We develop a categorical framework for reasoning about abstract properties of differentiation, based on the theory of fibrations. Our work encompasses the first-order fragments of several existing categorical structures for differentiation, including cartesian differential categories, generalised cartesian differential categories, tangent categories, as well as the versions of these categories axiomatising reverse derivatives. We explain uniformly and concisely the requirements expressed by these structures, using sections of suitable fibrations as unifying concept. Our perspective sheds light on their similarities and differences, as well as simplifying certain constructions from the literature.

翻译：我们基于纤维化理论发展了一个用于推理微分抽象性质的范畴论框架。我们的工作涵盖了几种现有微分范畴结构的一阶片段，包括笛卡尔微分范畴、广义笛卡尔微分范畴、切范畴，以及这些范畴的公理化反向导数版本。通过使用适当纤维化的截面作为统一概念，我们以统一且简洁的方式阐释了这些结构所表达的要求。我们的视角揭示了这些结构的相似性与差异性，同时简化了文献中的某些构造。