This paper develops some ideas from previous work (coauthored, mostly with C.J.Isham). In that work, the main proposal is to assign as the value of a physical quantity in quantum theory (or classical physics), not a real number, but a certain kind of set (a sieve) of quantities that are functions of the given quantity. The motivation was in part physical---such a valuation illuminates the Kochen-Specker theorem; in part mathematical---the valuations arise naturally in the theory of presheaves; and (...) in part conceptual---the valuations arise from applying to propositions about the values of physical quantities some general axioms governing partial truth for any kind of proposition. In this paper, I give another conceptual motivation for the proposal. I develop (in Sections 2 and 3) the notion of a topos (of which presheaves give just one kind of example); and explain how this notion gives a satisfactory general framework for making sense of the idea of partial truth. Then I review (in Section 4) how our proposal applies this framework to the case of physical theories. (shrink)

We investigate a certain system of intuitionistic set theory from three points of view: an elementary set theory with bounded separation, a topos with distinguished inclusions, and a category of classes with a system of small maps. The three presentations are shown to be equivalent in a strong sense.

Volume 40, Issue 4, November 2019, Page 389-402.

We discuss some ways in which topos theory (a branch of category theory) can be applied to interpretative problems in quantum theory and quantum gravity. In Sec.1, we introduce these problems. In Sec.2, we introduce topos theory, especially the idea of a topos of presheaves. In Sec.3, we discuss several possible applications of topos theory to the problems in Sec.1. In Sec.4, we draw some conclusions.

We give a formal interpretation of Martin-Löf's Constructive Theory of Types in Elementary Topos Theory which is presented as a formalised theory with intensional equality of objects. Types are interpreted as arrows and variables as sections of their types. This is necessary to model correctly the working of the assumption x ∈ A. Then intensional equality interprets equality of types. The normal form theorem which asserts that the interpretation of a type is intensional equal to the (...) pullback of its “alignment” along some “base” arrow relates this interpretation to categorical semantic of types. (shrink)

The view that toposes originated as generalized set theory is a figment of set theoretically educated common sense. This false history obstructs understanding of category theory and especially of categorical foundations for mathematics. Problems in geometry, topology, and related algebra led to categories and toposes. Elementary toposes arose when Lawvere's interest in the foundations of physics and Tierney's in the foundations of topology led both to study Grothendieck's foundations for algebraic geometry. I end with remarks on a categorical (...) view of the history of set theory, including a false history plausible from that point of view that would make it helpful to introduce toposes as a generalization from set theory. (shrink)

Topos Theory is an important branch of mathematical logic of interest to theoretical computer scientists, logicians and philosophers who study the foundations of mathematics, and to those working in differential geometry and continuum physics. This compendium contains material that was previously available only in specialist journals. This is likely to become the standard reference work for all those interested in the subject.

Topos Theory is a subject that stands at the junction of geometry, mathematical logic and theoretical computer science, and it derives much of its power from the interplay of ideas drawn from these different areas. Because of this, an account of topos theory which approaches the subject from one particular direction can only hope to give a partial picture; the aim of this compendium is to present as comprehensive an account as possible of all the main (...) approaches and thereby to demonstrate the overall unity of the subject. The material is organised in such a way that readers interested in following a particular line of approach may do so by starting at an appropriate point in the text. (shrink)

The Topos of Music is the upgraded and vastly deepened English extension of the seminal German Geometrie der Töne. It reflects the dramatic progress of mathematical music theory and its operationalization by information technology since the publication of Geometrie der Töne in 1990. The conceptual basis has been vastly generalized to topos-theoretic foundations, including a corresponding thoroughly geometric musical logic. The theoretical models and results now include topologies for rhythm, melody, and harmony, as well as a classification (...) theory of musical objects that comprises the topos-theoretic concept framework. Classification also implies techniques of algebraic moduli theory. The classical models of modulation and counterpoint have been extended to exotic scales and counterpoint interval dichotomies. The probably most exciting new field of research deals with musical performance and its implementation on advanced object-oriented software environments. This subject not only uses extensively the existing mathematical music theory, it also opens the language to differential equations and tools of differential geometry, such as Lie derivatives. Mathematical performance theory is the key to inverse performance theory, an advanced new research field which deals with the calculation of varieties of parameters which give rise to a determined performance. This field uses techniques of algebraic geometry and statistics, approaches which have already produced significant results in the understanding of highest-ranked human performances. The book's formal language and models are currently being used by leading researchers in Europe and Northern America and have become a foundation of music software design. This is also testified by the book's nineteen collaborators and the included CD-ROM containing software and music examples. (shrink)

