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...

For a complete cartesian-closed category V with coproducts, and for any pointed endofunctor T of the category of sets satisfying a suitable Beck-Chevalley-type condition, it is shown that the category of lax reflexive (T,V)-algebras is a quasitopos. This result encompasses many known and new examples of quasitopoi.

For Denjoy–Carleman differentiable 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...

Recent works dealing with new considerations about the notion centralizer of equivalence relations gave the opportunity to reveal beyond this construction widerranging phenomenons of functorial nature which do not belong, as it is well-known, to this notion by itself. And this was done in two distinct ways: on the one hand through the notion of action accessible category [2] and on the other ha...

We define the notion of a (P, P̃ )-structure on a universe p in a locally cartesian closed category category with a binary product structure and construct a (Π, λ)-structure on the C-systems CC(C, p) from a (P, P̃ )-structure on p. We then define homomorphisms of C-systems with (Π, λ)-structures and functors of universe categories with (P, P̃ )-structures and show that our construction is functori...

We propose a semantics for permutation equivalence in higher-order rewriting. This semantics takes place in cartesian closed 2-categories, and is proved sound and complete.

BLOCKINy the construction of the PL-category in 3 directly from a given PL-category A without any reference to an ambient locally cartesian closed category L. An object in the bre over U n is a pair hT; T 0 i such that (i) T is an object in A(U n)|let t: U n-U be such that A(t)(X) = T |

The projective tensor product in a category of topologicalR-modules (where R is a topological ring) can be defined in Top, the category of topological spaces, by the same universal property used to define the tensor product of R-modules in Set. In this article, we extend this definition to an arbitrary topological category X and study how the cartesian closedness of X is related to the monoidal...