Since its formal definition over sixty years ago, category theory has been increasingly recognized as having a foundational role in mathematics. It provides the conceptual lens to isolate and characterize the structures with importance and universality in mathematics. The notion of an adjunction (a pair of adjoint functors) has moved to center-stage as the principal lens. The central feature of an adjunction is what might be called “determination through universals” based on universal mapping properties. A recently developed “heteromorphic” (...) theory about adjoints suggests a conceptual structure, albeit abstract and atemporal, for how new relatively autonomous behavior can emerge within a system obeying certain laws. The focus here is on applications in the life sciences (e.g., selectionist mechanisms) and human sciences (e.g., the generative grammar view of language). (shrink)

There is some consensus among orthodox category theorists that the concept of adjoint functors is the most important concept contributed to mathematics by category theory. We give a heterodox treatment of adjoints using heteromorphisms that parses an adjunction into two separate parts. Then these separate parts can be recombined in a new way to define a cognate concept, the brain functor, to abstractly model the functions of perception and action of a brain. The treatment uses relatively simple category (...) theory and is focused on the interpretation and application of the mathematical concepts. (shrink)

We prove, by using the concept of schematic interpretation, that the natural embedding from the category ISL, of intuitionistic sentential pretheories and i-congruence classes of morphisms, to the category CSL, of classical sentential pretheories and c-congruence classes of morphisms, has a left adjoint, which is related to the double negation interpretation of Gödel-Gentzen, and a right adjoint, which is related to the Law of Excluded Middle. Moreover, we prove that from the left to the right adjoint there (...) is a pointwise epimorphic natural transformation and that since the two endofunctors at CSL, obtained by adequately composing the aforementioned functors, are naturally isomorphic to the identity functor for CSL, the string of adjunctions constitutes an adjoint cylinder. On the other hand, we show that the operators of Lindenbaum-Tarski of formation of algebras from pretheories can be extended to equivalences of categories from the category CSL, respectively, ISL, to the category Bool, of Boolean algebras, respectively, Heyt, of Heyting algebras. Finally, we prove that the functor of regularization from Heyt to Bool has, in addition to its well-known right adjoint (that is, the canonical embedding of Bool into Heyt) a left adjoint, that from the left to the right adjoint there is a pointwise epimorphic natural transformation, and, finally, that such a string of adjunctions constitutes an adjoint cylinder. (shrink)

Since its formal definition over sixty years ago, category theory has been increasingly recognized as having a foundational role in mathematics. It provides the conceptual lens to isolate and characterize the structures with importance and universality in mathematics. The notion of an adjunction (a pair of adjoint functors) has moved to center-stage as the principal lens. The central feature of an adjunction is what might be called "internalization through a universal" based on universal mapping properties. A recently developed (...) "heteromorphic" theory of adjoint functors allows the concepts to be more easily applied empirically. This suggests a conceptual structure, albeit abstract, to model a range of selectionist mechanisms (e.g., in evolution and in the immune system). Closely related to adjoints is the notion of a "brain functor" which abstractly models structures of cognition and action (e.g., the generative grammar view of language). (shrink)

Saunders Mac Lane famously remarked that "Bourbaki just missed" formulating adjoints in a 1948 appendix (written no doubt by Pierre Samuel) to an early draft of Algebre--which then had to wait until Daniel Kan's 1958 paper on adjoint functors. But Mac Lane was using the orthodox treatment of adjoints that only contemplates the object-to-object morphisms within a category, i.e., homomorphisms. When Samuel's treatment is reconsidered in view of the treatment of adjoints using heteromorphisms or hets (object-to-object morphisms between (...) objects in different categories), then he, in effect, isolated the concept of a left representation solving a universal mapping problem. When dualized to obtain the concept of a right representation, the two halves only need to be united to obtain an adjunction. Thus Samuel was only a now-simple dualization away for formulating adjoints in 1948. Apparently, Bodo Pareigis' 1970 text was the first and perhaps only text to give the heterodox "new characterization" (i.e., heteromorphic treatment) of adjoints. Orthodox category theory uses various relatively artificial devices to avoid formally recognizing hets--even though hets are routinely used by the working mathematician. Finally we consider a "philosophical" question as to whether the most important concept in category theory is the notion of an adjunction or the notion of a representation giving a universal mapping property (where adjunctions arise as the special case of a bi-representation of dual universal mapping problems). (shrink)

The fact that many modal operators are part of an adjunction is probably folklore since the discovery of adjunctions. On the other hand, the natural idea of a minimal propositional calculus extended with a pair of adjoint operators seems to have been formulated only very recently. This recent research, mainly motivated by applications in computer science, concentrates on technical issues related to the calculi and not on the significance of adjunctions in modal logic. It then seems a worthy enterprise (...) (both for these contemporary topical pursuits and also for historical interest) to trace the concept of adjunction back to the origins of the algebraic semantics of modal logic and to make explicit its ubiquity in this branch of mathematics. (shrink)

