Preface; 1. Prelude; 2. Basic homotopical algebra; 3. The homotopy theory of ∞-categories; 4. Presheaves: externally; 5. Presheaves: internally; 6. Adjoints, limits and Kan extensions; 7. Homotopical algebra; References; Notation; Index.
At last, a friendly introduction to modern homotopy theory after Joyal and Lurie, reaching advanced tools and starting from scratch.
Denis-Charles Cisinski is Professor of Mathematics at the Universität Regensburg, Germany. His research focuses on homotopical algebra, category theory, K-theory and the cohomology of schemes. He is also the author of a monograph entitled Les préfaisceaux comme modèles des types d'homotopie (2007).
'Category theory is concerned with the organisation and
construction of general mathematical structures, while homotopy
theory is devoted to the study of abstract shapes associated to
geometric forms. This book is a window into the new field of
mathematics emerging from the convergence of these two branches of
mathematics … It was conjectured a few decades ago that category
theory has a natural extension to quasi-categories (also called
infinity-category), a notion introduced by Michael Boardman and
Reiner Vogt in the early nineteen-seventies … This book widens and
deepens the extension with the addition of a new theory of
presheaves inspired by type theory and a new theory of
localisation; it proposes an extension of homotopical algebra to
quasi-categories, offers new applications, and brings important
simplification to earlier works. It is an excellent introduction to
the subject and may be used for an advanced course.' André Joyal,
Université du Québec à Montréal, Canada
'Denis-Charles Cisinski offers a masterful introduction to the
world of infinity-categories, illustrating the necessary intuition
all throughout. A complete and clear exposition of the foundations
leads naturally to a full course teaching us how to handle all
aspects of homotopical algebra within the theory.' Carlos Simpson,
French National Center for Scientific Research, Université Côte
d'Azur, France
'In recent years the methods of homotopy theory have seen
increasingly wide applications in mathematics, and the framework of
abstract homotopy theory has been found to be an important lens
through which to view many mathematical structures. This book
offers a single, self-contained place to learn about the extensive
modern facets of abstract homotopy theory. Readers will appreciate
Cisinski's thoughtful choice of details and his carefully
articulated philosophical point of view. This is an excellent
resource for mathematicians experiencing first contact with the
subject and for more seasoned researchers in the area.' Michael
Hopkins, Harvard University
'As someone doing research and advising graduate students in a
closely-related area, I am happy to see a book like this in the
literature. It will help readers to learn this subject, and to gain
a deep understanding of the foundational ideas.' Julie Bergner, MAA
Reviews
|
Ask a Question About this Product More... |
|
