In algebraic geometry, the problem of classifying geometric objects or isomorphism classes of such objects is rather hard, since one wants to give some geometric structure to the set of such isomorphism classes. This is natural, since one wants the points of this classifying or moduli space to parametrize the isomorphism classes.
Moduli spaces encode all data related to the objects and the various isomorphisms between them. In general, these moduli spaces are not schemes or algebraic spaces, but many in interesting cases they are “algebraic stacks.” Grothendieck’s categorical approach to the construction of moduli spaces is to encode all data related to a given object (objects isomorphic to the given one and the various isomorphisms between all these objects) in terms of fibered categories over the given object. This approach, after several abstraction steps, provides a framework for descent theory, where the problem of the geometric existence of a geometric structure on a quotient object is rather difficult.
Classifying spaces come also from algebraic topology, where for a topological group G its classifying space BG is a topological space that represents the functor that assigns to any topological space X the set of isomorphism classes of principal G-bundles on X. When G is discrete group, its classifying space BG has fundamental group isomorphic to G and its higher homotopy groups are trivial. There are many applications of classifying spaces: characteristic classes, cobordism, group cohomology, topological K-theory and other generalized cohomology theories.
There are several constructions of classifying spaces, but one of the earlier ones gives a description of classifying spaces as simplicial sets. This is quite interesting, since the category of simplicial sets has a purely algebraic description, making it suitable for applications in algebraic geometry. There is a categorical definition of simplicial objects as contravariant functors with values in any category and with domain the category of simplices, whose objects are totally ordered finite sets and whose morphisms are order-preserving functions between them. In this general situation, simplicial sets are just contravariant functors from the category of simplices to the category of sets. In the 1960s, D. Quillen proved that the category of simplicial sets is a closed model category.
It is a remarkable fact that the two tracks sketched above, which were developing somehow independently, were to merge after some further categorical abstraction. Moduli spaces can be constructed using the nerve of the underlying category and this construction keeps track of all the combinatorial data coming from the isomorphic objects to a given object and the isomorphisms between them. But the nerve of a category is a simplicial object, and it is a generalization of the classifying space of a (discrete) group. Furthermore, Quillen’s construction of the higher algebraic K-theory groups of an exact category uses the nerve of a category associated to the given exact category.
In this context the main example is the exact category of locally free coherent sheaves on a fixed scheme. Moreover, this construction generalizes his earlier construction of the higher algebraic K-groups of a ring using a space associated to classifying space of the general linear group over the given ring. Wagoner proved that this space is an infinite loop space and interest in delooping spaces boomed for a while; so-called “delooping machines” were constructed in several guises.
It is in this framework that we can see the pioneering work of D. Quillen on model categories, P. May on operads to study delooping machines, and G. Segal on simplicial delooping machines, which introduced the notion of a category weakly enriched over the category of simplicial sets, building on Grothendieck’s characterization of the nerve of a category. And it was again Grothendieck who would set the general frame to look at these problems. In Pursuing Stacks, a now famous manuscript that started as a collection of letters to several colleagues, Grothendieck raised the problem of setting the correct framework to create a homotopical algebra that would recover, amongst other things, the higher homotopy groups of algebraic topology but in such a general way to be useful in algebraic geometry. In particular, Grothendieck singles out the problem of defining the notion of a weakly associative category and shows that strict associativity of the composition of morphisms would not give the correct definition. This brings us to the main purpose of the book under review: The homotopy theory of higher categories.
Higher dimensional categories are generalizations of the notion of category. A 0-category is a set and an (n+1)-category is a category enriched over an n-category, where enriching means that the Homs are objects of an n-category. One has two options for the usual properties of composition of morphisms in n-categories. For the associativity and identity conditions one may require strict equality or a weak form of it, i.e., that these conditions are satisfied up to isomorphism in the next level. The subject has seen a spectacular growth in the last decade and there are now several approaches to construct weakly associative higher categories, reflecting the rich roots of the subject: algebraic topology and algebraic geometry.
The book under review starts with an overview of the various approaches to the construction of weakly associative categories. Part I starts with a review of some of the motivation for introducing higher categories, gives some historical remarks and points to recent developments. Then, in chapter 2, we have a detailed exposition of strict categories, strict groupoids (categories where all morphisms are invertible), realization functors and Grothendieck’s argument that strict groupoids are not sufficient to model homotopy types.
The general theory of n-categories is introduced in Chapter 3 ending with some comments on ∞-categories. Chapter four gives a succinct overview of the operadic approaches towards higher categories, recalling first P. May’s delooping machines and then the Baez-Nolan approach. A good source for a more detailed discussion is T. Leinster’s book Higher Operads, Higher Categories (LMS-Cambridge, 2004). Chapter five is devoted to the simplicial approach to the theory of higher categories. It introduces Segal’s simplicial delooping machine, Segal categories and their realizations. Next, following Leinster, Bergner and Bacard, the author introduces a method of weak enrichment over a monoidal category where the important case is when the monoid structure corresponds to the Cartesian product. Chapter six gives an outline of the basic properties of Segal categories and Tamsamani and Pellissier’s approach, which is developed in more detail in the rest of the book.
Part II of the book sets up the categorical framework that will be used later on starting with Quillen’s theory of model categories, Cartesian model categories and enriched categories, cell complexes and Bousfield localizations. The remaining three parts are the main parts of the book. Part III is devoted to the process, analogous to the classical process of describing an algebraic object by generators and relations, of passing from a precategory to an associated weakly enriched category by requiring the Segal condition to hold, with detailed discussions of some important instances, in particular categories weakly enriched over the model category of simplicial sets of Kan and Quillen. Part IV is devoted to the construction of the Cartesian model category and Part V applies the formalisms developed previously to obtain various versions of model categories for n-categories and groupoids. The last chapters include some variants of localizations techniques: inversion of morphisms, limits and adjunctions.
This is a well-organized monograph on a topic that is central for both algebraic geometry and algebraic topology, which after several decades of lying still is now attracting a well-deserved interest of researchers and students. There are now a few more monographs with emphases on alternative approaches, among them J. Lurie’s Higher Topos Theory (Princeton, 2009) and the collection of surveys in Towards Higher Categories edited by J. C. Baez and P. May (Springer, 2009) and the already mentioned T. Leinster’s book Higher Operads, Higher Categories (LMS-Cambridge, 2004).
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is firstname.lastname@example.org.
Part I. Higher Categories: 1. History and motivation
2. Strict n-categories
3. Fundamental elements of n-categories
4. The need for weak composition
5. Simplicial approaches
6. Operadic approaches
7. Weak enrichment over a Cartesian model category: an introduction
Part II. Categorical Preliminaries: 8. Some category theory
9. Model categories
10. Cartesian model categories
11. Direct left Bousfield localization
Part III. Generators and Relations: 12. Precategories
13. Algebraic theories in model categories
14. Weak equivalences
16. Calculus of generators and relations
17. Generators and relations for Segal categories
Part IV. The Model Structure: 18. Sequentially free precategories
21. The model category of M-enriched precategories
22. Iterated higher categories
Part V. Higher Category Theory: 23. Higher categorical techniques
24. Limits of weak enriched categories