Looking at the operation of forming neat $\alpha $ -reducts as a functor, with $\alpha $ an infinite ordinal, we investigate when such a functor obtained by truncating $\omega $ dimensions, has a right adjoint. We show that the neat reduct functor for representable cylindric algebras does not have a right adjoint, while that of polyadic algebras is an equivalence. We relate this categorial result to several amalgamation properties for classes of representable algebras. We show that the variety (...) of cylindric algebras of infinite dimension, endowed with the merry go round identities, fails to have the amalgamation property answering a question of Németi’s. We also study two variants of the so-called cylindric polyadic algebras introduced by Ferenczi (all are reducts of polyadic equality algebras, that are also varieties). We show that one is more cylindric than polyadic, and that the other is more polyadic than cylindric. Our classification is determined by results on neat embeddings and amalgamation expressed from the point of view of category theory, thereby witnessing, and, indeed, further emphasizing, the dichotomy between the cylindric and polyadic paradigms. For example, the first class does not have the unique neat embedding property, fails to have the amalgamation property and the neat reduct functor does not have a right adjoint, while the second class has the unique neat emdedding property, the superamalgamation property and the neat reduct functor is strongly invertible. Other results, like first order definability of the class of neat reducts and the class of completely representable algebras, confirming our classification along these lines are presented. (shrink)

We develop a general covariant categorical modeling theory of natural systems’ behavior based on the fundamental functorial processes of representation and localization-globalization. In the first part of this study we analyze the process of representation. Representation constitutes a categorical modeling relation that signifies the semantic bidirectional process of correspondence between natural systems and formal symbolic systems. The notion of formal systems is substantiated by algebraic rings of observable attributes of natural systems. In this perspective, the distinction between simple and complex (...) systems is reflected in the appropriate qualification of their corresponding rings of observables. The crucial distinguishing requirement with respect to the coordinatizing rings of observables has to do with the property of global commutativity. The global information structure representing the behavior of a complex system is modeled functorially in terms of its Spectrum functor. Due to the fact that, the fundamental process of representation is bidirectional by construction, it admits a precise categorical formulation in terms of the syntactic language of adjoint functors, constituting thus, a categorical adjunction. The left adjoint functor of this adjunction signifies the process of encoding the information related with phenomena of natural systems in terms of coordinatizing rings of observables, whereas, the right adjoint functor signifies the inverse process of information decoding, which, can be used for making predictions about the behavior of natural systems. (shrink)

This unique book provides a self-contained conceptual and technical introduction to the theory of differential sheaves. This serves both the newcomer and the experienced researcher in undertaking a background-independent, natural and relational approach to "physical geometry". In this manner, this book is situated at the crossroads between the foundations of mathematical analysis with a view toward differential geometry and the foundations of theoretical physics with a view toward quantum mechanics and quantum gravity. The unifying thread is provided by the theory (...) of adjoint functors in category theory and the elucidation of the concepts of sheaf theory and homological algebra in relation to the description and analysis of dynamically constituted physical geometric spectrums. (shrink)

Rosen's modelling relations constitute a conceptual schema for the understanding of the bidirectional process of correspondence between natural systems and formal symbolic systems. The notion of formal systems used in this study refers to information structures constructed as algebraic rings of observable attributes of natural systems, in which the notion of observable signifies a physical attribute that, in principle, can be measured. Due to the fact that modelling relations are bidirectional by construction, they admit a precise categorical formulation in terms (...) of the category-theoretic syntactic language of adjoint functors, representing the inverse processes of information encoding/decoding via adjunctions. As an application, we construct a topological modelling schema of complex systems. The crucial distinguishing requirement between simple and complex systems in this schema is reflected with respect to their rings of observables by the property of global commutativity. The global information structure representing the behaviour of a complex system is modelled functorially in terms of its spectrum functor. An exact modelling relation is obtained by means of a complex encoding/decoding adjunction restricted to an equivalence between the category of complex information structures and the category of sheaves over a base category of partial or local information carriers equipped with an appropriate topology. (shrink)

We construct a sheaf-theoretic representation of quantum probabilistic structures, in terms of covering systems of Boolean measure algebras. These systems coordinatize quantum states by means of Boolean coefficients, interpreted as Boolean localization measures. The representation is based on the existence of a pair of adjoint functors between the category of presheaves of Boolean measure algebras and the category of quantum measure algebras. The sheaf-theoretic semantic transition of quantum structures shifts their physical significance from the orthoposet axiomatization at the (...) level of events, to the sheaf-theoretic gluing conditions at the level of Boolean localization systems. (shrink)

