Purpose. Cubical diagrams have become increasingly important over the last two decades, both as a powerful organizational tool and because of their many applications. They provide the language necessary for the Blakers–Massey Theorems, which unify many classical results; they lie at the heart of calculus of functors, which has many uses in algebraic and geometric topology; and they are intimately related to homotopy (co)limits of diagrams and (co)simplicial spaces. The growing importance of cubical diagrams demands an up-to-date, comprehensive introduction to this subject.
In addition, self-contained, expository accounts of homotopy (co)limits and (co)simplicial spaces do not appear to exist in the literature. Most standard references on these subjects adopt the language of model categories, thereby usually sacrificing concreteness for generality. One of the goals of this book is to provide an introductory treatment to the theory of homotopy (co)limits in the category of topological spaces.
This book makes the case for adding the homotopy limit and colimit of a punctured square (homotopy pullback and homotopy pushout) to the essential toolkit for a homotopy theorist. These elementary constructions unify many basic concepts and endow the category of topological spaces with a sophisticated way to “add” (pushout) and “multiply” (pullback) spaces, and so “do algebra”. Homotopy pullbacks and pushouts lie at the core of much of what we do and they build a foundation for the homotopy theory of cubical diagrams, which in turn provides a concrete introduction to the theory of general homotopy (co)limits and (co)simplicial spaces.
Features. We develop the homotopy theory of cubical diagrams in a gradual way, starting with squares and working up to cubes and beyond. Along the way, we show the reader how to develop competence with these topics with over 300 worked examples. Fully worked proofs are provided for the most part, and the reader will be able to fill in those that are not provided or have only been sketched. Many results in this book are known, but their proofs do not appear to exist. If we were not able to find a proof in the literature, we have indicated that this is the case. The reader will also benefit from an abundance of suggestions for further reading.
Cubical diagrams are an essential concept for stating and understanding the generalized Blakers–Massey Theorems, fundamental results lying at the intersection of stable and unstable homotopy theory.