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 A-directed diagram A-presentable A-presentable objects A-pure A-small accessible category adjoint algebras analogous assigns assume bounded called canonical characterization choose closed cocomplete cocone complete concept condition cone Consequently consider consisting contains Corollary corresponding deﬁned Deﬁnition denote dense directed colimits domain element embedding epimorphism equations equivalent essentially Example Exercise exists extension fact factorizes ﬁnitary ﬁnitely presentable ﬁrst follows formula full subcategory function given graphs groups homomorphism implies injectivity isomorphism less limits locally presentable category logic maps monomorphisms morphism f natural object obviously operation ordered orthogonality pair poset precisely Pres preserves projectives PROOF Proposition prove quotient reﬂective regular cardinal relation Remark satisﬁes signature sketch small category sort spaces split statement strong structures subobjects subset symbol Theorem theory unique variables variety verify Vopénka’s principle weak weakly