In this paper we adopt a category-theoretic viewpoint in order to analyze the semantics of complementarity for quantum systems. Based on the existence of a pair of adjoint functors between the topos of presheaves of the Boolean kind of structure and the category of the quantum kind of structure, we establish a twofold complementarity scheme which constitutes an instance of the concept of adjunction. It is further argued that the established scheme is inextricably connected with a realistic philosophical (...) attitude, although substantially different from the classical one. (shrink)

This volume is the first ever collection devoted to the field of proof-theoretic semantics. Contributions address topics including the systematics of introduction and elimination rules and proofs of normalization, the categorial characterization of deductions, the relation between Heyting's and Gentzen's approaches to meaning, knowability paradoxes, proof-theoretic foundations of set theory, Dummett's justification of logical laws, Kreisel's theory of constructions, paradoxical reasoning, and the defence of model theory. The field of proof-theoretic semantics has existed for almost 50 years, but the term (...) itself was proposed by Schroeder-Heister in the 1980s. Proof-theoretic semantics explains the meaning of linguistic expressions in general and of logical constants in particular in terms of the notion of proof. This volume emerges from presentations at the Second International Conference on Proof-Theoretic Semantics in Tübingen in 2013, where contributing authors were asked to provide a self-contained description and analysis of a significant research question in this area. The contributions are representative of the field and should be of interest to logicians, philosophers, and mathematicians alike. (shrink)

Category Theory has developed rapidly. This book aims to present those ideas and methods which can now be effectively used by Mathe­ maticians working in a variety of other fields of Mathematical research. This occurs at several levels. On the first level, categories provide a convenient conceptual language, based on the notions of category, functor, natural transformation, contravariance, and functor category. These notions are presented, with appropriate examples, in Chapters I and II. Next comes the fundamental idea of an (...) class='Hi'>adjoint pair of functors. This appears in many substantially equivalent forms: That of universal construction, that of direct and inverse limit, and that of pairs offunctors with a natural isomorphism between corresponding sets of arrows. All these forms, with their interrelations, are examined in Chapters III to V. The slogan is "Adjoint functors arise everywhere". Alternatively, the fundamental notion of category theory is that of a monoid -a set with a binary operation of multiplication which is associative and which has a unit; a category itself can be regarded as a sort of general­ ized monoid. Chapters VI and VII explore this notion and its generaliza­ tions. Its close connection to pairs of adjoint functors illuminates the ideas of universal algebra and culminates in Beck's theorem characterizing categories of algebras; on the other hand, categories with a monoidal structure lead inter alia to the study of more convenient categories of topological spaces. (shrink)

Fundamental properties of N-valued logics are compared and eleven theorems are presented for their Logic Algebras, including Łukasiewicz–Moisil Logic Algebras represented in terms of categories and functors. For example, the Fundamental Logic Adjunction Theorem allows one to transfer certain universal, or global, properties of the Category of Boolean Algebras,, (which are well-understood) to the more general category \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal L}$$\end{document}Mn of Łukasiewicz–Moisil Algebras. Furthermore, the relationships of LMn-algebras to other many-valued logical (...) structures, such as the n-valued Post, MV and Heyting logic algebras, are investigated and several pertinent theorems are derived. Applications of Łukasiewicz–Moisil Algebras to biological problems, such as nonlinear dynamics of genetic networks – that were previously reported – are also briefly noted here, and finally, probabilities are precisely defined over LMn-algebras with an eye to immediate, possible applications in biostatistics. (shrink)

The overwhelming majority of the attempts in exploring the problems related to quantum logical structures and their interpretation have been based on an underlying set-theoretic syntactic language. We propose a transition in the involved syntactic language to tackle these problems from the set-theoretic to the category-theoretic mode, together with a study of the consequent semantic transition in the logical interpretation of quantum event structures. In the present work, this is realized by representing categorically the global structure of a quantum algebra (...) of events (or propositions) in terms of sheaves of local Boolean frames forming Boolean localization functors. The category of sheaves is a topos providing the possibility of applying the powerful logical classification methodology of topos theory with reference to the quantum world. In particular, we show that the topos-theoretic representation scheme of quantum event algebras by means of Boolean localization functors incorporates an object of truth values, which constitutes the appropriate tool for the definition of quantum truth-value assignments to propositions describing the behavior of quantum systems. Effectively, this scheme induces a revised realist account of truth in the quantum domain of discourse. We also include an Appendix, where we compare our topos-theoretic representation scheme of quantum event algebras with other categorial and topos-theoretic approaches. (shrink)

