What does it mean for a category to be endowed with a compatible differentiable structure? In this talk, we will discuss the interplay of a categorical structure with that of a smooth manifold, and show how to describe such categories infinitesimally, similarly as to how we construct the Lie algebra of a Lie group. We will generalise the notion of rank from linear algebra to morphisms of Lie categories, and introduce the notion of an extension of a Lie category to a groupoid. Examples of Lie categories arising in differential geometry and in physics will be highlighted.