The study of higher categories is attracting growing interest for
its many applications in topology, algebraic geometry, mathematical
physics and category theory. In this highly readable book, Carlos
Simpson develops a full set of homotopical algebra techniques and
proposes a working theory of higher categories. Starting with a
cohesive overview of the many different approaches currently used
by researchers, the author proceeds with a detailed exposition of
one of the most widely used techniques: the construction of a
Cartesian Quillen model structure for higher categories. The fully
iterative construction applies to enrichment over any Cartesian
model category, and yields model categories for weakly associative
n-categories and Segal n-categories. A corollary is the
construction of higher functor categories which fit together to
form the (n+1)-category of n-categories. The approach uses
Tamsamani's definition based on Segal's ideas, iterated as in
Pelissier's thesis using modern techniques due to Barwick, Bergner,
Lurie and others.
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