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.
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