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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.

Categories (Mathematics) --- Representations of categories. --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Topology --- Functor theory --- Representations of categories --- 512.58 --- 512.58 Categories. Category theory --- Categories. Category theory

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The first two chapters of this book offer a modern, self-contained exposition of the elementary theory of triangulated categories and their "ients. The simple, elegant presentation of these known results makes these chapters eminently suitable as a text for graduate students. The remainder of the book is devoted to new research, providing, among other material, some remarkable improvements on Brown's classical representability theorem. In addition, the author introduces a class of triangulated categories"--the "well generated triangulated categories"--and studies their properties. This exercise is particularly worthwhile in that many examples of triangulated categories are well generated, and the book proves several powerful theorems for this broad class. These chapters will interest researchers in the fields of algebra, algebraic geometry, homotopy theory, and mathematical physics.

Categories (Mathematics). --- Category theory. Homological algebra --- Categories (Mathematics) --- 512.58 --- 512.58 Categories. Category theory --- Categories. Category theory --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Topology --- Functor theory --- Abelian category. --- Abelian group. --- Additive category. --- Adjoint functors. --- Adjoint. --- Adjunction (field theory). --- Associative property. --- Axiom. --- Basis (linear algebra). --- Bijection. --- Biproduct. --- Brown's representability theorem. --- Cardinal number. --- Cardinality. --- Category of abelian groups. --- Chain complex. --- Class (set theory). --- Cohomology. --- Computation. --- Coproduct. --- Corollary. --- Countable set. --- Counterexample. --- Derived category. --- Derived functor. --- Diagram (category theory). --- Direct limit. --- Direct sum. --- Discrete valuation ring. --- Duality (mathematics). --- Embedding. --- Equivalence class. --- Equivalence of categories. --- Exact functor. --- Exact sequence. --- Existence theorem. --- Existential quantification. --- Factorization. --- Finitely generated abelian group. --- Functor category. --- Functor. --- Grothendieck category. --- Grothendieck's Tôhoku paper. --- Group homomorphism. --- Homological algebra. --- Homotopy category of chain complexes. --- Homotopy category. --- Homotopy colimit. --- Homotopy. --- I0. --- Injective function. --- Injective object. --- Integer. --- Isomorph. --- Isomorphism class. --- Jack Morava. --- K-theory. --- Limit (category theory). --- Limit of a sequence. --- Limit ordinal. --- Linear map. --- Mapping cone (homological algebra). --- Mathematical induction. --- Maximal ideal. --- Module (mathematics). --- Monomorphism. --- Moore space. --- Morphism. --- N0. --- Natural transformation. --- Open set. --- Partially ordered set. --- Pierre Deligne. --- Prime number. --- Projective object. --- Proportionality (mathematics). --- Quotient category. --- Regular cardinal. --- Representable functor. --- Sheaf (mathematics). --- Special case. --- Spectral sequence. --- Subcategory. --- Subobject. --- Subsequence. --- Subset. --- Successor ordinal. --- Summation. --- Tautology (logic). --- Tensor product. --- Theorem. --- Theory. --- Topological group. --- Transfinite induction. --- Transfinite. --- Triangulated category. --- Universal property. --- Vector space. --- Vladimir Voevodsky. --- Yoneda lemma.