Elements of ?-Category Theory

The language of ?-categories provides an insightful new way of expressing many results in higher-dimensional mathematics but can be challenging for the uninitiated. To explain what exactly an ?-category is requires various technical models, raising the question of how they might be compared. To overcome this, a model-independent approach is desired, so that theorems proven with any model would apply to them all. This text develops the theory of ?-categories from first principles in a model-independent fashion using the axiomatic framework of an ?-cosmos, the universe in which ?-categories live as objects. An ?-cosmos is a fertile setting for the formal category theory of ?-categories, and in this way the foundational proofs in ?-category theory closely resemble the classical foundations of ordinary category theory. Equipped with exercises and appendices with background material, this first introduction is meant for students and researchers who have a strong foundation in classical 1-category theory.