Locally Presentable and Accessible Categories
The concepts of a locally presentable category and an accessible category are extremely 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. The concepts of lambda-presentable objects, locally lambda-presentable categories, and lambda-accessible categories are discussed in detail. The authors prove that Freyd's essentially algebraic categories are precisely the locally presentable categories. In the final chapter, they treat some advanced topics in model theory.
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A-accessible A-directed diagram A-presentable objects A-pure Abelian groups accessible category accessible functor accessibly embedded Adamek adjoint analogous arity axiomatizable binary relation canonical diagram category of models closed under A-directed closed under products cocomplete codomain coequalizer coﬁnal colim comma-category compatible cocone cone cone-reﬂective Corollary deﬁned Deﬁnition denote dense subcategory directed colimits E-algebra E-structure epimorphism equations equivalent essentially algebraic Example fact factorizes ﬁltered ﬁnitary ﬁnitely accessible ﬁrst follows forgetful functor formula full embedding full subcategory functor F given graph homomorphism graphs hom-functors huge cardinals implies isomorphism Lemma limit sketch locally ﬁnitely presentable locally presentable category many-sorted measurable cardinals monomorphisms morphism f natural transformation ordinal orthogonality class phism poset preserves A-directed colimits PROOF Proposition prove quasivariety quotient reﬂective subcategory regular cardinal RelE Remark Rosicky S-sorted satisﬁes sentable signature small category split idempotents split subobjects StrE subobjects subset suﬂicient Theorem variables weakly reﬂective