Type Theory - Science topic

This paper explores a novel synthesis between ancient mystical traditions and contemporary mathematical thought. Specifically, it reinterprets the chakra system—a spiritual framework for understanding the human subtle body and its evolution toward divine consciousness—through the rigorous formalisms of modern mathematics. Drawing on sheaf theory, c...

A bstract Recently a one-dimensional Schwarzian type theory was proposed as an effective dual theory of pure gravity in (2+1) dimensional asymptotically flat spacetimes [1]. This codimension-two ‘celestial’ dual captures the Bekenstein-Hawking entropy of bulk flat cosmologies in semiclassical limit. In this paper, we extend this analysis beyond sem...

Canonical is a solver for type inhabitation in dependent type theory, that is, the problem of producing a term of a given type. We present a Lean tactic which invokes Canonical to generate proof terms and synthesize programs. The tactic supports higher-order and dependently-typed goals, structural recursion over indexed inductive types, and definit...

Meta-Epistemic Framework for Inferential Pluralism: A Logic-Consistent Model Beyond Totalization presents a groundbreaking theoretical architecture that challenges classical assumptions in logic, semantics, and epistemology. In an age of increasing complexity and interdisciplinary entanglement, this book proposes a pluralistic, non-totalizing appro...

This is a review of Benjamin, Thibaut; Finster, Eric; Mimram, Samuel Globular weak ω-categories as models of a type theory. (English) Zbl 08006148 High. Struct. 8, No. 2, 1-69 (2024)

This paper critically examines the ongoing debate over the legitimacy of visual arguments and proposes a resolution to this issue. Using a type-theory framework, the legitimacy of visual arguments is addressed through two key sub-problems. First, the paper argues that visual arguments exist, with their existence grounded in dynamic existentialism....

We present Dependent Lambek Calculus, a domain-specific dependent type theory for verified parsing and formal grammar theory. In Dependent Lambek Calculus, linear types are used as a syntax for formal grammars, and parsers can be written as linear terms. The linear typing restriction provides a form of intrinsic verification that a parser yields on...

This paper explores a recurring yet often overlooked feature of mathematics, physics, and theology: the structural and generative role of the midpoint—frequently expressed as the value ½. From the spin-½ behavior of fundamental particles and the critical line in the Riemann zeta function to the equator of the Bloch Sphere and the balance of 0 and 1...

This paper presents a formal framework for understanding emergent intelligence and meaning as structural outcomes of informational systems. By modeling systems that compress, store, and predict information—such as human cognition, artificial intelligence, and cosmological processes—within a categorical and homotopical structure, we show that meanin...

We extend the model structure on the category $\mathbf{Cat}(\mathcal{E})$ of internal categories studied by Everaert, Kieboom and Van der Linden to an algebraic model structure. Moreover, we show that it restricts to the category of internal groupoids. We show that in this case, the algebraic weak factorisation system that consists of the algebraic...

Mathematics, logic, and art converge in their exploration of self-reference, paradox, and incompleteness. Kurt Gödel’s Incompleteness Theorems revealed fundamental limitations within formal mathematical systems, demonstrating that certain truths remain unprovable. M.C. Escher’s artwork visually encapsulates these ideas, using impossible figures and...

Financial markets exhibit complex, multi-scale behaviors that challenge traditional mathematical modeling techniques. Existing tools, such as moving averages, momentum indicators, and statistical models, often fail to capture the continuous transformations, abrupt market shifts, and higher-order dependencies that characterize real-world financial s...

Blockchains are formal systems for equipping objects with value, transacting their exchange, and creating domain-specific event histories. Categorical cryptoeconomics is the application of category-theoretic methods to blockchain study with formalisms which pertain to blockchains and generalize to the programmable computational infrastructure more...

Mathematics has long been considered a universal language, yet its development has been shaped by human cognition, intuition, and historical necessity. This paper explores whether artificial intelligence (AI), when trained on restricted mathematical foundations, reconstructs known mathematical structures or invents alternative formalisms. By limiti...