In this essay Elizabeth de Freitas follows Tim Ingold's groundbreaking anthropological work on lines and their cultural and material significance to argue that the line is the engine of theory, be it the drawn line of inscription or mathematical measure, the exclusionary line of delineation, or the undulating generative line of flight. De Freitas focuses on contemporary theories of perception that deploy the line — and mobilize the force of theory — so as to encode and reconfigure the (...) student's body. She draws on the work of Gilles Deleuze to develop a topological approach to the line, and argues that such an approach aligns with current philosophical approaches to social interaction that might be termed “topo-philosophy.”. (shrink)

This book introduces a set of methods and techniques for studying mathematical theories and relating them to each other through the use of Grothendieck toposes.

ABSTRACT Topos quantum theory is standardly portrayed as a kind of ‘neo-realist’ reformulation of quantum mechanics.1 1 In this article, I study the extent to which TQT can really be characterized as a realist formulation of the theory, and examine the question of whether the kind of realism that is provided by TQT satisfies the philosophical motivations that are usually associated with the search for a realist reformulation of quantum theory. Specifically, I show that the notion (...) of the quantum state is problematic for those who view TQT as a realist reformulation of quantum theory. 1Introduction 2Topos Quantum Theory 2.1Phase space 2.2Hilbert space 2.3Beyond Hilbert space 2.4Defining realism 2.5The spectral presheaf 2.6The logic of topos quantum theory 3Interpreting States in Topos Quantum Theory 4Interpreting Truth Values and Clopen Subobjects in Topos Quantum Theory 4.1Interpreting the truth values 4.2Interpreting Subcl 5Neo-realism 5.1The covariant approach 6Conclusion. (shrink)

Following the book Algebraic Set Theory from André Joyal and leke Moerdijk [8], we give a characterization of the initial ZF-algebra, for Heyting pretoposes equipped with a class of small maps. Then, an application is considered (the effective topos) to show how to recover an already known model (McCarty [9]).

Topos quantum theory represents a whole new approach to the formalization of non-relativistic quantum theory. It is well known that TQT replaces the orthomodular quantum logic of the traditional Hilbert space formalism with a new intuitionistic logic that arises naturally from the topos theoretic structure of the theory. However, it is less well known that TQT also has a dual logical structure that is paraconsistent. In this paper, we investigate the relationship between these two logical (...) structures and study the implications of this relationship for the definition of modal operators in TQT. (shrink)

Starting from logical structures of classical and quantum mechanics we reconstruct the logic of so-called no-signaling theories, where the correlations among subsystems of a composite system are restricted only by a simplest form of causality forbidding an instantaneous communication. Although such theories are, as it seems, irrelevant for the description of physical reality, they are helpful in understanding the relevance of quantum mechanics. The logical structure of each theory has an epistemological flavor, as it is based on analysis of (...) possible results of experiments. In this note we emphasize that not only logical structures of classical, quantum and no-signaling theory may be treated on the same ground but it is also possible to give to all of them a common ontological basis by constructing a “phase space” in all cases. In non-classical cases the phase space is not a set, as in classical theory, but a more general object obtained by means of category theory, but conceptually it plays the same role as the phase space in classical physics. (shrink)

The daseinisation is a mapping from an orthomodular lattice in ordinary quantum theory into a Heyting algebra in topos quantum theory. While distributivity does not always hold in orthomodular lattices, it does in Heyting algebras. We investigate the conditions under which negations and meets are preserved by daseinisation, and the condition that any element in the Heyting algebra transformed through daseinisation corresponds to an element in the original orthomodular lattice. We show that these conditions are equivalent, and (...) that, not only in the case of non-distributive orthomodular lattices but also in the case of Boolean algebras containing more than four elements, the Heyting algebra transformed from the orthomodular lattice through daseinisation will contain an element that does not correspond to any element of the original orthomodular lattice. (shrink)

