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 "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，大量综述强调，关键XAI概念——尤其是仍缺乏精确定义的"解释"一词——尚未得到严谨的数学形式化。为填补这一空白，本文利用范畴论这一基础扎实的形式化工具，首次给出了关键XAI概念及过程的数学严格定义。我们证明，该范畴论框架能够：（i）对现有学习方案与架构进行建模，（ii）形式化定义"解释"术语，（iii）为XAI分类学建立理论基础，以及（iv）分析解释方法中常被忽视的方面。因此，该范畴论框架促进了人工智能技术的伦理与安全部署，标志着可解释人工智能向严谨理论基础迈出了重要一步。