Given a real reductive group $G$, the purpose of this paper is to show an asymptotic formula of the large-time behavior of the $G$-trace of the heat operator on the associated symmetric spaces. Together with Carmona's proof on Vogan's lambda map, our results provide a geometric counterpart of Vogan's minimal $K$-type theory.

The program of internal type theory seeks to develop the categorical model theory of dependent type theory using the language of dependent type theory itself. In the present work we study internal homotopical type theory by relaxing the notion of a category with families (cwf) to that of a wild, or precoherent higher cwf, and determine coherence co...

In dependent type theory, being able to refer to a type universe as a term itself increases its expressive power, but requires mechanisms in place to prevent Girard's paradox from introducing logical inconsistency in the presence of type-in-type. The simplest mechanism is a hierarchy of universes indexed by a sequence of levels, typically the natur...

Modular development of programs relies on the principle that library code may be freely replaced without affecting client behavior. While an interface mediating this interaction should require a precise behavior of its implementations, allowing for downstream verification of client code, it should do so in a manner that allows private algorithmic a...

This is a review of Benjamin, Thibaut Monoidal weak ω-categories as models of a type theory. (English) £ ¢ ¡ Zbl 07813367 Math. Struct. Comput. Sci. 33, No. 8, 744-780 (2023).

We report on a detailed exploration of the properties of conversion (definitional equality) in dependent type theory, with the goal of certifying decision procedures for it. While in that context the property of normalisation has attracted the most light, we instead emphasize the importance of injectivity properties, showing that they alone are bot...

In this paper, we present a construction from a Reedy category $C$ of a direct category $\operatorname{Down}(C)$ and a functor $\operatorname{Down}(C) \to C$, which exhibits $C$ as an $(\infty,1)$-categorical localization of $\operatorname{Down}(C)$. This result refines previous constructions in the literature by ensuring finiteness of the direct c...

We characterize the epimorphisms in homotopy type theory (HoTT) as the fiberwise acyclic maps and develop a type-theoretic treatment of acyclic maps and types in the context of synthetic homotopy theory as developed in univalent foundations. We present examples and applications in group theory, such as the acyclicity of the Higman group, through th...

Civilizations evolve through complex, nonlinear processes shaped by economic, political, cultural, and technological forces. Traditional historical analysis often relies on reductionist models that focus on isolated causes, failing to capture the emergent, self-organizing nature of societal dynamics. In this paper, we introduce a Homotopy Type Theo...

Modalities in homotopy type theory are used to create and access subuniverses of a given type universe. These have significant applications throughout mathematics and computer science, and in particular can be used to create universes in which certain logical principles are true. We define presentations of topological modalities, which act as an in...

Secure Multi-Party Computation (MPC) is an important enabling technology for data privacy in modern distributed applications. We develop a new type theory to automatically enforce correctness,confidentiality, and integrity properties of protocols written in the \emph{Prelude/Overture} language framework. Judgements in the type theory are predicated...

We use a type theory for omega-categories to produce higher-dimensional generalisations of the Eckmann-Hilton argument. The heart of our construction is a family of padding and repadding techniques, which give a notion of congruence between cells of different types. This gives explicit witnesses in all dimensions that, for cells with degenerate bou...

Replication is an alternative construct to recursion for describing infinite behaviours in the pi-calculus. In this paper we explore the implications of including type-level replication in Multiparty Session Types (MPST), a behavioural type theory for message-passing programs. We introduce MPST!, a session-typed multiparty process calculus with rep...

While ordinals have traditionally been studied mostly in classical frameworks, constructive ordinal theory has seen significant progress in recent years. However, a general constructive treatment of ordinal exponentiation has thus far been missing. We present two seemingly different definitions of constructive ordinal exponentiation in the setting...

Riehl and Shulman introduced simplicial type theory (STT), a variant of homotopy type theory which aimed to study not just homotopy theory, but its fusion with category theory: $(\infty,1)$-category theory. While notoriously technical, manipulating $\infty$-categories in simplicial type theory is often easier than working with ordinary categories,...

