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The concepts of a locally presentable category and an accessible
category have turned out to be useful in formulating connections
between universal algebra, model theory, logic and computer
science. The aim of this book is to provide an exposition of both
the theory and the applications of these categories at a level
accessible to graduate students. Firstly the properties of
l-presentable objects, locally l-presentable categories, and
l-accessible categories are discussed in detail, and the
equivalence of accessible and sketchable categories is proved. The
authors go on to study categories of algebras and prove that
Freyd's essentially algebraic categories are precisely the locally
presentable categories. In the final chapters they treat some
topics in model theory and some set theoretical aspects. For
researchers in category theory, algebra, computer science, and
model theory, this book will be a necessary purchase.
Algebraic theories, introduced as a concept in the 1960s, have been
a fundamental step towards a categorical view of general algebra.
Moreover, they have proved very useful in various areas of
mathematics and computer science. This carefully developed book
gives a systematic introduction to algebra based on algebraic
theories that is accessible to both graduate students and
researchers. It will facilitate interactions of general algebra,
category theory and computer science. A central concept is that of
sifted colimits - that is, those commuting with finite products in
sets. The authors prove the duality between algebraic categories
and algebraic theories and discuss Morita equivalence between
algebraic theories. They also pay special attention to one-sorted
algebraic theories and the corresponding concrete algebraic
categories over sets, and to S-sorted algebraic theories, which are
important in program semantics. The final chapter is devoted to
finitary localizations of algebraic categories, a recent research
area.
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