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.

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.

A Collection of Lectures by Variuos Authors

In Enthymemes and Topoi in Dialogue, Ellen Breitholtz presents a novel and precise account of reasoning from an interactional perspective. The account draws on the concepts of enthymemes and topoi, originating in Aristotelian rhetoric and dialectic, and integrates these in a formal dialogue semantic account using TTR, a type theory with records. Argumentation analysis and formal approaches to reasoning often focus the logical validity of arguments on inferences made in discourse from a god’s-eye perspective. In contrast, Breitholtz’s account emphasises the individual perspectives of interlocutors and the function and acceptability of their reasoning in context. This provides an analysis of interactions where interlocutors have access to different topoi and therefore make different inferences.

We give a general method of forcing over categories as a category-theoretic universal construction which subsumes, on one hand, all known instances of forcing in set theory, Boolean and Heyting valued models and sheaf interpretations for both classical and intuitionistic formal systems; and, on the other hand, constructions of classifying topoi in topos theory.

The author develops a universal framework to study smooth higher orbifolds on the one hand and higher Deligne-Mumford stacks (as well as their derived and spectral variants) on the other, and use this framework to obtain a completely categorical description of which stacks arise as the functor of points of such objects. He chooses to model higher orbifolds and Deligne-Mumford stacks as infinity-topoi equipped with a structure sheaf, thus naturally generalizing the work of Lurie, but his approach applies not only to different settings of algebraic geometry such as classical algebraic geometry, derived algebraic geometry, and the algebraic geometry of commutative ring spectra but also to differential topology, complex geometry, the theory of supermanifolds, derived manifolds etc., where it produces a theory of higher generalized orbifolds appropriate for these settings. This universal framework yields new insights into the general theory of Deligne-Mumford stacks and orbifolds, including a representability criterion which gives a categorical characterization of such generalized Deligne-Mumford stacks. This specializes to a new categorical description of classical Deligne-Mumford stacks, which extends to derived and spectral Deligne-Mumford stacks as well.

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.

The series Studies and Texts in Scepticism contains monographs, translations, and collected essays exploring scepticism in its dual manifestation as a purely philosophical tradition and as a set of sceptical strategies, concepts, and attitudes in the cultural field - especially in religions, perhaps most notably in Judaism. In such cultural contexts scepticism manifests as a critical attitude towards different dimensions and systems of secular or revealed knowledge and towards religious and political authorities. It is not merely an intellectual or theoretical worldview, but a critical form of life that expresses itself in such diverse phenomena as religion, literature, and society. Further book series of the Maimonides Centre for Advanced Studies are Jewish Thought, Philosophy, and Religion and the Yearbook of the Maimonides Centre for Advances Studies.

Traditionen leben von der Dialektik der Wiederholung, die das Gleiche stets anders inszeniert. Sie speisen sich aus den Erinnerungen; an der Grenze von bedacht und selbstverständlich getan lassen sie sich nur als selbstverständliche Überzeugungen bestimmen. Aber wie ist es dennoch möglich, Traditionen zu beeinflussen? Der vorliegende Band widmet sich der Frage, wie man mit Traditionen Symbolpolitik machen kann. Was sind die Zumutungen der Traditionen, wenn sie politisch instrumentalisiert werden? Gibt es Grenzen der Manipulation, die im Wesen der jeweiligen Traditionen liegen und sie folglich definieren? Die BeiträgerInnen geben eine Vielzahl von Antworten, indem sie sich Topoi aus Mittelalter und Früher Neuzeit aus interdisziplinärer Perspektive widmen. Traditions thrive on the dialectic of repetition. Drawing their topoi from the well of memory, they are situated on the very border between the deliberate and the habitual. Yet how is it possible to influence traditions? The present collection of essays studies the ways in which traditions are employed in the service of symbolic politics. What are the burdens and impositions of traditions, when their topoi are consciously exploited in the service of ideological purposes? Are there certain limits to manipulation that lie in the very nature of the traditions in question, a nature which therefore defines them? The contributors give a host of answers, studying topoi in medieval and early modern Europe from an interdisciplinary perspective.

Ars Topica is the first full-length study of the nature and development of topoi, the conceptual ancestors of modern argument schemes, between Aristotle and Cicero. Aristotle and Cicero configured topoi in a way that influenced the subsequent tradition. Their work on the topos-system grew out of an interest in creating a theory of argumentation which could stand between the rigour of formal logic and the emotive potential of rhetoric. This system went through a series of developments and transformations resulting from the interplay between the separate aims of gaining rhetorical effectiveness and of maintaining dialectical standards. Ars Topica presents a comprehensive treatment of Aristotle’s and Cicero’s methods of topoi and, by exploring their relationship, it illuminates an area of ancient rhetoric and logic which has been obscured for more than two thousand years. Through an interpretation which is philologically rooted in the historical context of topoi, the book lays the ground for evaluating the relevance of the classical approaches to modern research on arguments, and at the same time provides an introduction to Greek and Roman theory of argumentation focussed on its most important theoretical achievements.