The category TOP of topological spaces is not cartesian closed, but can be embedded into the cartesian closed category ASSM of assemblies over algebraic lattices, which is a generalisation of Scott's category EQU of equilogical spaces. In this paper, we identify cartesian closed subcategories of assemblies which correspond to well-known separation properties of topology: T 0 ,

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...

Abstract We prove a strictification theorem for cartesian closed bicategories. First, we adapt Power’s proof of coherence bicategories with finite bilimits to show that every bicategory bicategorical structure is biequivalent 2-category 2-categorical structure. Then how extend this result Mac Lane-style “all pasting diagrams commute” theorem: precisely, in the free on graph, there at most one 2...

We prove in a uniform way that all Denjoy–Carleman differentiable function classes of Beurling type C(M) and of Roumieu type C{M}, admit a convenient setting if the weight sequence M = (Mk) is log-convex and of moderate growth: For C denoting either C(M) or C{M}, 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. Application...

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 |