. In this article, we show how the characterization of extreme Gibbs measures (Theorem 7.7 [1]), one of the steps for proving a zero-one law for the extreme Gibbs measures, transfers in the categorical setting. In the classical theory of rigorous statistical mechanics, the tail σ-algebra generates the observables for which a ’generalized’ law of large numbers (zero-one law) holds. In this article, we give a candidate for such a σ-algebra in the categor- ical setting and show the associated extreme Gibbs measures decomposition. We will discuss in a follow-up paper the generalization of the zero-one law of specifications to A -specifications.