Any attempt to construct a realist interpretation of quantum theory founders on the Kochen-Specker theorem, which asserts the impossibility of assigning values to quantum quantities in a way that preserves functional relations between them. We construct a new type of valuation which is defined on all operators, and which respects an appropriate version of the functional composition principle. The truth-values assigned to propositions are (i) contextual; and (ii) multi-valued, where the space of contexts and the multi-valued logic for each (...) context come naturally from the topos theory of presheaves. The first step in our theory is to demonstrate that the Kochen-Specker theorem is equivalent to the statement that a certain presheaf defined on the category of self-adjoint operators has no global elements. We then show how the use of ideas drawn from the theory of presheaves leads to the definition of a generalised valuation in quantum theory whose values are sieves of operators. In particular, we show how each quantum state leads to such a generalised valuation. A key ingredient throughout is the idea that, in a situation where no normal truth-value can be given to a proposition asserting that the value of a physical quantity A lies in a set D of real numbers , it is nevertheless possible to ascribe a partial truth-value which is determined by the set of all coarse-grained propositions that assert that some function f(A) lies in f(D), and that are true in a normal sense. The set of all such coarse-grainings forms a sieve on the category of self-adjoint operators, and is hence fundamentally related to the theory of presheave. (shrink)

In a previous paper, we have proposed assigning as the value of a physical quantity in quantum theory, a certain kind of set (a sieve) of quantities that are functions of the given quantity. The motivation was in part physical---such a valuation illuminates the Kochen-Specker theorem; and in part mathematical---the valuation arises naturally in the topos theory of presheaves. This paper discusses the conceptual aspects of this proposal. We also undertake two other tasks. First, we explain how (...) the proposed valuations could arise much more generally than just in quantum physics; in particular, they arise as naturally in classical physics. Second, we give another motivation for such valuations (that applies equally to classical and quantum physics). This arises from applying to propositions about the values of physical quantities some general axioms governing partial truth for any kind of proposition. (shrink)

This paper focuses on the significance of the concept of noise for cognition and computation. The concept of noise was massively transformed in the twentieth century with the advent of information theory, cybernetics, and computer science, all of which provide formal accounts of information and noise centrally concerned with contingency. We show how the concept has changed from these classical formulations, through developments in mathematics (topology and topos theory), computing (interactive computing and univalent foundations), and cognitive science (...) (predictive processing and cognitive morphodynamics). Ultimately it argues for the central importance of noise not only within a topological conception of cognition and computation, but also in the transcendental-empirical torsion of image schemata and the social interactive elaboration of freedom. (shrink)

We extend the topos-theoretic treatment given in previous papers of assigning values to quantities in quantum theory, and of related issues such as the Kochen-Specker theorem. This extension has two main parts: the use of von Neumann algebras as a base category (Section 2); and the relation of our generalized valuations to (i) the assignment to quantities of intervals of real numbers, and (ii) the idea of a subobject of the coarse-graining presheaf (Section 3).

We extend the topos-theoretic treatment given in previous papers of assigning values to quantities in quantum theory. In those papers, the main idea was to assign a sieve as a partial and contextual truth-value to a proposition that the value of a quantity lies in a certain set D of real numbers. Here we relate such sieve-valued valuations to valuations that assign to quantities subsets, rather than single elements, of their spectrum (we call these interval valuations). There are (...) two main results. First, there is a natural correspondence between these two kinds of valuation, which uses the notion of a state's support for a quantity (Section 3). Second, if one starts with a more general notion of interval valuation, one sees that our interval valuations based on the notion of support (and correspondingly, our sieve-valued valuations) are a simple way to secure certain natural properties of valuations, such as monotonicity (Section 4). (shrink)