In this paper I compare two well studied approaches to topological semantics – the domain-theoretic approach, exemplified by the category of countably based equilogical spaces, Equ and Typ Two Effectivity, exemplified by the category of Baire space representations, Rep . These two categories are both locally cartesian closed extensions of countably based T0-spaces. A natural question to ask is how they are related.First, we show that Rep is equivalent to a full coreflective subcategory of Equ, consisting of the so-called 0-equilogical (...) spaces. This establishes a pair of adjoint functors between Rep and Equ. The inclusion Rep → Equ and its coreflection have many desirable properties, but they do not preserve exponentials in general. This means that the cartesian closed structures of Rep and Equ are essentially different. How ever, in a second comparison we show that Rep and Equ do share a common cartesian closed subcategory that contains all countably based-T0 spaces. Therefore, the domain-theoretic approach and TTE yield equivalent topological semantics of computation for all higher-order types over countably based T0-spaces. We consider several examples involving the natural numbers and the real numbers to demonstrate how these comparisons make it possible to transfer results from one setting to another. (shrink)

In this paper we present a general theory of normal forms, based on a categorial result for the free monoid construction. We shall use the theory mainly for proposictional modal logic, although it seems to have a wider range of applications. We shall formally represent normal forms as combinatorial objects, basically labelled trees and forests. This geometric conceptualization is implicit in and our approach will extend it to other cases and make it more direct: operations of a purely geometric and (...) combinatorial nature will be introduced in order to give a mathematical description of the basic logical/algebraic constructions .We begin by recalling the above-mentioned categorial construction: we need a careful inspection of it because in the various examples considered later we plan to deduce from it in a uniform way the normal forms and the description of finitely generated free algebras. This method always works whenever we can describe the category of algebras corresponding to the logic under consideration as a T-objects category. When this simple description seems not to be available, still the general theory might be of some interest, because a description of the category of algebras as a T-objects category plus equation is possible .The central part of the paper is more advanced and specific: we show how the general approach presented here can provide some insights even in the basic case of the modal system K. Section 4 contains a contribution to the theory of normal forms, namely the description of the uniform substitution. This result will enable us to give a duality theorem for the category of finitely generated free modal algebras and in Section 5 to provide a characterization of the collections of normal forms which happen to be normal forms for a logic, thus giving a description of the lattice of modal logics.Section 6 deals with some applications: we shall show how to use normal forms in order to prove for the modal system K the definability of higher-order propositional quantifiers and of the tense operator F .As to the prerequisites, the paper is almost self-contained. The reader is only assumed to have familiarity with standard techniques in algebraic logic ). Knowledge of the basic facts about adjoint functors is required too, see e.g. McLane or the appendix. (shrink)

The interaction between syntax and its semantics is one which has been well studied in categorical logic. The results of this particular study are employed to understand how the brain is able to create meanings. To emphasize the toy character of the proposed model, we prefer to speak of the homunculus brain rather than the brain per se. The homunculus brain consists of neurons, each of which is modeled by a category, and axons between neurons, which are modeled by (...) class='Hi'>functors between the corresponding neuron-categories. Each neuron has its own program enabling its working, i.e. a theory of this neuron. In analogy to what is known from categorical logic, we postulate the existence of a pair of adjoint functors, called Lang and Syn, from a category, now called BRAIN, of categories, to a category, now called MIND, of theories. Our homunculus is a kind of “mathematical robot”, the neuronal architecture of which is not important. Its only aim is to provide us with the opportunity to study how such a simple brain-like structure could “create meanings” and perform abstraction operations out of its purely syntactic program. The pair of adjoint functors Lang and Syn model the mutual dependencies between the syntactical structure of a given theory of MIND and the internal logic of its semantics given by a category of BRAIN. In this way, a formal language and its meanings are interwoven with each other in a manner corresponding to the adjointness of the functors Lang and Syn. Higher cognitive functions of abstraction and realization of concepts are also modelled by a corresponding pair of adjoint functors. The categories BRAIN and MIND interact with each other with their entire structures and, at the same time, these very structures are shaped by this interaction. (shrink)

Among the main concerns of 20th century philosophy was that of the foundations of mathematics. But usually not recognized is the relevance of the choice of a foundational approach to the other main problems of 20th century philosophy, i.e., the logical structure of language, the nature of scientific theories, and the architecture of the mind. The tools used to deal with the difficulties inherent in such problems have largely relied on set theory and its “received view”. There are specific issues, (...) in philosophy of language, epistemology and philosophy of mind, where this dependence turns out to be misleading. The same issues suggest the gain in understanding coming from category theory, which is, therefore, more than just the source of a “non-standard” approach to the foundations of mathematics. But, even so conceived, it is the very notion of what a foundation has to be that is called into question. The philosophical meaning of mathematics is no longer confined to which first principles are assumed and which “ontological” interpretation is given to them in terms of some possibly updated version of logicism, formalism or intuitionism. What is central to any foundational project proper is the role of universal constructions that serve to unify the different branches of mathematics, as already made clear in 1969 by Lawvere. Such universal constructions are best expressed by means of adjoint functors and representability up to isomorphism. In this lies the relevance of a category-theoretic perspective, which leads to wide-ranging consequences. One such is the presence of functorial constraints on the syntax–semantics relationships; another is an intrinsic view of (constructive) logic, as arises in topoi and, subsequently, in more general fibrations. But as soon as theories and their models are described accordingly, a new look at the main problems of 20th century’s philosophy becomes possible. The lack of any satisfactory solution to these problems in a purely logical and set-theoretic setting is the result of too circumscribed an approach, such as a static and punctiform view of objects and their elements, and a misconception of geometry and its historical changes before, during, and after the foundational “crisis”, as if algebraic geometry and synthetic differential geometry – not to mention algebraic topology – were secondary sources for what concerns foundational issues. The objectivity of basic geometrical intuitions also acts against the recent version of structuralism proposed as ‘the’ philosophy of category theory. On the other hand, the need for a consistent and adequate conceptual framework in facing the difficulties met by pre-categorical theories of language and scientific knowledge not only provides the basic concepts of category theory with specific applications but also suggests further directions for their development (e.g., in approaching the foundations of physics or the mathematical models in the cognitive sciences). This ‘virtuous’ circle is by now largely admitted in theoretical computer science; the time is ripe to realise that the same holds for classical topics of philosophy. (shrink)

