Exponentiation in V-categories

Abstract

For a Heyting algebra V which, as a category, is monoidal closed, we obtain characterizations of exponentiable objects and morphisms in the category of V-categories and apply them to some well-known examples. In the case V = R + these characterizations of exponentiable morphisms and objects in the categories (P)Met of (pre)metric spaces and non-expansive maps show in particular that exponentiable metric spaces are exactly the almost convex metric spaces, while exponentiable complete metric spaces are the complete totally convex ones.

Exponentiable Functors Between Quantaloid-Enriched Categories

“…The second condition is precisely what [3] had too, albeit in their more restrictive setting; but they did not discover the first condition an sich: because it is obviously always true if the base category is a locale. The proof of our theorem goes as follows.…”

“…For a detailed treatment of generalized metric spaces and non-expansive maps (categories and functors enriched over the quantale of positive real numbers) we refer to [3]. This example is quite different from the ones given above: the quantale of positive real numbers does not satisfy the interchange law (even though its underlying suplattice is a locale).…”

“…Conduché fibrations), has a long history; see [7] for a short account. Recently M. M. Clementino and D. Hofmann [3] found simple necessary-and-sufficient conditions for the exponentiability of a functor between Venriched categories, where V is a symmetric quantale which has its top element as unit for its multiplication and whose underlying sup-lattice is a locale. Our aim here is to prove the following characterization of the exponentiable functors between Q-enriched categories, where now Q is any (small) quantaloid, thus considerably generalizing the aforementioned result of [3].…”

On some special classes of continuous maps

“…Exponentiable and effective descent maps between metric -and more generally premetric -spaces are characterised in [7] and [6] respectively. We list here the results which might serve, together with the corresponding results for topological spaces, as a guideline for the study of these classes of maps in approach spaces as outlined below.…”

Axioms for Sequential Convergence

“…In fact, Barr [1] describes topological spaces as relational algebras for the ultrafilter monad. Recently the theory of relational algebras gained renewed interest (see [8,9,20]), and was used to study uniformly various important categories of topology (see [6,7,13], for instance).…”