We define a naturality construction for the operations of weak {\omega}-categories, as a meta-operation in a dependent type theory. Our construction has a geometrical motivation as a local tensor product, and we realise it as a globular analogue of Reynolds parametricity. Our construction operates as a "power tool" to support construction of terms...

We present a "homotopification" of fundamental concepts from information theory. Using homotopy type theory, we define homotopy types that behave analogously to probability spaces, random variables, and the exponentials of Shannon entropy and relative entropy. The original analytic theories emerge through homotopy cardinality, which maps homotopy t...

The Dependent Calculus of Indistinguishability (DCOI) uses dependency tracking to identify irrelevant arguments and uses indistinguishability during type conversion to enable proof irrelevance, supporting run-time and compile-time irrelevance with the same uniform mechanism. DCOI also internalizes reasoning about indistinguishability through the us...

We introduce a model of simple type theory with potential infinite carrier sets. The functions in this model are automatically continuous, as defined in this paper. This notion of continuity does not rely on topological concepts, including domain theoretic concepts, which essentially use actual infinite sets. The model is based on the concept of a...

Formal reasoning with non-denoting terms, esp. non-referring descriptions such as "the King of France", is still an under-investigated area. The recent exception being a series of papers e.g. by Indrzejczak, Zawidzki and K\"rbis. The present paper offers an alternative to their approach since instead of free logic and sequent calculus, it's framed...

Solution of any engineering problem starts with a modelling process, which provides a model representing a system under consideration. The critical question for practical use of models is to assess if the model and results of using it can be trusted. To address this question, several approaches have been proposed in the past. However, some of these...

Background English complex long sentence optimization in English, a machine translation algorithm (MTA) is a computing system created to mechanically translate intricate, protracted statements from one language to another. The term MTA based on a big data corpus refers to a translation system that uses a large and varied set of textual materials fo...

At first glance, the question of whether AI logic, represented through Topos theory, can outperform Homotopy Type Theory (HoTT) algorithms in automating banana peeling seems whimsical, perhaps even absurd. Yet, beneath the surface lies a profound intersection of mathematics, artificial intelligence, and robotics. The act of peeling a banana—a task...

Mathematics often carries an aura of complexity, deterring many from exploring its beauty and relevance. Yet, at its core, mathematics is a universal language that describes the patterns and principles governing our world. The challenge lies in bridging the gap between abstract mathematical ideas and everyday experiences—a task that can be accompli...

In a recent paper, we highlighted in $w$CDM models derived from general relativity (GR) (with Dark Energy Universe numerical simulation data), a cosmological invariance of the distribution of dark-matter (DM) halo shapes when expressed in terms of the nonlinear fluctuations of the cosmic matter field. This paper shows that this invariance persists...

The Lean mathematical library Mathlib features extensive use of the typeclass pattern for organising mathematical structures, based on Lean’s mechanism of instance parameters. Related mechanisms for typeclasses are available in other provers including Agda, Coq and Isabelle with varying degrees of adoption. This paper analyses representative exampl...

Parametricity is a key metatheoretic property of type systems, which implies strong uniformity & modularity properties of the structure of types within systems possessing it. In recent years, various systems of dependent type theory have emerged with the aim of expressing such parametric reasoning in their internal logic, toward the end of solving...

Polynomials in a category have been studied as a generalization of the traditional notion in mathematics. Their construction has recently been extended to higher groupoids, as formalized in homotopy type theory, by Finster, Mimram, Lucas and Seiller, thus resulting in a cartesian closed bicategory. We refine and extend their work in multiple direct...

Two novel descriptions of weak {\omega}-categories have been recently proposed, using type-theoretic ideas. The first one is the dependent type theory CaTT whose models are {\omega}-categories. The second is a recursive description of a category of computads together with an adjunction to globular sets, such that the algebras for the induced monad...

Pitts generalized nominal sets to finitely supported Cb-sets by utilizing the monoid Cb of name substitutions instead of the monoid of finitary permutations over names. Finitely supported Cb-sets provide a framework for studying essential ideas of models of homotopy type theory at the level of convenient abstract categories. Here, the interplay of...