Transitive extensional well founded relations provide an intuitionistic notion of ordinals which admits transfinite induction. However these ordinals are not directed and their successor operation is poorly behaved, leading to problems of functoriality. We show how to make the successor monotone by introducing plumpness, which strengthens transitivity. This clarifies the traditional development of successors and unions, making it intuitionistic; even the (classical) proof of trichotomy is made simpler. The definition is, however, recursive, and, as their name suggests, the plump ordinals (...) grow very rapidly. Directedness must be defined hereditarily. It is orthogonal to the other four conditions, and the lower powerdomain construction is shown to be the universal way of imposing it. We treat ordinals as order-types, and develop a corresponding set theory similar to Osius' transitive set objects. This presents Mostowski's theorem as a reflection of categories, and set-theoretic union is a corollary of the adjoint functor theorem. Mostowski's theorem and the rank for some of the notions of ordinal are formulated and proved without the axiom of replacement, but this seems to be unavoidable for the plump rank. The comparison between sets and toposes is developed as far as the identification of replacement with completeness, and there are some suggestions for further work in this area. Each notion of set or ordinal defines a free algebra for one of the theories discussed by Joyal and Moerdijk, namely joins of a family of arities together with an operation s satisfying conditions such as x ≤ sx, monotonicity or s(x ∨ y) ≤ sx ∨ sy. Finally we discuss the fixed point theorem for a monotone endofunction s of a poset with least element and directed joins. This may be proved under each of a variety of additional hypotheses. We explain why it is unlikely that any notion of ordinal obeying the induction scheme for arbitrary predicates will prove the pure result. (shrink)

The simple connection of completeness and cocompleteness of lattices grows in categories into the Adjoint Functor Theorem. The connection of completeness and cocompleteness of Boolean algebras — even simpler — is similarly related to Paré's Theorem for toposes. We explain these relations, and then study the fibrational versions of both these theorems — for small complete categories. They can be interpreted as definability results in logic with proofs-as-constructions, and transferred to type theory.

The category-theoretic representation of quantum event structures provides a canonical setting for confronting the fundamental problem of truth valua- tion in quantum mechanics as exemplified, in particular, by Kochen-Specker’s theorem. In the present study, this is realized on the basis of the existence of a categorical adjunction between the category of sheaves of variable local Boolean frames, constituting a topos, and the category of quantum event al- gebras. We show explicitly that the latter category is equipped with an object of (...) truth values, or classifying object, which constitutes the appropriate tool for assigning truth values to propositions describing the behavior of quantum systems. Effectively, this category-theoretic representation scheme circumvents consistently the semantic ambiguity with respect to truth valuation that is in- herent in conventional quantum mechanics by inducing an objective contextual account of truth in the quantum domain of discourse. The philosophical im- plications of the resulting account are analyzed. We argue that it subscribes neither to a pragmatic instrumental nor to a relative notion of truth. Such an account essentially denies that there can be a universal context of reference or an Archimedean standpoint from which to evaluate logically the totality of facts of nature. In this light, the transcendence condition of the usual concep- tion of correspondence truth is superseded by a reflective-like transcendental reasoning of the proposed account of truth that is suitable to the quantum domain of discourse. (shrink)

We propose a sheaf-theoretic framework for the representation of a quantum observable structure in terms of Boolean information sieves. The algebraic representation of a quantum observable structure in the relational local terms of sheaf theory effectuates a semantic transition from the axiomatic set-theoretic context of orthocomplemented partially ordered sets, la Birkhoff and Von Neumann, to the categorical topos-theoretic context of Boolean information sieves, la Grothendieck. The representation schema is based on the existence of a categorical adjunction, which is used as (...) a theoretical platform for the development of a functorial formulation of information transfer, between quantum observables and Boolean localisation devices in typical quantum measurement situations. We also establish precise criteria of integrability and invariance of quantum information transfer by cohomological means. (shrink)

