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)

Category theory has foundational importance because it provides conceptual lenses to characterize what is important and universal in mathematics—with adjunctions being the primary lens. If adjunctions are so important in mathematics, then perhaps they will isolate concepts of some importance in the empirical sciences. But the applications of adjunctions have been hampered by an overly restrictive formulation that avoids heteromorphisms or hets. By reformulating an adjunction using hets, it is split into two parts, a left and a right semiadjunction. Semiadjunctions (...) (essentially a formulation of universal mapping properties using hets) can then be combined in a new way to define the notion of a brain functor that provides an abstract model of the intentionality of perception and action (as opposed to the passive reception of sense-data or the reflex generation of behavior). (shrink)

In this document, we introduce a novel formalism for any field theory and apply it to the effective field theories of large-scale structure. The new formalism is based on functors of actions composing those theories. This new formalism predicts the actionic fields. We discuss our findings in a cosmological gravitology framework. We present these results with a cosmological inference approach and give guidelines on how we can choose the best candidate between those models with some latest understanding of model selection (...) using Bayesian inference. (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 construct a 2-equivalence \(\mathfrak {CohTheory}^{op }\simeq \mathfrak {TypeSpaceFunc}\). Here \(\mathfrak {CohTheory}\) is the 2-category of positive theories and \(\mathfrak {TypeSpaceFunc}\) is the 2-category of type space functors. We give a precise definition of interpretations for positive logic, which will be the 1-cells in \(\mathfrak {CohTheory}\). The 2-cells are definable homomorphisms. The 2-equivalence restricts to a duality of categories, making precise the philosophy that a theory is ‘the same’ as the collection of its type spaces (i.e. its type space (...) class='Hi'>functor). In characterising those functors that arise as type space functors, we find that they are specific instances of (coherent) hyperdoctrines. This connects two different schools of thought on the logical structure of a theory. The key ingredient, the Deligne completeness theorem, arises from topos theory, where positive theories have been studied under the name of coherent theories. (shrink)

Fried and Kollár constructed a fully faithful functor from the category of graphs to the category of fields. We give a new construction of such a functor and use it to resolve a longstanding open problem in computable model theory, by showing that for every nontrivial countable structure${\cal S}$, there exists a countable field${\cal F}$of arbitrary characteristic with the same essential computable-model-theoretic properties as${\cal S}$. Along the way, we develop a new “computable category theory”, and prove that our (...) functor and its partially defined inverse are computable functors. (shrink)

This paper is an attempt to solve the following problem: given a logic, how to turn it into a paraconsistent one? In other words, given a logic in which ex falso quodlibet holds, how to convert it into a logic not satisfying this principle? We use a framework provided by category theory in order to define a category of consequence structures. Then, we propose a functor to transform a logic not able to deal with contradictions into a paraconsistent one. (...) Moreover, we study the case of paraconsistentization of propositional classical logic. (shrink)

Several Gentzen-style syntactic type calculi with product are considered. They form a hierarchy in such a way that one calculus results from another by imposing a new condition upon the sequent-forming operation. It turns out that, at some steps of this process, two different functors collapse to a single one. For the remaining stages of the hierarchy, analogues of Wajsbergs's theorem on non-mutual-definability are proved.

Given some links between Lyndon’s Interpolation Theorem, term distribution, and Sommers and Englebretsen’s logic, in this contribution we attempt to capture a sense of interpolation for Sommers and Englebretsen’s Term Functor Logic. In order to reach this goal we first expound the basics of Term Functor Logic, together with a sense of term distribution, and then we offer a proof of our main contribution.

We prove a version of the associated sheaf functor theorem in Algebraic Set Theory. The proof is established working within a Heyting pretopos equipped with a system of small maps satisfying the axioms originally introduced by Joyal and Moerdijk. This result improves on the existing developments by avoiding the assumption of additional axioms for small maps and the use of collection sites.

It is often said that a person P knows the meaning of a sentence S if P knows S’ s truth-conditions, in the sense that given any possible world, P knows whether S is true in that world. This idea of sentence-meaning corresponds fairly closely to what Frege, Russell, Carnap, and other philosophers have had in mind in speaking of the senses, propositional contents, or “locutionary” meanings of sentences; and, not unnaturally, it has encouraged semanticists such as David Lewis, Robert (...) Stalnaker and Max Cresswell to suggest that sentence-meanings or propositions simply are functions from the set of possible worlds onto the truth-values. (shrink)

This work gives an extended presentation of the treatment of variable-binding operators adumbrated in [3:1993d]. Illustrative examples include elementary languages with quantifiers and lambda-equipped categorial languages. Some remarks are also offered to illustrate the philosophical import of the resulting picture. Particularly, a certain conception of logic emerges from the account: the view that logics are true theories in the model-theoretic sense, i.e. the result of selecting a certain class of models as the only “admissible” interpretation structures (for a given language).

This article suggests a novel way to advance a current debate in the philosophy of mathematics. The debate concerns the role of diagrams and visual reasoning in proofs—which I take to concern the criteria of legitimate representation of mathematical thought. Drawing on the so-called ‘maverick’ approach to philosophy of mathematics, I turn to mathematical practice itself to adjudicate in this debate, and in particular to category theory, because there (a) diagrams obviously play a major role, and (b) category theory itself (...) addresses questions of representation and information preservation over mappings. We obtain a mathematical answer to a philosophical question: a good mathematical representation can be characterized as a category theoretic natural transformation. (shrink)

Alain Badiou’s treatment of objects in Logics of Worlds is both rich and highly technical, though its terminological challenges are softened by his use of illuminating examples. This article takes a twofold approach to the topic. In a first sense, the theory of objects developed in Logics of Worlds by way of an imagined protest at the Place de la République in Paris exhibits two questionable aspects: (1) the notion that the object is a bundle of qualities (found proverbially in (...) Hume, but also in Kant’s “transcendental object=X”), and (2) the ultimately idealist assumption of a possible isomorphy between appearance and reality. But in a second sense, Badiou’s transcendental account of worlds leads him to a fascinating theory of exemplary entities, one that is immune to the critiques of onto-theology made by Heidegger and Derrida. This can be found in his account of the “transcendental functor” in Alexander the Great’s decisive victory over Darius III at the Battle of Gaugamela in 331 B.C.E. (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 “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)

In this article, we study the commutativity between the pull-back and the push-forward functors on constructible functions in Cluckers–Loeser motivic integration.

The most difficult problem that Leniewski came across in constructing his system of the foundations of mathematics was the problem of defining definitions, as he used to put it. He solved it to his satisfaction only when he had completed the formalization of his protothetic and ontology. By formalization of a deductive system one ought to understand in this context the statement, as precise and unambiguous as possible, of the conditions an expression has to satisfy if it is added to (...) the system as a new thesis. Now, some protothetical theses, and some ontological ones, included in the respective systems, happen to be definitions. In the present essay I employ Leniewski's method of terminological explanations for the purpose of formalizing ukasiewicz's system of implicational calculus of propositions, which system, without having recourse to quantification, I first extended some time ago into a functionally complete system. This I achieved by allowing for a rule of implicational definitions, which enabled me to define any propositionforming functor for any finite number of propositional arguments. (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)