283 research outputs found
Free constructions and coproducts of d-frames
A general theory of presentations for d-frames does not yet exist. We review
the difficulties and give sufficient conditions for when they can be overcome.
As an application we prove that the category of d-frames is closed under
coproducts
Free Constructions and Coproducts of d-Frames
A general theory of presentations for d-frames does not yet exist. We review the difficulties and give sufficient conditions for when they can be overcome. As an application we prove that the category of d-frames is closed under coproducts
Relational parametricity for higher kinds
Reynolds’ notion of relational parametricity has been extremely influential and well studied for polymorphic programming languages and type theories based on System F. The extension of relational parametricity to higher kinded polymorphism, which allows quantification over type operators as well as types, has not received as much attention. We present a model of relational parametricity for System Fω, within the impredicative Calculus of Inductive Constructions, and show how it forms an instance of a general class of models defined by Hasegawa. We investigate some of the consequences of our model and show that it supports the definition of inductive types, indexed by an arbitrary kind, and with reasoning principles provided by initiality
A uniqueness theorem for stable homotopy theory
In this paper we study the global structure of the stable homotopy theory of
spectra. We establish criteria for when the homotopy theory associated to a
given stable model category agrees with the classical stable homotopy theory of
spectra. One sufficient condition is that the associated homotopy category is
equivalent to the stable homotopy category as a triangulated category with an
action of the ring of stable homotopy groups of spheres. In other words, the
classical stable homotopy theory, with all of its higher order information, is
determined by the homotopy category as a triangulated category with an action
of the stable homotopy groups of spheres. Another sufficient condition is the
existence of a small generating object (corresponding to the sphere spectrum)
for which a specific `unit map' from the infinite loop space QS^0 to the
endomorphism space is a weak equivalence
Beyond topological persistence: Starting from networks
Persistent homology enables fast and computable comparison of topological
objects. However, it is naturally limited to the analysis of topological
spaces. We extend the theory of persistence, by guaranteeing robustness and
computability to significant data types as simple graphs and quivers. We focus
on categorical persistence functions that allow us to study in full generality
strong kinds of connectedness such as clique communities, -vertex and
-edge connectedness directly on simple graphs and monic coherent categories.Comment: arXiv admin note: text overlap with arXiv:1707.0967
The connected Vietoris powerlocale
The connected Vietoris powerlocale is defined as a strong monad Vc on the category of locales. VcX is a sublocale of Johnstone's Vietoris powerlocale VX, a localic analogue of the Vietoris hyperspace, and its points correspond to the weakly semifitted sublocales of X that are “strongly connected”. A product map ×:VcX×VcY→Vc(X×Y) shows that the product of two strongly connected sublocales is strongly connected. If X is locally connected then VcX is overt. For the localic completion of a generalized metric space Y, the points of are certain Cauchy filters of formal balls for the finite power set with respect to a Vietoris metric. \ud
Application to the point-free real line gives a choice-free constructive version of the Intermediate Value Theorem and Rolle's Theorem. \ud
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The work is topos-valid (assuming natural numbers object). Vc is a geometric constructio
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