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.
The intuitions of higher dimensional algebra for the study of structured space
