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Higher Categories and Homotopical Algebra

Denis-Charles Cisinski
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Studies in Advanced Mathematics
[Reviewed by
Julie Bergner
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For many math graduate students, an algebraic topology course provides a first exposure to the language of categories and functors.  Classical invariants such as the fundamental group and homology groups are defined in terms of functors between topological spaces and (abelian) groups. Although category theory was developed with close ties to topology, and the branch of algebraic topology known as homotopy theory has always made heavy use of categorical machinery, in many ways the two fields went in different directions. Abstract homotopy theory, in which the structure of the category of topological spaces is axiomatized to be used in other settings, is, at its heart, interested in categories in which some of the functions are designated as weak equivalences. Category theory has developed other directions, including the study of higher categories, in which there are functions between functions, and so forth.
However, the two subjects have come together in a deep way in the development of what one might call higher homotopical categories. The idea is to consider something like a category, but whose morphisms from one object to another form a topological space, rather than simply a set, and for which composition might only be defined up to homotopy. Such a structure turns out to have several other interpretations: a certain kind of higher category for which various higher morphisms are invertible (often called an \( (\infty,1) \)-category or simply \( \infty \)-category), or even as a category with weak equivalences in the sense of abstract homotopy theory.
Furthermore, these categories up to homotopy can be modeled in different ways, in the sense that different kinds of mathematical objects sensibly encode such a structure, yet are all known to be equivalent to one another. A good deal of attention has focused on the model of quasi-categories, and many classical constructions in category theory have been translated into this more homotopical framework. For example, one can talk about limits and colimits in a quasi-category, or what it means to have an adjoint pair of functors between two quasi-categories. 
This subject has received much attention for over a decade now, but one obstacle has been its accessibility.  In particular, Lurie's Higher Topos Theory provides a comprehensive treatment of the subject, but can be intimidating for beginners. The purpose of Cisinski's book is to provide a gentler introduction which prepares a reader to jump into Lurie's book or even into some of the current literature. For example, Cisinski begins with material on simplicial sets and model categories that Lurie takes for granted, and he focuses on essential categorical constructions rather than the broad theory. As such, this book is a valuable contribution to the subject and will be extremely helpful for graduate students and researchers in related fields.
That said, this book is still not elementary. In the preface, the author claims that there are no formal prerequisites but that "a solid background in algebraic topology or category theory would certainly help the reader."  I would go further and say that it would be very difficult to read this book without an understanding of basic category theory, along the lines of at least the first few chapters of Riehl's Category Theory in Context, and that many of the methods used would likely feel unmotivated for a reader without some knowledge of classical homotopy theory, say from Chapter 4 of Hatcher's Algebraic Topology
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 is an associate professor at the University of Virginia, and her research is in homotopy theory.