We develop a relativistic perspective on structures of quantum observables, in terms of localization systems of Boolean coordinatizing charts. This perspective implies that the quantum world is comprehended via Boolean reference frames for measurement of observables, pasted together along their overlaps. The scheme is formalized categorically, as an instance of the adjunction concept. The latter is used as a framework for the specification of a categorical equivalence signifying an invariance in the translational code of communication between Boolean localizing contexts and (...) quantum systems. Aspects of the scheme semantics are discussed in relation to logic. The interpretation of coordinatizing localization systems, as structure sheaves, provides the basis for the development of an algebraic differential geometric machinery suited to the quantum regime. (shrink)

In this paper we analyze the physical semantics and propose an interpretation of quantum event structures from the perspective offered by the categorical scheme of Part I.

In an attempt to probe the objects belonging to the quantum species of structure,we develop the idea of using observables of the Boolean species of structures,as coordinatizing objects in the quantum world. This results in a contextualisticperspective on the latter through local Boolean measurement reference frames.The semantics of this representation is discussed extensively.

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)

In abstract algebraic logic, many systems, such as those paraconsistent logics taking inspiration from da Costa's hierarchy, are not algebraizable by even the broadest standard methodologies, as that of Blok and Pigozzi. However, these logics can be semantically characterized by means of non-deterministic algebraic structures such as Nmatrices, RNmatrices and swap structures. These structures are based on multialgebras, which generalize algebras by allowing the result of an operation to assume a non-empty set of values. This leads to an interest in (...) exploring the foundations of multialgebras applied to the study of logic systems. -/- It is well known from universal algebra that, for every signature Sigma, there exist algebras over Sigma which are absolutely free, meaning that they do not satisfy any identities or, alternatively, satisfy the universal mapping property for the class of Sigma-algebras. Furthermore, once we fix a cardinality of the generating set, they are, up to isomorphisms, unique, and equal to algebras of terms (or propositional formulas, in the context of logic). Equivalently, the forgetful functor, from the category of Sigma-algebras to Set, has a left adjoint. This result does not extend to multialgebras. Not only multialgebras satisfying the universal mapping property do not exist, but the forgetful functor U, from the category of Sigma-multialgebras to Set, does not have a left adjoint. -/- In this paper we generalize, in a natural way, algebras of terms to multialgebras of terms, whose family of submultialgebras enjoys many properties of the former. One example is that, to every pair consisting of a function, from a submultialgebra of a multialgebra of terms to another multialgebra, and a collection of choices (which selects how a homomorphism approaches indeterminacies), there corresponds a unique homomorphism, what resembles the universal mapping property. Another example is that the multialgebras of terms are generated by a set that may be viewed as a strong basis, which we call the ground of the multialgebra. Submultialgebras of multialgebras of terms are what we call weakly free multialgebras. Finally, with these definitions at hand, we offer a simple proof that multialgebras with the universal mapping property for the class of all multialgebras do not exist and that U does not have a left adjoint. (shrink)

Motivated by an old construction due to J. Kalman that relates distributive lattices and centered Kleene algebras we define the functor K • relating integral residuated lattices with 0 with certain involutive residuated lattices. Our work is also based on the results obtained by Cignoli about an adjunction between Heyting and Nelson algebras, which is an enrichment of the basic adjunction between lattices and Kleene algebras. The lifting of the functor to the category of residuated lattices leads us to study (...) other adjunctions and equivalences. For example, we treat the functor C whose domain is cuRL, the category of involutive residuated lattices M whose unit is fixed by the involution and has a Boolean complement c. If we restrict to the full subcategory NRL of cuRL of those objects that have a nilpotent c, then C is an equivalence. In fact, C M is isomorphic to C e M, and C e is adjoint to, where assigns to an object A of IRL0 the product A × A 0 which is an object of NRL. (shrink)

