Topoi: The Categorial Analysis of Logic
A classic introduction to mathematical logic from the perspective of category theory, this text is suitable for advanced undergraduates and graduate students and accessible to both philosophically and mathematically oriented readers. Its approach moves always from the particular to the general, following through the steps of the abstraction process until the abstract concept emerges naturally.
Beginning with a survey of set theory and its role in mathematics, the text proceeds to definitions and examples of categories and explains the use of arrows in place of set-membership. The introduction to topos structure covers topos logic, algebra of subobjects, and intuitionism and its logic, advancing to the concept of functors, set concepts and validity, and elementary truth. Explorations of categorial set theory, local truth, and adjointness and quantifiers conclude with a study of logical geometry.
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6-arrow 6-object algebra arrow f assigns axioms bijection Bn(I Boolean category theory characterisation classical co-product codomain colimits collection component concept construction corresponding deﬁned deﬁnition denoted diagram commute disjoint domain epic epimorphic equaliser equivalent EXAMPLE EXERCISE exists exponential extensional f and g ﬁnd ﬁnite ﬁrst ﬁrst-order formula function f functor F geometric morphism given Grothendieck topos hence Heyting Heyting algebra identiﬁed identity arrow iﬁ iﬂ implies inclusion inﬁnite interpretation intuitionistic inverse lattice left adjoint Lemma logic mathematical monic monoid natural numbers object natural transformation notion open sets pair partial elements poset precisely preserves PROOF pseudo-complement pullback Q-set quantiﬁers reﬂects relation satisﬁes semantics sentence set theory Sh(C sheaf sheaves Show singleton stalk structure Sub(d subobject subobject classiﬁer subset symbol terminal object Theorem Top(I topoi topological space topology true truth truth-values unique arrow valid variables well-pointed