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

Abstract

Abstract. 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 Lawvere's presentation of metric spaces and Barr's presentation of topological spaces. In this laxalgebraic 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. (2000): 18C20, 18B30, 54E15. Mathematics Subject Classifications

Universality of Coproducts in Categories of Lax Algebras

“…first group happen to be presentable as categories of lax algebras, that is: They are of the form Alg(T, V ), for a suitable extension T of a Set-monad T 0 = (T 0 , e, m) and a complete lattice V that comes with an associative and commutative binary operation ⊗ preserving suprema in each variable, and a ⊗-neutral element k (distinct from the bottom element ⊥), [5]. In this note, we show that this observation is not coincidental, that is: Coproducts in Alg(T, V ) are always stable under pullback, making Alg(T, V ) in fact an (infinitely) extensive category, provided that T 0 preserves inverse images.…”

On lax-algebraic (co)nuclei

“…Since one part of "⇒" has already been considered in, e.g., [4], we show thatŤ =T ϕ . Given a W -relation X s / / Y , on the one hand, compatibility of ϕ givesT (ϕ s) ϕ (Ť s), and thus, ϕT (ϕ s) Ť s by Definition 2.13; on the other hand,Ť s…”

“…This section briefly outlines the setting of lax-algebraic approach to topology as it is developed in, e.g., [4,7,12,13,24]. The theory was motivated by the result of M. Barr [2], who showed that the category Top of topological spaces and continuous maps is isomorphic to the category of lax Eilenberg-Moore algebras with respect to the canonical lax extension of the ultrafilter monad on the category Set of sets and maps to the category Rel of sets and relations.…”

“…P + -Cat is the category Met of premetric spaces (in the sense of F. W. Lawvere [17]) and non-expansive maps [4,6].…”

“…In a series of papers, M. M. Clementino, D. Hofmann, and W. Tholen [3,4,6,7,10,11] generalized this approach to an arbitrary monad T on Set and the category V -Rel of sets and V -relations, where V is an arbitrary unital quantale. In particular, they showed that many of the existing categories of topological structures (e.g., preordered sets, premetric spaces in the sense of F. W. Lawvere [17], approach spaces of R. Lowen [18], probabilistic metric spaces of B. Schweizer and A. Sklar [23]) can be represented as the categories (T, V )-Cat of lax algebras and lax homomorphisms with respect to a suitable monad T and a quantale V .…”