In abstract algebraic logic, many systems, such as those paraconsistent logics taking inspiration from da Costa's hierarchy, are not algebraizable by even the broadest standard methodologies, as that of Blok and Pigozzi. However, these logics can be semantically characterized by means of non-deterministic algebraic structures such as Nmatrices, RNmatrices and swap structures. These structures are based on multialgebras, which generalize algebras by allowing the result of an operation to assume a non-empty set of values. This leads to an interest in (...) exploring the foundations of multialgebras applied to the study of logic systems. It is well known from universal algebra that, for every signature \, there exist algebras over \ which are absolutely free, meaning that they do not satisfy any identities or, alternatively, satisfy the universal mapping property for the class of \-algebras. Furthermore, once we fix a cardinality of the generating set, they are, up to isomorphisms, unique, and equal to algebras of terms. Equivalently, the forgetful functor, from the category of \-algebras to Set, has a left adjoint. This result does not extend to multialgebras. Not only multialgebras satisfying the universal mapping property do not exist, but the forgetful functor \, from the category of \-multialgebras to Set, does not have a left adjoint. In this paper we generalize, in a natural way, algebras of terms to multialgebras of terms, whose family of submultialgebras enjoys many properties of the former. One example is that, to every pair consisting of a function, from a submultialgebra of a multialgebra of terms to another multialgebra, and a collection of choices, there corresponds a unique homomorphism, what resembles the universal mapping property. Another example is that the multialgebras of terms are generated by a set that may be viewed as a strong basis, which we call the ground of the multialgebra. Submultialgebras of multialgebras of terms are what we call weakly free multialgebras. Finally, with these definitions at hand, we offer a simple proof that multialgebras with the universal mapping property for the class of all multialgebras do not exist and that \ does not have a left adjoint. (shrink)

We show that for any uncountable cardinal λ, the category of sets of cardinality at least λ and monomorphisms between them cannot appear as the category of points of a topos, in particular is not the category of models of a ${L_{\infty,\omega }}$-theory. More generally we show that for any regular cardinal $\kappa < \lambda$ it is neither the category of κ-points of a κ-topos, in particular, nor the category of models of a ${L_{\infty,\kappa }}$-theory.The proof relies on the construction (...) of a categorified version of the Scott topology, which constitute a left adjoint to the functor sending any topos to its category of points and the computation of this left adjoint evaluated on the category of sets of cardinality at least λ and monomorphisms between them. The same techniques also apply to a few other categories. At least to the category of vector spaces of with bounded below dimension and the category of algebraic closed fields of fixed characteristic with bounded below transcendence degree. (shrink)

In order to understand negation as such, at least since Aristotle’s time, there have been many ways of conceptually modelling it. In particular, negation has been studied as inconsistency, contradictoriness, falsity, cancellation, an inversion of arrangements of truth values, etc. In this paper, making substantial use of category theory, we present three more conceptual and abstract models of negation. All of them capture negation as turning upside down the entire structure under consideration. The first proposal turns upside down the structure (...) almost literally; it is the well known construction of opposite category. The second one treats negation as a contravariant functor and the third one captures negation as adjointness. Traditionally, negation was investigated in the context of language as negation of sentences or parts of sentences, e.g. names. On the contrary we propose to negate structures globally. As a consequence of our approach we provide a solution to the ontological problem of the existence of negative states of affairs. (shrink)

This paper is about equality of proofs in which a binary predicate formalizing properties of equality occurs, besides conjunction and the constant true proposition. The properties of equality in question are those of a preordering relation, those of an equivalence relation, and other properties appropriate for an equality relation in linear logic. The guiding idea is that equality of proofs is induced by coherence, understood as the existence of a faithful functor from a syntactical category into a category whose arrows (...) correspond to diagrams. Edges in these diagrams join occurrences of variables that must remain the same in every generalization of the proof. It is found that assumptions about equality of proofs for equality are parallel to standard assumptions about equality of arrows in categories. They reproduce standard categorial assumptions on a different level. It is also found that assumptions for a preordering relation involve an adjoint situation. (shrink)

After proving, in a purely categorial way, that the inclusion functor |$\textrm {In}_{\textbf {Alg}(\varSigma )}$| from |$\textbf {Alg}(\varSigma )$|⁠, the category of many-sorted |$\varSigma $|-algebras, to |$\textbf {PAlg}(\varSigma )$|⁠, the category of many-sorted partial |$\varSigma $|-algebras, has a left adjoint |$\textbf {F}_{\varSigma }$|⁠, the (absolutely) free completion functor, we recall, in connection with the functor |$\textbf {F}_{\varSigma }$|⁠, the generalized recursion theorem of Schmidt, which we will also call the Schmidt construction. Next, we define a category |$\textbf {Cmpl}(\varSigma )$|⁠, (...) of |$\varSigma $|-completions, and prove that |$\textbf {F}_{\varSigma }$|⁠, labelled with its domain category and the unit of the adjunction of which it is a part, is a weakly initial object in it. Following this, we associate to an ordered pair |$(\boldsymbol {\alpha },f)$|⁠, where |$\boldsymbol {\alpha }=(K,\gamma,\alpha )$| is a morphism of |$\varSigma $|-completions from |${{\mathscr {F}}}=(\textbf {C},F,\eta )$| to |$\mathscr {G}= (\textbf {D},G,\rho )$| and |$f$| a homomorphism of |$\textbf {D}$| from the partial |$\varSigma $|-algebra |$\textbf {A}$| to the partial |$\varSigma $|-algebra |$\textbf {B}$|⁠, a homomorphism |$\varUpsilon ^{\mathscr {G},0}_{\boldsymbol {\alpha }}(f)\colon \textbf {Sch}_{\boldsymbol {\alpha }}(f)\longrightarrow \textbf {B}$|⁠. We then prove that there exists an endofunctor, |$\varUpsilon ^{\mathscr {G},0}_{\boldsymbol {\alpha }}$|⁠, of |$\textbf {Mor}_{\textrm {tw}}(\textbf {D})$|⁠, the twisted morphism category of |$\textbf {D}$|⁠, thus showing the naturalness of the previous construction. Afterwards, we prove that, for every |$\varSigma $|-completion |$\mathscr {G}=(\textbf {D},G,\rho )$|⁠, there exists a functor |$\varUpsilon ^{\mathscr {G}}$| from the comma category |$(\textbf {Cmpl}(\varSigma )\!\downarrow \!\mathscr {G})$| to |$\textbf {End}(\textbf {Mor}_{\textrm {tw}}(\textbf {D}))$|⁠, the category of endofunctors of |$\textbf {Mor}_{\textrm {tw}}(\textbf {D})$|⁠, such that |$\varUpsilon ^{\mathscr {G},0}$|⁠, the object mapping of |$\varUpsilon ^{\mathscr {G}}$|⁠, sends a morphism of |$\varSigma $|-completion of |$\textbf {Cmpl}(\varSigma )$| with codomain |$\mathscr {G}$|⁠, to the endofunctor |$\varUpsilon ^{\mathscr {G},0}_{\boldsymbol {\alpha }}$|⁠. (shrink)

