Maria Manuel Clementino

Topological Features of Lax Algebras

Having as starting point Barrs description of topological spaces as lax algebras for the ultrafilter monad, in this paper we present further topological examples of lax algebras – such as quasi-metric spaces, approach spaces and quasi-uniform spaces – and show that, in a suitable setting, the categories of lax algebras have indeed a topological nature. Furthermore, we generalize to this setting known properties of special categories of lax algebras and, extending the construction of Manes, we describe the Čech–Stone compactification of lax algebras.

Metric, topology and multicategory—a common approach

Abstract For a symmetric monoidal-closed category V and a suitable monad T on the category of sets, we introduce the notion of reflexive and transitive (T, V ) -algebra and show that various old and new structures are instances of such algebras. Lawveres presentation of a metric space as a V -category is included in our setting, via the Betti–Carboni–Street–Walters interpretation of a V -category as a monad in the bicategory of V -matrices, and so are Barrs presentation of topological spaces as lax algebras, Lowens approach spaces, and Lambeks multicategories, which enjoy renewed interest in the study of n -categories. As a further example, we introduce a new structure called ultracategory which simultaneously generalizes the notions of topological space and of category.

One Setting for All: Metric, Topology, Uniformity, Approach Structure

For a complete lattice V which, as a category, is monoidal closed, and for a suitable Set-monad T we consider (T,V)-algebras and introduce (T,V)-proalgebras, in generalization of Lawveres presentation of metric spaces and Barrs presentation of topological spaces. In this lax-algebraic setting, uniform spaces appear as proalgebras. Since the corresponding categories behave functorially both in T and in V, one establishes a network of functors at the general level which describe the basic connections between the structures mentioned by the title. Categories of (T,V)-algebras and of (T,V)-proalgebras turn out to be topological over Set.

Separation versus connectedness

Abstract For a closure operator c in the sense of Dikranjan and Giuli, the subcategory Δ(c) (▽(c)) of objects X with c-closed (c-dense) diagonal δX: X → X × X is known to give a general notion of separation (connectedness, respectively), with the expected closure properties under products and subspaces (images), etc. The purpose of this note is to fully characterize the notions of connectedness and disconnectedness in the sense of Arhangelskiǐ and Wiegandt and of separation by Pumplun and Rohrl in this context. Briefly, an AW-connectedness is a subcategory of type ▽(c) with c a regular closure operator, and an AW-disconnectedness is of type Δ(c) with c a coregular closure operator, as introduced in this paper. The latter subcategory is in particular PR-separated, i.e., a subcategory of type Δ(c) with c weakly hereditary. Categorical proofs and new applications are provided for the characterization theorems originally given by Arhangelskiǐ and Wiegandt in the context of topological spaces.

Lawvere Completeness in Topology

It is known since 1973 that Lawvere’s notion of Cauchy-complete enriched category is meaningful for metric spaces: it captures exactly Cauchy-complete metric spaces. In this paper, we introduce the corresponding notion of Lawvere completeness for

Triquotient maps via ultrafilter convergence

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Tychonoff’s Theorem in a category

-categories and show that it has an interesting meaning for topological spaces and quasi-uniform spaces: for the former ones it means weak sobriety while for the latter it means Cauchy completeness. Further, we show that

Effective Descent Morphisms in Categories of Lax Algebras

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What is a Quotient Map with Respect to a Closure Operator

has a canonical

Local homeomorphisms via ultrafilter convergence

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