Related papers: Cauchy completeness and causal spaces

In this paper we will characterize the completeness and barrelledness of a normed space through the strong p-Ces\`aro summability of series. A new characterization of weakly unconditionally Cauchy series and unconditionally convergent…

We set up a general theory of weak or homotopy-coherent enrichment in an arbitrary monoidal $\infty$-category $\mathcal{V}$. Our theory of enriched $\infty$-categories has many desirable properties; for instance, if the enriching…

This rough note describes some attempts to define a notion of enriched topology (and the associated theory of enriched stacks) on a category enriched over a symmetric monoidal model category, and poses some related questions.

We define a mapping space for Gray-enriched categories adapted to higher gauge theory. Our construction differs significantly from the canonical mapping space of enriched categories in that it is much less rigid. The two essential…

Differential completions and compactifications of differential spaces are introduced and investigated. The existence of the maximal differential completion and the maximal differential compactification is proved. A sufficient condition for…

We investigate an enriched-categorical approach to a field of discrete mathematics. The main result is a duality theorem between a class of enriched categories (called $\overline{\mathbb{Z}}$- or $\overline{\mathbb{R}}$-categories) and that…

We give a new account of the correspondence, first established by Nishizawa--Power, between finitary monads and Lawvere theories over an arbitrary locally finitely presentable base. Our account explains this correspondence in terms of…

This paper studies a notion of parameterized flatness in the enriched context: p-flatness where the parameter p stands for a class of presheaves. One obtains a completion of a category A by considering the category F_p(A) of p-flat…

We describe a general framework for notions of commutativity based on enriched category theory. We extend Eilenberg and Kelly's tensor product for categories enriched over a symmetric monoidal base to a tensor product for categories…

A metric space $\mathbf{X}$ is called densely complete if there exists a dense set $D$ in $\mathbf{X}$ such that every Cauchy sequence of points of $D $ converges in $\mathbf{X}$. One of the main aims of this work is to prove that the…

We apply some tools developed in categorical logic to give an abstract description of constructions used to formalize constructive mathematics in foundations based on intensional type theory. The key concept we employ is that of a Lawvere…

Our work over the past years shows that not only the collection of (for instance) all topological spaces gives rise to a category, but also each topological space can be seen individually as a category by interpreting the convergence…

This paper provides a comprehensive overview of some of the foundational properties of categories enriched over quantaloids, along with several new results. We demonstrate that the category whose objects are quantaloid-enriched categories…

Generalized metrics, arising from Lawvere's view of metric spaces as enriched categories, have been widely applied in denotational semantics as a way to measure to which extent two programs behave in a similar, although non equivalent, way.…

In the second section, we introduce dense unital magmas and show that a near-ring is dense if and only if it has a positive element smaller that unity. In the third section, we discuss magma-valued metric spaces. The density property of the…

Building on our previous work on enriched universal algebra, we define a notion of enriched language consisting of function and relation symbols whose arities are objects of the base of enrichment. In this context, we construct atomic…

In contrast to the fact that every completely distributive lattice is necessarily continuous in the sense of Scott, it is shown that complete distributivity of a category enriched over the closed category obtained by endowing the unit…

We introduce the notion of an enriched set, as an abstraction of enriched categories, and a category of enriched sets. The set of enriched sets is itself described as a set enriched over the category of enriched sets. We introduce a method…

Regular and exact categories were first introduced by Michael Barr in 1971; since then, the theory has developed and found many applications in algebra, geometry, and logic. In particular, a small regular category determines a certain…

We develop universal algebra over an enriched category $\mathcal K$ and relate it to finitary enriched monads over $\mathcal K$. Using it, we deduce recent results about ordered universal algebra where inequations are used instead of…