scholarly journals Structuralism, Invariance, and Univalence

2018 ◽  
Author(s):  
Steve Awodey
Keyword(s):  
Type Theory ◽  
Homotopy Theory ◽  
Mathematical Practice ◽  
New Approach ◽  
Constructive Type Theory ◽  
Homotopy Type Theory ◽  
Computational Implementation ◽  
Geometric Content ◽  
Abstract Homotopy Theory ◽  
New System

The recent discovery of an interpretation of constructive type theory into abstract homotopy theory suggests a new approach to the foundations of mathematics with intrinsic geometric content and a computational implementation. Voevodsky has proposed such a program, including a new axiom with both geometric and logical significance: the univalence axiom. It captures the familiar aspect of informal mathematical practice according to which one can identify isomorphic objects. This powerful addition to homotopy type theory gives the new system of foundations a distinctly structural character.

scholarly journals Introduction – from type theory and homotopy theory to univalent foundations

2015 ◽  
Vol 25 (5) ◽  
pp. 1005-1009
Author(s):  
STEVE AWODEY ◽  
NICOLA GAMBINO ◽  
ERIK PALMGREN
Keyword(s):  
Homotopy Type ◽  
Type Theory ◽  
Homotopy Theory ◽  
Foundations Of Mathematics ◽  
Special Issue ◽  
Research Papers ◽  
Homotopy Type Theory ◽  
Main Ideas ◽  
Univalent Foundations

We give an overview of the main ideas involved in the development of homotopy type theory and the univalent foundations of Mathematics programme. This serves as a background for the research papers published in the special issue.

scholarly journals Natural models of homotopy type theory

2016 ◽  
Vol 28 (2) ◽  
pp. 241-286 ◽  
Author(s):  
STEVE AWODEY
Keyword(s):  
Homotopy Type ◽  
Type Theory ◽  
Homotopy Theory ◽  
Inference Rules ◽  
Basic System ◽  
Homotopy Type Theory ◽  
Algebraic Formulation

The notion of a natural model of type theory is defined in terms of that of a representable natural transfomation of presheaves. It is shown that such models agree exactly with the concept of a category with families in the sense of Dybjer, which can be regarded as an algebraic formulation of type theory. We determine conditions for such models to satisfy the inference rules for dependent sums Σ, dependent products Π and intensional identity types Id, as used in homotopy type theory. It is then shown that a category admits such a model if it has a class of maps that behave like the abstract fibrations in axiomatic homotopy theory: They should be stable under pullback, closed under composition and relative products, and there should be weakly orthogonal factorizations into the class. It follows that many familiar settings for homotopy theory also admit natural models of the basic system of homotopy type theory.

RELATIONAL TOPOLOGY AS THE FOUNDATION FOR QUANTUM GRAVITY

2005 ◽  
Vol 20 (17n18) ◽  
pp. 1261-1269 ◽  
Author(s):  
LOUIS CRANE
Keyword(s):  
General Relativity ◽  
Algebraic Geometry ◽  
Quantum Theory ◽  
Homotopy Theory ◽  
Categorical Approach ◽  
New Approach ◽  
Spin Foam Models ◽  
Spin Foam ◽  
Theory Of Gravitation ◽  
Abstract Homotopy Theory

We propose a new approach to the quantum theory of gravitation, in which the point sets of regions of spacetime are replaced by objects in a category. The objects can be constructed as limits of spin foam models, thus directly connecting to approximations to general relativity which have already been studied, and leading to a mathematically natural interpretation of the meaning of the triangulations in the spin foam models. The physical motivation for our proposal is a connection between ideas about the limited amount of information which can flow from one region to another in General Relativity, an analysis of the problem of the infinities in QFT, and the relational or categorical approach to topology in abstract homotopy theory and algebraic geometry.

scholarly journals Brouwer's fixed-point theorem in real-cohesive homotopy type theory

2017 ◽  
Vol 28 (6) ◽  
pp. 856-941 ◽  
Author(s):  
MICHAEL SHULMAN
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Homotopy Type ◽  
Algebraic Topology ◽  
Type Theory ◽  
Homotopy Theory ◽  
Homotopy Type Theory ◽  
Brouwer’S Fixed Point Theorem ◽  
Brouwer's Fixed Point Theorem

