Categories, Allegories (Google eBook)
General concepts and methods that occur throughout mathematics – and now also in theoretical computer science – are the subject of this book. It is a thorough introduction to Categories, emphasizing the geometric nature of the subject and explaining its connections to mathematical logic. The book should appeal to the inquisitive reader who has seen some basic topology and algebra and would like to learn and explore further.
The first part contains a detailed treatment of the fundamentals of Geometric Logic, which combines four central ideas: natural transformations, sheaves, adjoint functors, and topoi.
A special feature of the work is a general calculus of relations presented in the second part. This calculus offers another, often more amenable framework for concepts and methods discussed in part one. Some aspects of this approach find their origin in the relational calculi of Peirce and Schroeder from the last century, and in the 1940's in the work of Tarski and others on relational algebras. The representation theorems discussed are an original feature of this approach.
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A Q B abelian category adjoint axiom of choice bicartesian binary products boolean algebra canonical cartesian category category of sets coequalizer cokernel complement composition construct contains contravariant coproducts coreﬂexive morphisms coterminator countable deﬁne deﬁnition denote diagram disjoint union distributive allegory division allegory element embedding equalizer equations equivalence functor equivalence relation exponential faithful representation faithfully represented ﬁnd ﬁnite set ﬁrst forgetful functor full subcategory function Grothendieck topos hence Heyting algebra homeomorphism Horn sentence idempotents implies inﬁnite intersection isomorphism last section lattice left-adjoint lemma local homeomorphism locally complete logos monic natural numbers non-empty Note object obtain one-object open subsets operation pair poset power allegory pre-topos predicate preserve and reﬂect proper subobject proto-morphisms pullbacks quantiﬁer quotient recursive reﬂexive regular category representable functors satisﬁes semi-simple source-target space split subobject subterminator sufﬁces to show tabular terminator theory topoi topological unique unitary variables well-supported