We present a Cartesian closed category ELoc of equilocales, which contains the category Loc of locales as a reflective full subcategory. The embedding of Loc into ELoc preserves products and all exponentials of exponentiable locales.

We show that the subconstruct Fing of Prtop, consisting of all finitely generated pretopological spaces, is the largest Cartesian closed coreflective subeonstmct of Prtop. This implies that in any coreflective subconstruct of Prtop, exponential objects are finitely generated. Moreover, in any finitely productive, coreflective subconstruct, exponential objects are precisely those objects of the ...

We present a generalization of cartesian closed categories (CCCs) for dependent types, called dependent cartesian closed categories (DCCCs), which also provides a reformulation of categories with families (CwFs), an abstract semantics for Martin-Löf type theory (MLTT) which is very close to the syntax. Thus, DCCCs accomplish mathematical elegance as well as a direct interpretation of the syntax...

We present an algebraic treatment of the correspondence of gaps and dualities in partial ordered classes induced by the morphism structures of certain categories which we call Heyting (such are for instance all cartesian closed categories, but there are other important examples). This allows to extend the results of [14] to a wide range of more general structures. Also, we introduce a notion of...

The Tietze-Urysohn Theorem states that every continuous real-valued function defined on a closed subspace of a normal space can be extended to a continuous function on the whole space. We prove an effective version of this theorem in the Type Two Model of Effectivity (TTE). Moreover, we introduce for qcb-spaces a slightly weaker notion of normality than the classical one and show that this prop...

For Denjoy–Carleman differential function classes C where the weight sequence M = (Mk) is logarithmically convex, stable under derivations, and non-quasianalytic of moderate growth, we prove the following: A mapping is C if it maps C -curves to C -curves. The category of C -mappings is cartesian closed in the sense that C (E, C (F, G)) = C (E × F, G) for convenient vector spaces. Applications t...

When Gödel developed his functional interpretation, also known as the Dialectica interpretation, his aim was to prove (relative) consistency of first order arithmetic by reducing it to a quantifier-free theory with finite types. Like other functional interpretations (e.g. Kleene’s realizability interpretation and Kreisel’s modified realizability) Gödel’s Dialectica interpretation gives rise to ...

There are two main approaches to obtaining \topological" cartesian-closed categories. Under one approach, one restricts to a full subcategory of topological spaces that happens to be cartesian closed | for example, the category of sequential spaces. Under the other, one generalises the notion of space | for example, to Scott's notion of equilogical space. In this paper we show that the two appr...