588 research outputs found
Undecidability of Equality in the Free Locally Cartesian Closed Category (Extended version)
We show that a version of Martin-L\"of type theory with an extensional
identity type former I, a unit type N1 , Sigma-types, Pi-types, and a base type
is a free category with families (supporting these type formers) both in a 1-
and a 2-categorical sense. It follows that the underlying category of contexts
is a free locally cartesian closed category in a 2-categorical sense because of
a previously proved biequivalence. We show that equality in this category is
undecidable by reducing it to the undecidability of convertibility in
combinatory logic. Essentially the same construction also shows a slightly
strengthened form of the result that equality in extensional Martin-L\"of type
theory with one universe is undecidable
Belief propagation in monoidal categories
We discuss a categorical version of the celebrated belief propagation
algorithm. This provides a way to prove that some algorithms which are known or
suspected to be analogous, are actually identical when formulated generically.
It also highlights the computational point of view in monoidal categories.Comment: In Proceedings QPL 2014, arXiv:1412.810
On generalized algebraic theories and categories with families
We give a syntax independent formulation of finitely presented generalized algebraic theories as initial objects in categories of categories with families (cwfs) with extra structure. To this end, we simultaneously define the notion of a presentation Σ of a generalized algebraic theory and the associated category CwFΣ of small cwfs with a Σ-structure and cwf-morphisms that preserve Σ-structure on the nose. Our definition refers to the purely semantic notion of uniform family of contexts, types, and terms in CwFΣ. Furthermore, we show how to syntactically construct an initial cwf with a Σ-structure. This result can be viewed as a generalization of Birkhoff’s completeness theorem for equational logic. It is obtained by extending Castellan, Clairambault, and Dybjer’s construction of an initial cwf. We provide examples of generalized algebraic theories for monoids, categories, categories with families, and categories with families with extra structure for some type formers of Martin-Löf type theory. The models of these are internal monoids, internal categories, and internal categories with families (with extra structure) in a small category with families. Finally, we show how to extend our definition to some generalized algebraic theories that are not finitely presented, such as the theory of contextual cwfs.publishedVersio
Modal dependent type theory and dependent right adjoints
In recent years we have seen several new models of dependent type theory
extended with some form of modal necessity operator, including nominal type
theory, guarded and clocked type theory, and spatial and cohesive type theory.
In this paper we study modal dependent type theory: dependent type theory with
an operator satisfying (a dependent version of) the K-axiom of modal logic. We
investigate both semantics and syntax. For the semantics, we introduce
categories with families with a dependent right adjoint (CwDRA) and show that
the examples above can be presented as such. Indeed, we show that any finite
limit category with an adjunction of endofunctors gives rise to a CwDRA via the
local universe construction. For the syntax, we introduce a dependently typed
extension of Fitch-style modal lambda-calculus, show that it can be interpreted
in any CwDRA, and build a term model. We extend the syntax and semantics with
universes
Recommended from our members
Modal dependent type theory and dependent right adjoints
In recent years we have seen several new models of dependent type theory
extended with some form of modal necessity operator, including nominal type
theory, guarded and clocked type theory, and spatial and cohesive type theory.
In this paper we study modal dependent type theory: dependent type theory with
an operator satisfying (a dependent version of) the K-axiom of modal logic. We
investigate both semantics and syntax. For the semantics, we introduce
categories with families with a dependent right adjoint (CwDRA) and show that
the examples above can be presented as such. Indeed, we show that any finite
limit category with an adjunction of endofunctors gives rise to a CwDRA via the
local universe construction. For the syntax, we introduce a dependently typed
extension of Fitch-style modal lambda-calculus, show that it can be interpreted
in any CwDRA, and build a term model. We extend the syntax and semantics with
universes
Topologie
The Oberwolfach conference “Topologie” is one of only a few opportunities for researchers from many different areas in algebraic and geometric topology to meet and exchange ideas. The program covered new developments in fields such as automorphisms of manifolds, applications of algebraic topology to differential geometry, quantum field theories, combinatorial methods in low-dimensional topology, abstract and applied homotopy theory and applications of L2-cohomology. We heard about new results describing the cohomology of the automorphism spaces of some smooth manifolds, progress on spaces of positive scalar curvature metrics, a variant of the Segal conjecture without completion, advances in classifying topological quantum field theories, and a new undecidability result in combinatorial group theory, to mention some examples. As a special attraction, the conference featured a series of three talks by Dani Wise on the combinatorics of CAT(0)-cube complexes and applications to 3-manifold topology
Semantics of multimodal adjoint type theory
We show that contrary to appearances, Multimodal Type Theory (MTT) over a
2-category M can be interpreted in any M-shaped diagram of categories having,
and functors preserving, M-sized limits, without the need for extra left
adjoints. This is achieved by a construction called "co-dextrification" that
co-freely adds left adjoints to any such diagram, which can then be used to
interpret the "context lock" functors of MTT. Furthermore, if any of the
functors in the diagram have right adjoints, these can also be internalized in
type theory as negative modalities in the style of FitchTT. We introduce the
name Multimodal Adjoint Type Theory (MATT) for the resulting combined general
modal type theory. In particular, we can interpret MATT in any finite diagram
of toposes and geometric morphisms, with positive modalities for inverse image
functors and negative modalities for direct image functors.Comment: 24 pages. v2: Improved notation; extended pre-proceedings version for
MFPS 202
Cubical Syntax for Reflection-Free Extensional Equality
We contribute XTT, a cubical reconstruction of Observational Type Theory
which extends Martin-L\"of's intensional type theory with a dependent equality
type that enjoys function extensionality and a judgmental version of the
unicity of identity types principle (UIP): any two elements of the same
equality type are judgmentally equal. Moreover, we conjecture that the typing
relation can be decided in a practical way. In this paper, we establish an
algebraic canonicity theorem using a novel cubical extension (independently
proposed by Awodey) of the logical families or categorical gluing argument
inspired by Coquand and Shulman: every closed element of boolean type is
derivably equal to either 'true' or 'false'.Comment: Extended version; International Conference on Formal Structures for
Computation and Deduction (FSCD), 201
- …