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Category theory provides structure for the mathematical world and
is seen everywhere in modern mathematics. With this book, the
author bridges the gap between pure category theory and its
numerous applications in homotopy theory, providing the necessary
background information to make the subject accessible to graduate
students or researchers with a background in algebraic topology and
algebra. The reader is first introduced to category theory,
starting with basic definitions and concepts before progressing to
more advanced themes. Concrete examples and exercises illustrate
the topics, ranging from colimits to constructions such as the Day
convolution product. Part II covers important applications of
category theory, giving a thorough introduction to simplicial
objects including an account of quasi-categories and Segal sets.
Diagram categories play a central role throughout the book, giving
rise to models of iterated loop spaces, and feature prominently in
functor homology and homology of small categories.
Within algebraic topology, the prominent role of multiplicative
cohomology theories has led to a great deal of foundational
research on ring spectra and in the 1990s this gave rise to
significant new approaches to constructing categories of spectra
and ring-like objects in them. This book contains some important
new contributions to the theory of structured ring spectra as well
as survey papers describing these and relationships between them.
One important aspect is the study of strict multiplicative
structures on spectra and the development of obstruction theories
to imposing strictly associative and commutative ring structures on
spectra. A different topic is the transfer of classical algebraic
methods and ideas, such as Morita theory, to the world of stable
homotopy.
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