The language of homotopy type theory has proved to be appropriate as an internal language for various higher toposes, for example with Synthetic Algebraic Geometry for the Zariski topos. In this paper we apply such techniques to the higher topos corresponding to the light condensed sets of Dustin Clausen and Peter Scholze. This seems to be an appro...

This is a foundation for algebraic geometry, developed internal to the Zariski topos, building on the work of Kock and Blechschmidt (Kock (2006) [I.12], Blechschmidt (2017)). The Zariski topos consists of sheaves on the site opposite to the category of finitely presented algebras over a fixed ring, with the Zariski topology, that is, generating cov...

This article constructs the moduli stack of torsion-free $G$ -jet-structures in homotopy type theory with one monadic modality. This yields a construction of this moduli stack for any $\infty$ -topos equipped with any stable factorization systems. In the intended applications of this theory, the factorization systems are given by the deRham-Stack c...

Recent trends in biblical scholarship that have generated new interest in the book of Psalms and in the voice of lamentation may in turn present new opportunities for the liturgical use of psalms of lament. Drawing on the SIFT approach to biblical hermeneutics, the present study tested the ways in which feeling types and thinking types may evaluate...

Finster and Mimram have defined a dependent type theory called CaTT, which describes the structure of omega-categories. Types in homotopy type theory with their higher identity types form weak omega-groupoids, so they are in particular weak omega-categories. In this article, we show that this principle makes homotopy type theory into a model of CaT...

Emergent phenomena in artificial intelligence (AI) systems—such as unexpected strategies, self-organization, and novel behaviors—pose significant challenges for prediction and control. These behaviors arise from complex interactions between components, often defying analysis by traditional tools. Addressing this complexity requires a robust mathema...

In the rapidly evolving landscape of knowledge and innovation, the boundaries between disciplines are becoming increasingly porous. Interdisciplinary collaboration often reveals unexpected equivalences between ideas from seemingly disparate fields, such as metaphysics, artificial intelligence (AI), and quantum mechanics. These connections suggest t...

Despite persistent challenges in women’s political participation, online political participation (OPP) has created new opportunities, with women’s efficacy in changing political processes. However, the results of the studies on OPP and political efficacy (PE) among women are varied. While existing literature suggests a strong correlation between po...

The Barber's Paradox, attributed to Bertrand Russell, exemplifies self-referential inconsistencies in formal systems, questioning whether the barber who shaves everyone not shaving themselves shaves himself. This paper revisits the paradox within the frameworks of set theory, logic, and graph theory. We introduce mathematical formalisms to model se...

We contribute to the theory of (homotopy) colimits inside homotopy type theory. The heart of our work characterizes the connection between colimits in coslices of a universe, called coslice colimits, and colimits in the universe (i.e., ordinary colimits). To derive this characterization, we find an explicit construction of colimits in coslices that...

When working in homotopy type theory and univalent foundations, the traditional role of the category of sets, $\mathcal{Set}$, is replaced by the category $\mathcal{hSet}$ of homotopy sets (h-sets); types with h-propositional identity types. Many of the properties of $\mathcal{Set}$ hold for $\mathcal{hSet}$ ((co)completeness, exactness, local cart...

Polynomials in a category have been studied as a generalization of the traditional notion in mathematics. Their construction has recently been extended to higher groupoids, as formalized in homotopy type theory, by Finster, Mimram, Lucas and Seiller, thus resulting in a cartesian closed bicategory. We refine and extend their work in multiple direct...

Static single assignment form, or SSA, has been the dominant compiler intermediate representation for decades. In this paper, we give a type theory for a variant of SSA, including its equational theory, which are strong enough to validate a variety of control and data flow transformations. We also give a categorical semantics for SSA, and show that...

The aim of this article is to give an expository account of the equivalence between modest sets and partial equivalence relations. Our proof is entirely self-contained in that we do not assume any knowledge of categorical realizability. At the heart of the equivalence lies the subquotient construction on a partial equivalence relation. The subquoti...

