Explainable AI (XAI) aims to address the human need for safe and reliable AI systems. However, numerous surveys emphasize the absence of a sound mathematical formalization of key XAI notions -- remarkably including the term ``\textit{explanation}'' which still lacks a precise definition. To bridge this gap, this paper presents the first mathematically rigorous definitions of key XAI notions and processes, using the well-funded formalism of Category theory. We show that our categorical framework allows to: (i) model existing learning schemes and architectures, (ii) formally define the term ``explanation'', (iii) establish a theoretical basis for XAI taxonomies, and (iv) analyze commonly overlooked aspects of explaining methods. As a consequence, our categorical framework promotes the ethical and secure deployment of AI technologies as it represents a significant step towards a sound theoretical foundation of explainable AI.

翻译：可解释人工智能（XAI）旨在满足人类对安全可靠人工智能系统的需求。然而，众多综述强调，关键XAI概念缺乏严谨的数学形式化——值得注意的是，"解释"一词至今仍无精确的定义。为弥合这一差距，本文利用范畴理论这一基础深厚的形式化体系，首次提出了XAI核心概念与过程的数学严谨定义。我们证明，该分类框架可：(i) 建模现有学习方案及架构，(ii) 正式定义"解释"一词，(iii) 为XAI分类法建立理论基础，以及(iv) 分析解释方法中常被忽视的方面。由此，我们的分类框架推动了AI技术安全可靠的部署，因为它为可解释人工智能奠定了坚实的理论基础，迈出了关键一步。