Elements of ∞-Category Theory

Emily Riehl · Dominic Verity

feb 2022 · Cambridge Studies in Advanced Mathematics Libro 194 · Cambridge University Press

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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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Dominic Verity is a professor of mathematics at Macquarie University in Sydney and is a director of the Centre of Australian Category Theory. While he is a leading proponent of 'Australian-style' higher category theory, he received his PhD from the University of Cambridge and migrated to Australia in the early 1990s. Over the years he has pursued a career that has spanned the academic and non-academic worlds, working at times as a computer programmer, quantitative analyst, and investment banker. He has also served as the Chair of the Academic Senate of Macquarie University, the principal academic governance and policy body.