## Toposes, Triples and TheoriesAs its title suggests, this book is an introduction to three ideas and the connections between them. Before describing the content of the book in detail, we describe each concept briefly. More extensive introductory descriptions of each concept are in the introductions and notes to Chapters 2, 3 and 4. A topos is a special kind of category defined by axioms saying roughly that certain constructions one can make with sets can be done in the category. In that sense, a topos is a generalized set theory. However, it originated with Grothendieck and Giraud as an abstraction of the of the category of sheaves of sets on a topological space. Later, properties Lawvere and Tierney introduced a more general id~a which they called "elementary topos" (because their axioms did not quantify over sets), and they and other mathematicians developed the idea that a theory in the sense of mathematical logic can be regarded as a topos, perhaps after a process of completion. The concept of triple originated (under the name "standard construc in Godement's book on sheaf theory for the purpose of computing tions") sheaf cohomology. Then Peter Huber discovered that triples capture much of the information of adjoint pairs. Later Linton discovered that triples gave an equivalent approach to Lawverc's theory of equational theories (or rather the infinite generalizations of that theory). Finally, triples have turned out to be a very important tool for deriving various properties of toposes. |

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algebra apply arrow Boolean called Chapter closed coequalizer colimits commutes complete component composition concept condition cone consider construction contains Corollary corresponding cover defined definition denote dense describe diagram easy element embedding epimorphic equalizer equivalence relation example Exercise exists fact factorization faithful finite limits follows function functor geometric given gives global graph Grothendieck hence identity implies inclusion induced inverse isomorphism kernel pair left adjoint left exact Lemma limits logical means models monic mono morphism natural transformation notation Note object Observe operation preserves projection Proof Proposition prove pullback reflects regular regular epi result right adjoint satisfies Section sheaf sheaves Show sieve sketch space split square structure subobject sums Suppose takes Theorem theory topology topos toposes triple tripleable true underlying unique universal