We combine homotopy type theory with axiomatic cohesion, expressing the latter internally with a version of ‘adjoint logic’ in which the discretization and codiscretization modalities are characterized using a judgemental formalism of ‘crisp variables.’ This yields type theories that we call ‘spatial’ and ‘cohesive,’ in which the types can be viewed as having independent topological and homotopical structure. These type theories can then be used to study formally the process by which topology gives rise to homotopy theory (the ‘fundamental ∞-groupoid’ or ‘shape’), disentangling the ‘identifications’ of homotopy type theory from the ‘continuous paths’ of topology. In a further refinement called ‘real-cohesion,’ the shape is determined by continuous maps from the real numbers, as in classical algebraic topology. This enables us to reproduce formally some of the classical applications of homotopy theory to topology. As an example, we prove Brouwer's fixed-point theorem.

scholarly journals Homotopical patch theory

2016 ◽  
Vol 26 ◽  
Author(s):  
CARLO ANGIULI ◽  
EDWARD MOREHOUSE ◽  
DANIEL R. LICATA ◽  
ROBERT HARPER
Keyword(s):  
Control Systems ◽  
Homotopy Type ◽  
Category Theory ◽  
Type Theory ◽  
Homotopy Theory ◽  
New Class ◽  
Homotopy Type Theory ◽  
Inductive Type ◽  
Higher Category Theory ◽  
Inductive Types

AbstractHomotopy type theory is an extension of Martin-Löf type theory, based on a correspondence with homotopy theory and higher category theory. In homotopy type theory, the propositional equality type is proof-relevant, and corresponds to paths in a space. This allows for a new class of datatypes, called higher inductive types, which are specified by constructors not only for points but also for paths. In this paper, we consider a programming application of higher inductive types. Version control systems such as Darcs are based on the notion of patches—syntactic representations of edits to a repository. We show how patch theory can be developed in homotopy type theory. Our formulation separates formal theories of patches from their interpretation as edits to repositories. A patch theory is presented as a higher inductive type. Models of a patch theory are given by maps out of that type, which, being functors, automatically preserve the structure of patches. Several standard tools of homotopy theory come into play, demonstrating the use of these methods in a practical programming context.

scholarly journals Homotopy type-theoretic interpretations of constructive set theories

2019 ◽  
pp. 1-32
Author(s):  
Cesare Gallozzi
Keyword(s):  
Comparative Analysis ◽  
Set Theory ◽  
Homotopy Type ◽  
Type Theory ◽  
Axiom Of Choice ◽  
Homotopy Type Theory ◽  
Constructive Set Theory

Abstract We introduce a family of (k, h)-interpretations for 2 ≤ k ≤ ∞ and 1 ≤ h ≤ ∞ of constructive set theory into type theory, in which sets and formulas are interpreted as types of homotopy level k and h, respectively. Depending on the values of the parameters k and h, we are able to interpret different theories, like Aczel’s CZF and Myhill’s CST. We also define a proposition-as-hproposition interpretation in the context of logic-enriched type theories. The rest of the paper is devoted to characterising and analysing the interpretations considered. The formulas valid in the prop-as-hprop interpretation are characterised in terms of the axiom of unique choice. We also analyse the interpretations of CST into homotopy type theory, providing a comparative analysis with Aczel’s interpretation. This is done by formulating in a logic-enriched type theory the key principles used in the proofs of the two interpretations. Finally, we characterise a class of sentences valid in the (k, ∞)-interpretations in terms of the ΠΣ axiom of choice.

scholarly journals Quotienting the delay monad by weak bisimilarity

2017 ◽  
Vol 29 (1) ◽  
pp. 67-92 ◽  
Author(s):  
JAMES CHAPMAN ◽  
TARMO UUSTALU ◽  
NICCOLÒ VELTRI
Keyword(s):  
Computer Science ◽  
Equivalence Relation ◽  
Type Theory ◽  
Partial Functions ◽  
Homotopy Type Theory ◽  
Inductive Type ◽  
Alternative Approach ◽  
The One ◽  
Axiom Of Countable Choice