This book studies the foundations of quantum theory through its relationship to classical physics. This idea goes back to the Copenhagen Interpretation (in the original version due to Bohr and Heisenberg), which the author relates to the mathematical formalism of operator algebras originally created by von Neumann. The book therefore includes comprehensive appendices on functional analysis and C*-algebras, as well as a briefer one on logic, category theory, and topos theory. Matters of foundational as well as (...) mathematical interest that are covered in detail include symmetry (and its "spontaneous" breaking), the measurement problem, the Kochen-Specker, Free Will, and Bell Theorems, the Kadison-Singer conjecture, quantization, indistinguishable particles, the quantum theory of large systems, and quantum logic, the latter in connection with the topos approach to quantum theory. This book is Open Access under a CC BY licence. (shrink)

This dissertation is concerned with one of the central but most perplexing theories in Plotinus's metaphysics, namely the nature and origin of time. In contrast to those interpretations of Neoplatonism that treat time as an imperfect image and passive product of eternity, I argue for a much more subtle and multifaceted concept that makes the human observer central to Plotinus's account of how time is actualized and thus passes. His emphasis on mystical experience and the individual soul's journey toward the (...) One and the realm of eternity strengthens this account. By reading the treatise On Eternity and Time in this light, I demonstrate the importance of embodied human consciousness for the Plotinian theory of time. ;This emphasis invites comparison with the one contemporary Continental thinker who devoted countless pages in both his published work and his manuscripts to the relationship between time and consciousness. In the second part of the dissertation I argue that the Husserlian and Plotinian subjects have analogous characteristics, in accordance with their parallel roles as sources of time, and that the Husserlian notion of omnitemporality offers the best way of understanding the internal connection between Plotinus's notion of eternity and his notion of actualization through human experience. At the same time, Plotinus challenges a phenomenological account by giving the eternal ontological priority over the human realm of temporality. Nevertheless, these intriguing affinities and oppositions suggest that Plotinus's theory of time and eternity can best be understood and even expanded by uncovering its phenomenological underpinnings. (shrink)

The concept of love has emerged as a central topic for philosophical and theoretical discussion over the past few years. Whereas the dominant ideology of love in contemporary culture insists that love be understood as a discovery of my long lost second half with whom I merge and finally recreate a whole, contemporary philosophers and theorists have stressed the need to reconsider the concept of love within an ethical framework that can sustain the idea of the other as forever different (...) and separate. Taking as a starting point two different taxi scenes offered by Foucault and Woolf respectively, the article outlines and discusses those two contemporary love grammars, focusing on the concepts of lack and fantasy. Whereas Woolf’s taxi scene is reminiscent of a notion of love as a fantasy of unification, Foucault’s notion of love not only observes the need to maintain the other’s specificity and separateness, but also considers lack and fantasy to be primary and fundamental parts of love as lived experience. While Foucault insists on pleasure and action he also observes the central and unavoidable power of imagination that is attributed the name and function of recollection. The article finally addresses the question of if and how we might be responsible for our daydreams and fantasies. (shrink)

We construct a sheaf-theoretic representation of quantum observables algebras over a base category equipped with a Grothendieck topology, consisting of epimorphic families of commutative observables algebras, playing the role of local arithmetics in measurement situations. This construction makes possible the adaptation of the methodology of Abstract Differential Geometry (ADG), à la Mallios, in a topos-theoretic environment, and hence, the extension of the “mechanism of differentials” in the quantum regime. The process of gluing information, within diagrams of commutative algebraic localizations, (...) generates dynamics, involving the transition from the classical to the quantum regime, formulated cohomologically in terms of a functorial quantum connection, and subsequently, detected via the associated curvature of that connection. (shrink)

Conceptualizing the nature of reality and the way the world functions, Ekaterina Yahyaoui Krivenko analyzes the foundations of human rights law in the strict subject/object dichotomy. Seeking to dismantle this dichotomy using topo-logic, a concept developed by Japanese philosopher Nishida Kitaro, this topical book formulates ways to operationalize alternative visions of human rights practice. Subject/object dichotomy, Yahyaoui Krivenko demonstrates, emerges from and reflects a particular Western worldview through a quest for rationality and formal logic. Taking a metaphysical and epistemological perspective, (...) this book explores the alternative views of reality and logic, developed by Kitaro, to demonstrate how topo-logic can enable both a theoretical and a practical renewal of human rights and overcome the subject/object dichotomy. Examining the recent growth of social movements, decolonization and diversification of discourses about human rights, and substantive equality, the book identifies these developments in contemporary human rights as indications of a movement towards a topo-logical view beyond the subject/object dichotomy. Students and scholars of critical legal studies, legal theory and philosophy, and international human rights law will find this book to be an invigorating read. Laying ground for the possible renewal and enhancement of human rights law, it will also be a useful resource for practitioners of human rights law. (shrink)

We present a topos-theoretic interpretation of (a categorical generalization of) Fraïssé’s construction in Model Theory, with applications to homogeneous models and countably categorical theories.