Internal language theorems are fundamental in categorical logic, since they express an equivalence between syntax and semantics. One of such theorems was proven by Clairambault and Dybjer, who corrected the result originally by Seely. More specifically, they constructed a biequivalence between the bicategory of locally Cartesian closed categories a...

The Riemann zeta function, a fundamental object in mathematics, bridges number theory, complex analysis, and mathematical physics. Initially studied by Euler and extended by Riemann, it is deeply connected to the distribution of prime numbers and unsolved problems like the Riemann Hypothesis. While its classical domain spans integers, real numbers,...

Recently a one dimensional Schwarzian type theory was proposed as an effective dual theory of pure gravity in (2+1) dimensional asymptotically flat spacetimes. This codimension two `celestial' dual captures the Bekenstein-Hawking entropy of bulk flat cosmologies in semiclassical limit. In this paper, we extend this analysis beyond semiclassical app...

We develop realizability models of intensional type theory, based on groupoids, wherein realizers themselves carry non-trivial (non-discrete) homotopical structure. In the spirit of realizability, this is intended to formalize a homotopical BHK interpretation, whereby evidence for an identification is a path. Specifically, we study partitioned grou...

This paper explain how the geometric notions of local contractibility and properness are related to the $\Sigma$-types and $\Pi$-types constructors of dependent type theory. We shall see how every Grothendieck fibration comes canonically with such a pair of notions—called smooth and proper maps—and how this recovers the previous examples and many m...

In homotopy type theory, few constructions have proved as troublesome as the smash product. While its definition is just as direct as in classical mathematics, one quickly realises that in order to define and reason about functions over iterations of it, one has to verify an exponentially growing number of coherences. This has led to crucial result...

The MBTI (Myers-Briggs Type Indicator) is one of the most widely used personality tests in the world over the past two decades. It is an optional, self-report personality assessment tool that measures and describes people's patterns of mental activity and behavioural tendencies in the areas of information acquisition, decision-making, and attitudes...

This paper critically examines the ongoing debate concerning the existence of visual arguments and proposes a solution to this existential problem. Using the type-theory framework we introduce, the existential problem of visual arguments is discussed in two senses. Firstly, this paper argues that visual arguments exist while their existence is dyna...

The field of directed type theory seeks to design type theories capable of reasoning synthetically about (higher) categories, by generalizing the symmetric identity types of Martin-L\"of Type Theory to asymmetric hom-types. We articulate the directed type theory of the category model, with appropriate modalities for keeping track of variances and a...

Dependent type theory gives an expressive type system facilitating succinct formalizations of mathematical concepts. In practice, it is mainly used for interactive theorem proving with intensional type theories, with PVS being a notable exception. In this paper, we present native rules for automated reasoning in a dependently-typed version (DHOL) o...

Kandoushi or interjection is part of the Japanese word class or 品詞分類 (hinshi bunrui) which functions to express emotions or feelings, such as surprise, happiness, admiration, doubt, and so on. Kandoushi is often found in Japanese language communication and Japanese language media such as anime, comics, films, dramas, and even in games. In Japanese...

Building on our prior work on axiomatization of exact real computation by formalizing nondeterministic first-order partial computations over real and complex numbers in a constructive dependent type theory, we present a framework for certified computation on hyperspaces of subsets by formalizing various higher-order data types and operations. We fi...

We explore recursive programming with extensible data types. Row types make the structure of data types first class, and can express a variety of type system features from subtyping to modular combination of case branches. Our goal is the modular combination of recursive types and of recursive functions over them. The most significant challenge is...

Recently an extension to higher-order logic -- called DHOL -- was introduced, enriching the language with dependent types, and creating a powerful extensional type theory. In this paper we propose two ways how choice can be added to DHOL. We extend the DHOL term structure by Hilbert's indefinite choice operator $\epsilon$, define a translation of t...

The Fuzzy-HoTT Consciousness Model (FHTC) presents a novel approach to understanding the complexity of human thought by integrating fuzzy logic and Homotopy Type Theory (HoTT). Traditional models of cognition often treat ideas and emotions as fixed, binary entities. However, human consciousness is far more nuanced, with ideas existing in ambiguous,...

