Simplicial Methods for Higher Categories

Category theory today is a very important framework and language for understanding mathematics. It originated in the early twentieth century in the work of Eilenberg and MacLane as a way to understand topological invariants as part of the development of algebraic topology. It quickly became a way to capture the notions of structure and relationships between structures akin to Klein’s Erlangen program for classifying geometries through transformation groups. (See Marquis’ book From a Geometrical Point of View for a nice historical and philosophical account of this perspective.) The basic idea of a category is to organize mathematical objects according to a particular structure they share and account for how to relate those objects through an abstract way of specifying what are the functions between objects. For example, linear algebra is the study of vector spaces and linear transformations between vector spaces.

 

One can create a category where the objects are (small) categories themselves and where the functions between two categories are natural transformations. The natural transformations associated with an ordered pair of categories itself is a category here. This is a (strict) 2-category. Strict 2-categories can be seen to generalize to versions of 3-categories, 4-categories, and to n-categories in a straightforward way. We may weaken the categorical perspective by using homotopy equivalences in our definition of 2-category. A weak 2-category may be viewed as capturing the low homotopy group information (\( \pi_{0} \), \( \pi_{1} \),  and \( \pi_{2} \)) for \( X \) and so should determine the homotopy 2-type of \( X \). Continuing naively in this way, there should be a system of weak n-categories that capture the homotopy n-type of \( X \).  Such a program was initially crystalized in an extended letter from Grothendieck to Quillen which has become referred to as “Pursuing Stacks” and is today referred to as the homotopy hypothesis. See the paper “Lectures on n-categories and cohomology” by Baez and Shulman for an engaging account of early developments toward addressing this hypothesis.

 

The last two decades have seen many quite remarkable efforts to create frameworks for weak higher categories. Dr. Paoli’s approach to this utilizes one way to implement a powerful technical tool in homotopy theory: simplicial methods. This theory originated in the early developments of algebraic topology, beginning with Poincare, as a result of the effort to define triangulations for more general topological spaces as accomplished with the method of simplicial approximations. Simplicial sets were developed as a refinement of this method.  

 

In this book, the author succeeds in constructing a general model of weak n-categories in the notion of weakly globular n-categories and showing that this provides a solution to the homotopy hypothesis for general n. Moreover, this is conducted through a tour-de-force of higher category and simplicial homotopy techniques such as multiple categories, higher Tamasamani categories, multi-simplicial sets and categories, pseudo-functors, etc. The need to develop and apply such powerful tools is indicative of the complexity of n-categories and the homotopy of multi-simplicial sets and the careful attention needed to relate the two in order to properly capture the structure appropriate to weak n-categories by an inductive construction the author develops and analyzes. The author clearly demonstrates a deep understanding of both (higher) category theory and simplicial homotopy theory to develop the highly technical constructions and arguments needed to carry out this project.

 

The book is organized into four parts. The first part begins with an introduction to higher categories, describing the relevant history and motivations, as well as providing a summary of the current state-of-affairs. It then continues by giving an introduction to multi-simplicial techniques, Segal-type models, and deeper 2-category techniques. It also gives an overview of the important categorical notions of Tamasamani and weakly globular categories and the analysis of them through multi-simplicial methods that will be developed throughout the

book. (With beautiful diagrams along the way to help develop the geometric intuition.) The main results are summarized as well. The second part of the book describes in greater depth three Segal-type models for higher categories and Segalic pseudo-functors that would serve to generalize Segal’s condition to the higher category setting. The third part describes and analyzes rigidification of weak higher categories.  Finally, in the fourth part, weakly globular n-categories are properly defined, developed, and shown to provide a good model for weak n-categories and provide a justification for the homotopy hypothesis. This part closes with some conclusions and further directions.

 

This book is a research monograph and is primarily aimed primarily at professionals and advanced graduate students working in topology or category theory. It would also be useful to those working in theoretical physics or algebraic geometry who use higher category methods. It succeeds in solving a deep and challenging problem at the intersection of homotopy theory and higher category theory. It should be noted that, besides homotopy theory, weak higher categories have currently found applications also in algebraic geometry, mathematical logic, and mathematical physics. The book is organized in such a way as to give the necessary background and history needed to understand the problem, as well as provide a guide to the overall definitions, technology, and argument before entering the more technically challenging discussions delved into in the later parts. The only wish the reviewer had while reading this book was a place where a thorough discussion of the theory, techniques, and developments were discussed for weak n-categories in the lower dimensional (n = 2, 3) cases so as to help the reader better understand the resulting inductive construction the author gives. Other than that, this reviewer found the book, while challenging, a very rewarding journey in coming to understand the complexities in the development of higher category theory and their interactions with homotopy theory. The author’s solution to the homotopy hypothesis is appealing as well as geometrically and categorically insightful.

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James Turner is Professor of Mathematics at Calvin University. jturner@calvin.edu