2,138 research outputs found
C-system of a module over a monad on sets
This is the second paper in a series that aims to provide mathematical
descriptions of objects and constructions related to the first few steps of the
semantical theory of dependent type systems.
We construct for any pair , where is a monad on sets and is
a left module over , a C-system (contextual category) and
describe a class of sub-quotients of in terms of objects directly
constructed from and . In the special case of the monads of expressions
associated with nominal signatures this construction gives the C-systems of
general dependent type theories when they are specified by collections of
judgements of the four standard kinds
CoLoR: a Coq library on well-founded rewrite relations and its application to the automated verification of termination certificates
Termination is an important property of programs; notably required for
programs formulated in proof assistants. It is a very active subject of
research in the Turing-complete formalism of term rewriting systems, where many
methods and tools have been developed over the years to address this problem.
Ensuring reliability of those tools is therefore an important issue. In this
paper we present a library formalizing important results of the theory of
well-founded (rewrite) relations in the proof assistant Coq. We also present
its application to the automated verification of termination certificates, as
produced by termination tools
The Univalence Principle
The Univalence Principle is the statement that equivalent mathematical
structures are indistinguishable. We prove a general version of this principle
that applies to all set-based, categorical, and higher-categorical structures
defined in a non-algebraic and space-based style, as well as models of
higher-order theories such as topological spaces. In particular, we formulate a
general definition of indiscernibility for objects of any such structure, and a
corresponding univalence condition that generalizes Rezk's completeness
condition for Segal spaces and ensures that all equivalences of structures are
levelwise equivalences.
Our work builds on Makkai's First-Order Logic with Dependent Sorts, but is
expressed in Voevodsky's Univalent Foundations (UF), extending previous work on
the Structure Identity Principle and univalent categories in UF. This enables
indistinguishability to be expressed simply as identification, and yields a
formal theory that is interpretable in classical homotopy theory, but also in
other higher topos models. It follows that Univalent Foundations is a fully
equivalence-invariant foundation for higher-categorical mathematics, as
intended by Voevodsky.Comment: A short version of this book is available as arXiv:2004.06572. v2:
added references and some details on morphisms of premonoidal categorie
Cartesian institutions with evidence: Data and system modelling with diagrammatic constraints and generalized sketches
Data constraints are fundamental for practical data modelling, and a
verifiable conformance of a data instance to a safety-critical constraint
(satisfaction relation) is a corner-stone of safety assurance. Diagrammatic
constraints are important as both a theoretical concepts and a practically
convenient device. The paper shows that basic formal constraint management can
well be developed within a finitely complete category (hence the reference to
Cartesianity in the title). In the data modelling context, objects of such a
category can be thought of as graphs, while their morphisms play two roles: of
data instances and (when being additionally labelled) of constraints.
Specifically, a generalized sketch consists of a graph and a set of
constraints declared over , and appears as a pattern for typical
data schemas (in databases, XML, and UML class diagrams). Interoperability of
data modelling frameworks (and tools based on them) very much depends on the
laws regulating the transformation of satisfaction relations between data
instances and schemas when the schema graph changes: then constraints are
translated co- whereas instances contra-variantly. Investigation of this
transformation pattern is the main mathematical subject of the paperComment: 35 pages. The paper will be presented at the conference on Applied
Category Theory, ACT'2
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