Awodey, later with Newstead, showed how polynomial pseudomonads $(u,1,\Sigma)$ with extra structure (termed "natural models" by Awodey) hold within them the categorical semantics for dependent type theory. Their work presented these ideas clearly but ultimately led them outside of the category of polynomial functors in order to explain all of the s...

Homotopy Type Theory (HoTT) offers a new perspective for understanding and modeling the complex relationships between musical intervals, harmonics, and resonance in both traditional music theory and modern acoustics. By treating musical elements as types and the relationships between them as paths, HoTT provides a framework for analyzing harmonic s...

The goal of this dissertation is to present results from synthetic homotopy theory based on homotopy type theory (HoTT). After an introduction to Martin-L\"of's dependent type theory and homotopy type theory, key results include a synthetic construction of the Hopf fibration, a proof of the Blakers--Massey theorem, and a derivation of the Freudenth...

Mathematical practice by working mathematicians is rapidly reshaping the structure and form of modern mathematics. Old intuitions relying on set-theoretic mathematics are receding, replaced by new mathematics based on (higher) category theory. Vladimir Voevodsky, a Fields Medalist, proposed in 2010s a project for reconstructing mathematical proofs...

We take another look at the construction by Hofmann and Streicher of a universe $(U,{\mathcal{E}l})$ for the interpretation of Martin-Löf type theory in a presheaf category $[{{{\mathbb{C}}}^{\textrm{op}}},\textsf{Set}]$ . It turns out that $(U,{\mathcal{E}l})$ can be described as the nerve of the classifier $\dot{{\textsf{Set}}}^{\textsf{op}} \rig...

We show how dinaturality plays a central role in the interpretation of directed type theory where types are interpreted as (1-)categories and directed equality is represented by $\hom$-functors. We present a general elimination principle based on dinaturality for directed equality which very closely resembles the $J$-rule used in Martin-L\"of type...

The combination of Topological Superconductors and Homotopy Type Theory (HoTT) presents a groundbreaking framework for achieving fault-tolerant quantum computing. In this framework, Majorana zero modes (MZMs) in topological superconductors provide naturally protected qubits that are resistant to local errors, while HoTT offers a rich mathematical s...

As artificial intelligence (AI) systems become increasingly complex, the need for rigorous mathematical frameworks to improve explainability, reliability, and verification has become more pressing. Traditional approaches, such as set theory, struggle to fully capture the intricate spaces and transformations inherent in modern neural networks. Homot...

Using the language of homotopy type theory (HoTT), we 1) prove a synthetic version of the classification theorem for covering spaces, and 2) explore the existence of canonical change-of-basepoint isomorphisms between homotopy groups. There is some freedom in choosing how to translate concepts from classical algebraic topology into HoTT. The final t...

With the accelerating process of China’s internationalization, Chinese enterprises have more and more opportunities to participate in international competition. Foreign publicity of the enterprises plays an important role in this process. This paper takes Huawei’s 2022 Sustainability Report as the research object. From the perspective of Newmark’s...

Containers capture the concept of strictly positive data types in programming. The original development of containers is done in the internal language of Locally Cartesian Closed Categories (LCCCs) with disjoint coproducts and W-types. Although it is claimed that these developments can also be interpreted in extensional Martin-L\"of type theory, th...

In a recent paper (Lacombe, Mukohyama, and Seitz, JCAP 2024, 05, 064 (2024)), the authors provided an in-depth analysis of a class of modified gravity theories, generally called f (R, Matter) theories, which assume the existence of a non-minimal coupling between geometry and matter. It was argued that if the matter sector consists of Standard Model...

Chapter 1: Introduction to CEO in Organization: (04 Hours) Desired Characteristics, Skills, & Attributes of CEO, Responsibilities of CEO, Performance Areas and Roles of CEOs. Chapter 2: CEO & Leadership: (04 Hours) Leadership Types, Theories, and their relevance. Leadership Theories – Continued. AB Theory of Leadership, Leadership Strategies – Typ...

