# Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids

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*Ronald Brown; Philip J. Higgins; Rafael Sivera*

A publication of the European Mathematical Society

The main theme of this book is that the use of filtered spaces rather
than just topological spaces allows the development of basic algebraic topology
in terms of higher homotopy groupoids; these algebraic structures better
reflect the geometry of subdivision and composition than those commonly in use.
Exploration of these uses of higher dimensional versions of groupoids has been
largely the work of the first two authors since the mid 1960s.

The structure of the book is intended to make it useful to a wide class of
students and researchers for learning and evaluating these methods,
primarily in algebraic topology but also in higher category theory and its
applications in analogous areas of mathematics, physics, and computer
science.

Part I explains the intuitions and theory in dimensions 1 and 2, with many
figures and diagrams, and a detailed account of the theory of crossed
modules. Part II develops the applications of crossed complexes. The engine
driving these applications is the work of Part III on cubical
\(\omega\)-groupoids,
their relations to crossed complexes, and their homotopically defined
examples for filtered spaces. Part III also includes a chapter suggesting
further directions and problems, and three appendices give accounts of some
relevant aspects of category theory. Endnotes for each chapter give further
history and references.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students and research mathematicians interested in algebraic topology.