The first of its kind, this book presents a widely accessible exposition of topos theory, aimed at the philosopher-logician as well as the mathematician. It is suitable for individual study or use in class at the graduate level (it includes 500 exercises). It begins with a fully motivated introduction to category theory itself, moving always from the particular example to the abstract concept. It then introduces the notion of elementary topos, with a wide range of examples and goes on to develop its theory in depth, and to elicit in detail its relationship to Kripke's intuitionistic semantics, models of classical set theory and the conceptual framework of sheaf theory (``localization'' of truth). Of particular interest is a Dedekind-cuts style construction of number systems in topoi, leading to a model of the intuitionistic continuum in which a ``Dedekind-real'' becomes represented as a ``continuously-variable classical real number''. The second edition contains a new chapter, entitled Logical Geometry, which introduces the reader to the theory of geometric morphisms of Grothendieck topoi, and its model-theoretic rendering by Makkai and Reyes. The aim of this chapter is to explain why Deligne's theorem about the existence of points of coherent topoi is equivalent to the classical Completeness theorem for ``geometric'' first-order formulae.
Quasitopoi generalize topoi, a concept of major importance in the theory of Categoreis, and its applications to Logic and Computer Science. In recent years, quasitopoi have become increasingly important in the diverse areas of Mathematics such as General Topology and Fuzzy Set Theory. These Lecture Notes are the first comprehensive introduction to quasitopoi, and they can serve as a first introduction to topoi as well.
This monograph presents a new, systematic treatment of the relation between classifying topoi and classifying spaces of topological categories. Using a new generalized geometric realization which applies to topoi, a weak homotopy equival- ence is constructed between the classifying space and the classifying topos of any small (topological) category. Topos theory is then applied to give an answer to the question of what structures are classified by "classifying" spaces. The monograph should be accessible to anyone with basic knowledge of algebraic topology, sheaf theory, and a little topos theory.
This work provides some interesting new results on the notion of the topos and the theory of hypothetical syllogisms in Aristotle based on an incisive interpretation of Aristotle's "Topics" and certain passages of the "Analytics."
Linguists have become increasingly interested in examining how class culture is socially constructed and maintained through spoken language. Julie Lindquist's examination of the linguistic ethnography of a working-class bar in Chicago is an important and original contribution to the field. She examines how regular patrons argue about political issues in order to create a group identity centered around political ideology. She also shows how their political arguments are actually a rhetorical genre, one which creates a delicate balance between group solidarity and individual identity, as well as a tenuous and ambivalent sense of class identity.
"In Topoi/Graphein Christian Abrahamsson maps the paradoxical limit of the in-between to revealthat to be human is to know how tolive with the difference between the known and the unknown. Using filmic case studies, including CodeInconnu, Lord of the Flies, and Apocalypse Now,and focusing on key concerns developed in the works of the philosophers Deleuze, Olsson, and Wittgenstein, Abrahamsson starts within the notion of fixed spatiality, in whichhuman thought and action are anchored in the given of identity. He then movesthrough a social world in which spatiotemporal transformations are neitherfixed nor taken for granted. Finally he edges into the pure temporality that liesbeyond the maps of fixed points and social relations. Each chapter is organized into two subjects: topoi, orexcerpts from the films, and graphein, the author's interpretation ofpresented theoriesto mirror the displacements,transpositions, juxtapositions, fluctuations, and transformations between delimited categories. A landmark work in the study of human geography, Abrahamsson's book proposes that academic and intellectual attention should focus on the spatialization between meaning and its materialization in everyday life."