The delay datatype was introduced by Capretta (Logical Methods in Computer Science, 1(2), article 1, 2005) as a means to deal with partial functions (as in computability theory) in Martin-Löf type theory. The delay datatype is a monad. It is often desirable to consider two delayed computations equal, if they terminate with equal values, whenever one of them terminates. The equivalence relation underlying this identification is called weak bisimilarity. In type theory, one commonly replaces quotients with setoids. In this approach, the delay datatype quotiented by weak bisimilarity is still a monad–a constructive alternative to the maybe monad. In this paper, we consider the alternative approach of Hofmann (Extensional Constructs in Intensional Type Theory, Springer, London, 1997) of extending type theory with inductive-like quotient types. In this setting, it is difficult to define the intended monad multiplication for the quotiented datatype. We give a solution where we postulate some principles, crucially proposition extensionality and the (semi-classical) axiom of countable choice. With the aid of these principles, we also prove that the quotiented delay datatype delivers free ω-complete pointed partial orders (ωcppos).Altenkirch et al. (Lecture Notes in Computer Science, vol. 10203, Springer, Heidelberg, 534–549, 2017) demonstrated that, in homotopy type theory, a certain higher inductive–inductive type is the free ωcppo on a type X essentially by definition; this allowed them to obtain a monad of free ωcppos without recourse to a choice principle. We notice that, by a similar construction, a simpler ordinary higher inductive type gives the free countably complete join semilattice on the unit type 1. This type suffices for constructing a monad, which is isomorphic to the one of Altenkirch et al. We have fully formalized our results in the Agda dependently typed programming language.

scholarly journals Reverse vesicles as a new system for studying enzymes in organic solvents

1992 ◽  
Vol 285 (2) ◽  
pp. 373-376 ◽  
Author(s):  
A Sánchez-Ferrer ◽  
F García-Carmona
Keyword(s):  
Aqueous Solution ◽  
Organic Solvents ◽  
Reverse Micelles ◽  
Active Form ◽  
Ionic Surfactant ◽  
New Approach ◽  
Optically Transparent ◽  
Reverse Vesicles ◽  
Dodecyl Ether ◽  
New System

A new approach to the study of enzyme activity in organic solvents has been developed by using optically transparent reverse vesicles. Polyphenol oxidase was incorporated in an active form into the above ternary system formed by the non-ionic surfactant tetra(ethylene glycol) dodecyl ether/n-dodecane/water. The enzyme in this microenvironment, surprisingly, showed an apparent positive co-operativity which has never before been described either in aqueous solution or in reverse micelles. In addition, the Vmax. expressed was similar to that in water and twice that displayed in reverse micelles.

Formalization of metatheory of the Lambda Calculus in constructive type theory using the Barendregt variable convention

2021 ◽  
pp. 1-20
Author(s):  
Ernesto Copello ◽  
Nora Szasz ◽  
Álvaro Tasistro
Keyword(s):  
Type Theory ◽  
Lambda Calculus ◽  
Proof Assistant ◽  
First Order ◽  
Constructive Type Theory ◽  
Fundamental Part ◽  
Constructive Type ◽  
Free Variables ◽  
The Church ◽  
Theoretical Results

Abstarct We formalize in Constructive Type Theory the Lambda Calculus in its classical first-order syntax, employing only one sort of names for both bound and free variables, and with α-conversion based upon name swapping. As a fundamental part of the formalization, we introduce principles of induction and recursion on terms which provide a framework for reproducing the use of the Barendregt Variable Convention as in pen-and-paper proofs within the rigorous formal setting of a proof assistant. The principles in question are all formally derivable from the simple principle of structural induction/recursion on concrete terms. We work out applications to some fundamental meta-theoretical results, such as the Church–Rosser Theorem and Weak Normalization for the Simply Typed Lambda Calculus. The whole development has been machine checked using the system Agda.

Sign in / Sign up

Export Citation Format

Share Document