We show how concurrent quantales and concurrent Kleene algebras arise as convolution algebras of functions from relational structures with two ternary relations that satisfy relational interchange laws into concurrent quantales or Kleene algebras, among others. The elements of the quantales can be understood as weights; the case where weights are drawn from the booleans corresponds to languages. We develop a correspondence theory between properties of the relational structures and algebraic properties in the weight and convolution algebras in the sense of modal and substructural logics, or boolean algebras with operators. The resulting correspondence triangles yield in particular general construction principles for models of concurrent quantales and Kleene algebras as convolution algebras from much simpler relational structures, including weighted ones for quantitative applications. As examples, we construct the concurrent quantales and Kleene algebras of weighted words, digraphs, posets, isomorphism classes of finite digraphs and pomsets.

Using the category of metric spaces as a template, we develop a metric analogue of the categorical semantics of classical/intuitionistic logic, and show that the natural notion of predicate in this “continuous semantics” is equivalent to the a priori separate notion of predicate in continuous logic, a logic which is independently well-studied by model theorists and which finds various applications. We show this equivalence by exhibiting the real interval $[0,1]$ in the category of metric spaces as a “continuous subobject classifier” giving a correspondence not only between the two notions of predicate, but also between the natural notion of quantification in the continuous semantics and the existing notion of quantification in continuous logic.

Along the way, we formulate what it means for a given category to behave like the category of metric spaces, and afterwards show that any such category supports the aforementioned continuous semantics. As an application, we show that categories of presheaves of metric spaces are examples of such, and in fact even possess continuous subobject classifiers.