In 2016 Vladimir Voevodsky sent the author an email message where he explained his conception of mathematical structure using a historical example borrowed from the \emph{Commentary to the First Book of Euclid's Elements} by Proclus; this message was followed by a short exchange where Vladimir clarified his conception of structure. In this Chapter...

This expository note describes two convenient techniques in the context of homotopy type theory for proving and formalizing that a given map is an equivalence. The first technique decomposes the map as a series of basic equivalences, while the second refines this approach using the 3-for-2 property of equivalences. The techniques are illustrated by...

The Platonic solids, renowned for their symmetry and mathematical elegance, have been central to the study of geometry since ancient times. These five convex polyhedra—the tetrahedron, cube, octahedron, dodecahedron, and icosahedron—are distinguished by their unique structural properties and have been extensively studied within the frameworks of Eu...

In a recent paper in Astronomy & Astrophysics, Alimi & Koskas (2024) have highlighted in wCDM models derived from general relativity (with Dark Energy Universe numerical simulation data), a cosmological invariance of the distribution of dark matter (DM) halo shapes when expressed in terms of the non-linear fluctuations of the cosmic matter field. W...

Although the Linux kernel is widely used, its complexity makes errors common and potentially serious. Traditional formal verification methods often have high overhead and rely heavily on manual coding. They typically verify only specific functionalities of the kernel or target microkernels and do not support continuous verification of the entire ke...

Constructive type theory combines logic and programming in one language. This is useful both for reasoning about programs written in type theory, as well as for reasoning about other programming languages inside type theory. It is well-known that it is challenging to extend these applications to languages with recursion and computational effects su...

This is a review of Bidlingmaier, Martin E. An interpretation of dependent type theory in a model category of locally Cartesian closed categories. (English) Zbl 07527568 Math. Struct. Comput. Sci. 31, No. 5, 469-494 (2021)

We introduce a new method for precisely relating algebraic structures in a presheaf category and judgements of its internal type theory. The method provides a systematic way to organise complex diagrammatic reasoning and generalises the well-known Kripke-Joyal forcing for logic. As an application, we prove several properties of algebraic weak facto...

Presuppositions are one of the topics of pragmatic semantics. This presupposition is the speaker's assumption of the speaker. This presupposition is present in every speech act and is often encountered. As in the illocutionary speech act of boasting (TTIM), people usually use TTIM to brag about themselves and to exalt themselves. There are several...

We study exponentiable functors in the context of synthetic $\infty$-categories. We do this within the framework of simplicial Homotopy Type Theory of Riehl and Shulman. Our main result characterizes exponentiable functors. In order to achieve this, we explore Segal type completions. Moreover, we verify that our result is semantically sound.

We investigate two constructive approaches to defining quasi-compact and quasi-separated schemes (qcqs-schemes), namely qcqs-schemes as locally ringed lattices and as functors from rings to sets. We work in Homotopy Type Theory and Univalent Foundations, but reason informally. The main result is a constructive and univalent proof that the two defin...

We present an extension of simple type theory that incorporates types for any kind of mathematical structure (of any order). We further extend this system allowing isomorphic structures to be identified within these types thanks to some syntactical restrictions; for this purpose, we formally define what it means for two structures to be isomorphic....

Grothendieck fibrations are fundamental in capturing the concept of dependency, notably in categorical semantics of type theory and programming languages. A relevant instance are Dialectica fibrations which generalise G\"odel's Dialectica proof interpretation and have been widely studied in recent years. We characterise when a given fibration is a...

المقدمة: نجد أنفسنا أمام مفهوم يتجاوز حدود الفكر التقليدي، ألا وهو التفكير الإبداعي. يمثل هذا الجانب من التفكير جوهرًا مميزًا للإنسان، فهو يمكّنه من مواجهة التحديات وإيجاد حلول مبتكرة للمشكلات التي تواجهه في حياته اليومية. إن التفكير الإبداعي ليس مجرد عملية عقلية بسيطة، بل هو رحلة استكشافية مثيرة تتخطى الحدود المعتادة وتغوص في أعماق العقل البشري. و...