Elements of ∞-Category Theory

Inbunden, Engelska, 2022

Del 194 i serien Cambridge Studies in Advanced Mathematics

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Beskrivning

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.

Produktinformation

Utgivningsdatum:2022-02-10
Mått:160 x 232 x 43 mm
Vikt:1 180 g
Format:Inbunden
Språk:Engelska
Serie:Cambridge Studies in Advanced Mathematics
Antal sidor:770
Förlag:Cambridge University Press
ISBN:9781108837989

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Mer om författaren

Emily Riehl is an associate professor of mathematics at Johns Hopkins University. She received her PhD from the University of Chicago and was a Benjamin Peirce and NSF postdoctoral fellow at Harvard University. She is the author of Categorical Homotopy Theory (Cambridge, 2014) and Category Theory in Context (2016), and a co-author of Fat Chance: Probability from 0 to 1 (Cambridge, 2019). She and her present co-author have published ten articles over the course of the past decade that develop the new mathematics appearing in this book. 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.

Recensioner i media

'The book of Riehl and Verity is altogether a pedagogical introduction, a unified presentation and a foundation of higher category theory. The theory of ∞-cosmoi is an elegant way of organising and developing the subject. The extension of category theory to ∞-categories is by itself a miracle, vigorously presented in the book.' André Joyal, Université du Québec à Montréal

Innehållsförteckning

Part I. Basic ∞-Category Theory: 1. ∞-Cosmoi and their homotopy 2-categories; 2. Adjunctions, limits, and colimits I; 3. Comma ∞-categories; 4. Adjunctions, limits, and colimits II; 5. Fibrations and Yoneda's lemma; 6. Exotic ∞-cosmoi; Part II. The Calculus of Modules: 7. Two-sided fibrations and modules; 8. The calculus of modules; 9. Formal category theory in a virtual equipment; Part III. Model Independence: 10. Change-of-model functors; 11. Model independence; 12. Applications of model independence.