The electron double-slit interference is re-examined from the point of view of temporal topos. Temporal topos (or t-topos) is an abstract algebraic (categorical) method using the theory of sheaves. A brief introduction to t-topos is given. When the structural foundation for describing particles is based on t-topos, the particle-wave duality of electron is a natural consequence. A presheaf associated with the electron represents both particle-like and wave-like properties depending upon whether an object in the (...) site (t-site) is specified (particle-like) or not (wave-like). It is shown that the localization of the electron at one of the slits is equivalent to choosing a particular object in the t-site and that the electron behaves as a wave when it passes through a double-slit because there are more than one object in the t-site. Also, the single-slit diffraction is interpreted as a result of the possibility of many different ways of factoring a morphism between two objects. (shrink)

With regard to the importance of interrelations and interplays of topos and logos in ancient theory and practices, here I will appropriate Aristotle philosophizing of topos and logos and apply it for the ancient persian Empire as reflected in related inscriptions.

In his philosophy of nothingness, Kitarō Nishida illuminates the matrix of transformation of the world "from the Created to the Creating" (tsukuru mono kara tsukurareta mono e) through shintai, or the body. In this matrix, shintai enters into the stage of an action-sensation continuum and emerges as the immaculate iconic tool of nothingness to create new figures as extended self. This idea of shintai has resonance with the development of postwar art in Japan. The "Space of Transparency" put forth by (...) Ufan Lee, the leader of Monoha, is the principal example. This essay investigates Nishida's notion of shintai and its influence on Lee's theory of art. (shrink)

: In his philosophy of nothingness, Kitar Nishida illuminates the matrix of transformation of the world ‘‘from the Created to the Creating’’ (tsukuru mono kara tsukurareta mono e) through shintai, or the body. In this matrix, shintai enters into the stage of an action-sensation continuum and emerges as the immaculate iconic tool of nothingness to create new figures as extended self. This idea of shintai has resonance with the development of postwar art in Japan. The ‘‘Space of Transparency’’ put forth (...) by Ufan Lee, the leader of Monoha, is the principal example. This essay investigates Nishida’s notion of shintai and its influence on Lee’s theory of art. (shrink)

In his philosophy of nothingness, Kitar Nishida illuminates the matrix of transformation of the world ''from the Created to the Creating'' through shintai, or the body. In this matrix, shintai enters into the stage of an action-sensation continuum and emerges as the immaculate iconic tool of nothingness to create new figures as extended self. This idea of shintai has resonance with the development of postwar art in Japan. The ''Space of Transparency'' put forth by Ufan Lee, the leader of Monoha, (...) is the principal example. This essay investigates Nishida's notion of shintai and its influence on Lee's theory of art. (shrink)

A categorical, higher dimensional algebra and generalized topos framework for Łukasiewicz–Moisil Algebraic–Logic models of non-linear dynamics in complex functional genomes and cell interactomes is proposed. Łukasiewicz–Moisil Algebraic–Logic models of neural, genetic and neoplastic cell networks, as well as signaling pathways in cells are formulated in terms of non-linear dynamic systems with n-state components that allow for the generalization of previous logical models of both genetic activities and neural networks. An algebraic formulation of variable ‘next-state functions’ is extended to a (...) Łukasiewicz–Moisil Topos with an n-valued Łukasiewicz–Moisil Algebraic Logic subobject classifier description that represents non-random and non-linear network activities as well as their transformations in developmental processes and carcinogenesis. The unification of the theories of organismic sets, molecular sets and Robert Rosen’s (M,R)-systems is also considered here in terms of natural transformations of organismal structures which generate higher dimensional algebras based on consistent axioms, thus avoiding well known logical paradoxes occurring with sets. Quantum bionetworks, such as quantum neural nets and quantum genetic networks, are also discussed and their underlying, non-commutative quantum logics are considered in the context of an emerging Quantum Relational Biology. (shrink)

In this paper we develop the notion of formal precover in a topos by defining a relation between elements and sets in a local set theory. We show that such relations are equivalent to modalities and to universal closure operators. Finally we prove that these relations are well characterized by a convenient restriction to a particular set.