The aim of this paper is to contribute to the clarification of concepts usually found in books on quantum mechanics, aided by knowledge from the field of the theory of operators in Hilbert space. Frequently the basic distinction between bounded and unbounded operators is not established in books on quantum mechanics. It is repeatedly overlooked that the condition for an unbounded operator to be symmetric (Hermitian) is not sufficient to make it self-adjoint. To make things worse, nearly all operators (...) in quantum mechanics are unbounded. Often one finds statements such as: For any linear operator A we can write a Hermitian operator HA=(A+A+)/2, where Hermitian is thought to mean self-adjoint. Along these lines, self-adjointness of the momentum operator in generalized coordinates, taken from that expression, is questioned. In particular, the redescription in terms of spherical polar coordinates and its implications for the eventual loss of self-adjointness of the momenta conjugate to them are studied. (shrink)

The aim of this paper is to give a general background and a uniform treatment of several notions of mutual interpretability. Sentential calculi are treated as preorders and logical invariants of adjoint situations, i.e. Galois connections are investigated. The class of all sentential calculi is treated as a quasiordered class.Some methods of the axiomatization of the M-counterparts of modal systems are based on particular adjoints. Also, invariants concerning adjoints for calculi with implication are pointed out. Finally, the notion of (...) interpretability is generalized so that it may be applied to closure spaces as well. (shrink)

Following a proposal of Humberstone, this paper studies a semantics for modal logic based on partial “possibilities” rather than total “worlds.” There are a number of reasons, philosophical and mathematical, to find this alternative semantics attractive. Here we focus on the construction of possibility models with a finitary flavor. Our main completeness result shows that for a number of standard modal logics, we can build a canonical possibility model, wherein every logically consistent formula is satisfied, by simply taking each individual (...) finite formula (modulo equivalence) to be a possibility, rather than each infinite maximally consistent set of formulas as in the usual canonical world models. Constructing these locally finite canonical models involves solving a problem in general modal logic of independent interest, related to the study of adjoint pairs of modal operators: for a given modal logic L, can we find for every formula φ a formula f(φ) such that for every formula ψ, φ → []ψ is provable in L if and only if f(φ) → ψ is provable in L? We answer this question for a number of standard modal logics, using model-theoretic arguments with world semantics. This second main result allows us to build for each logic a canonical possibility model out of the lattice of formulas related by provable implication in the logic. (shrink)

Following a proposal of Humberstone, this paper studies a semantics for modal logic based on partial “possibilities” rather than total “worlds.” There are a number of reasons, philosophical and mathematical, to find this alternative semantics attractive. Here we focus on the construction of possibility models with a finitary flavor. Our main completeness result shows that for a number of standard modal logics, we can build a canonical possibility model, wherein every logically consistent formula is satisfied, by simply taking each individual (...) finite formula (modulo equivalence) to be a possibility, rather than each infinite maximally consistent set of formulas as in the usual canonical world models. Constructing these locally finite canonical models involves solving a problem in general modal logic of independent interest, related to the study of adjoint pairs of modal operators: for a given modal logic L, can we find for every formula phi a formula f(phi) such that for every formula psi, phi -> BOX psi is provable in L if and only if f(phi) -> psi is provable in L? We answer this question for a number of standard modal logics, using model-theoretic arguments with world semantics. This second main result allows us to build for each logic a canonical possibility model out of the lattice of formulas related by provable implication in the logic. (shrink)