A categorical, higher dimensional algebra and generalized topos framework for Łukasiewicz–Moisil Algebraic–Logic models of non-linear dynamics in complex functional genomes and cell interactomes is proposed. Łukasiewicz–Moisil Algebraic–Logic models of neural, genetic and neoplastic cell networks, as well as signaling pathways in cells are formulated in terms of non-linear dynamic systems with n-state components that allow for the generalization of previous logical models of both genetic activities and neural networks. An algebraic formulation of variable ‘next-state functions’ is extended to a (...) Łukasiewicz–Moisil Topos with an n-valued Łukasiewicz–Moisil Algebraic Logic subobject classifier description that represents non-random and non-linear network activities as well as their transformations in developmental processes and carcinogenesis. The unification of the theories of organismic sets, molecular sets and Robert Rosen’s (M,R)-systems is also considered here in terms of natural transformations of organismal structures which generate higher dimensional algebras based on consistent axioms, thus avoiding well known logical paradoxes occurring with sets. Quantum bionetworks, such as quantum neural nets and quantum genetic networks, are also discussed and their underlying, non-commutative quantum logics are considered in the context of an emerging Quantum Relational Biology. (shrink)

The older topos ›historia magistra vitae‹ – which was coined by Cicero, borrowing from a Hellenistic pattern – lasted almost unbroken into the eighteenth century, also thanks to its flexibility in accommodating the most different conclusions. As demonstrated by R. Koselleck, the common place was emptied of meaning just between the eighteenth and nineteenth century, at the same time and in connection with the invisible and sudden semantic process, through which in the German language area the naturalized foreign world (...) Historie and the plural ›die Geschichten‹ were displaced by the collective singular ›die Geschichte‹. This paper aims to frame Fichte’s position in the process of dissolution of the topos, described by Koselleck. For this purpose, the paper will be articulated in the following steps. Firstly, it reconstructs the most important and probable filters, through which Fichte has received Cicero’s topos. The second part will examine deeply the criticisms that Fichte moves to the different versions of the formula, during his whole speculative itinerary. Finally, the conclusive part considers how Fichte recovers and reworks the topos – since his first works – by transposing it in the completely different theoretical frame of his own philosophy of history and of his theory of the State.Die Wendung ›historia magistra vitae‹ – die in Anlehnung an hellenistische Vorbilder von Cicero geprägt wurde – währte fast ungebrochen bis in das achtzehnte Jahrhundert, wobei eine solche Dauerhaftigkeit auch und zunächst auf seiner Elastizität beruhte, die die unterschiedlichsten Schlüsse zuließ. Wie R. Koselleck gezeigt hat, wird die Formel lediglich zwischen dem achtzehnten und dem neunzehnten Jahrhundert – gleichzeitig und parallel mit dem unmerklichen und unbewussten Vorgang, der im deutschen Sprachgebiet die Pluralform von ›die Geschichte‹ zu einem Kollektivsingular verdichtet hat – sinnentleert. Der vorliegende Beitrag geht darauf aus, Fichtes eigentümliche Stellung in dieser Auflösung des alten Topos zu verdeutlichen. Dazu untersucht man zuerst die Quelle, die Fichtes Rezeption des Topos ermöglichen und vermitteln. Im zweiten Teil betrachtet man alle Stelle tiefgehend, wo Fichte im Lauf seiner ganzen philosophischen Untersuchung an den Topos kritisch aufweist. Am Schluss wird die radikale neue Auslegung illustriert, die Fichte auf Grund seiner Philosophie der Geschichte und seiner Auffassung der Staatslehre anbietet. (shrink)

Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis-a-vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves. We propose a synthesis of Bell's many-topoi view and modal-structuralism. Surprisingly, a combination of mereology and plural quantification suffices to describe hypothetical large domains, recovering the Grothendieck method of universes. Both topos theory and (...) set theory can be carried out relative to such domains; puzzles about ‘large categories’ and ‘proper classes’ are handled in a uniform way, by relativization, sustaining insights of Zermelo. (shrink)

Here, we present a category ${\mathbf {pEff}}$ which can be considered a predicative variant of Hyland's Effective Topos ${{\mathbf {Eff} }}$ for the following reasons. First, its construction is carried in Feferman’s predicative theory of non-iterative fixpoints ${{\widehat {ID_1}}}$. Second, ${\mathbf {pEff}}$ is a list-arithmetic locally cartesian closed pretopos with a full subcategory ${{\mathbf {pEff}_{set}}}$ of small objects having the same categorical structure which is preserved by the embedding in ${\mathbf {pEff}}$ ; furthermore subobjects in ${{\mathbf {pEff}_{set}}}$ are (...) classified by a non-small object in ${\mathbf {pEff}}$. Third ${\mathbf {pEff}}$ happens to coincide with the exact completion of the lex category defined as a predicative rendering in ${{\widehat {ID_1}}}$ of the subcategory of ${{\mathbf {Eff} }}$ of recursive functions and it validates the Formal Church’s thesis. Hence pEff turns out to be itself a predicative rendering of a full subcategory of ${{\mathbf {Eff} }}$. (shrink)

I see model theory as becoming increasingly detached from set theory, and the Tarskian notion of set-theoretic model being no longer central to model theory. In much of modern mathematics, the set-theoretic component is of minor interest, and basic notions are geometric or category-theoretic. In algebraic geometry, schemes or algebraic spaces are the basic notions, with the older “sets of points in affine or projective space” no more than restrictive special cases. The basic notions may be given (...) sheaf-theoretically, or functorially. To understand in depth the historically important affine cases, one does best to work with more general schemes. The resulting relativization and “transfer of structure” is incomparably more flexible and powerful than anything yet known in “set-theoretic model theory”.It seems to me now uncontroversial to see the fine structure of definitions as becoming the central concern of model theory, to the extent that one can easily imagine the subject being called “Definability Theory” in the near future.Tarski's set-theoretic foundational formulations are still favoured by the majority of model-theorists, and evolution towards a more suggestive language has been perplexingly slow. None of the main texts uses in any nontrivial way the language of category theory, far less sheaf theory or topos theory. Given that the most notable interactions of model theory with geometry are in areas of geometry where the language of sheaves is almost indispensable, this is a curious situation, and I find it hard to imagine that it will not change soon, and rapidly. (shrink)

We establish, generalizing Di Nola and Lettieri’s categorical equivalence, a Morita-equivalence between the theory of lattice-ordered abelian groups and that of perfect MV-algebras. Further, after observing that the two theories are not bi-interpretable in the classical sense, we identify, by considering appropriate topos-theoretic invariants on their common classifying topos, three levels of bi-interpretability holding for particular classes of formulas: irreducible formulas, geometric sentences, and imaginaries. Lastly, by investigating the classifying topos of the theory of perfect (...) MV-algebras, we obtain various results on its syntax and semantics also in relation to the cartesian theory of the variety generated by Chang’s MV-algebra, including a concrete representation for the finitely presentable models of the latter theory as finite products of finitely presentable perfect MV-algebras. Among the results established on the way, we mention a Morita-equivalence between the theory of lattice-ordered abelian groups and that of cancellative lattice-ordered abelian monoids with bottom element. (shrink)

Working in a fixed Grothendieck topos Sh(C, J C ) we generalize \({\mathcal{L}_{{\infty},\omega}}\) to allow our languages and formulas to make explicit reference to Sh(C, J C ). We likewise generalize the notion of model. We then show how to encode these generalized structures by models of a related sentence of \({\mathcal{L}_{{\infty},\omega}}\) in the category of sets and functions. Using this encoding we prove analogs of several results concerning \({\mathcal{L}_{{\infty},\omega}}\) , such as the downward Löwenheim–Skolem theorem, the completeness theorem (...) and Barwise compactness. (shrink)