This paper introduces a new approach to the theory of Ω-categories enriched by a frame. The approach combines ideas from various areas such as generalized ultrametric domains, Ω-categories, constructive analysis, and fuzzy mathematics. As the basic framework, we use the Wagner’s Ω-category [18,19] with a frame instead of a quantale with unit. The objects and morphisms in the category will be ca...

In this paper we unify the developments of [Batanin, 1998], [BataninWeber, 2011] and [Cheng, 2011] into a single framework in which the interplay between multitensors on a category V , and monads on the category GV of graphs enriched in V , is taken as fundamental. The material presented here is the conceptual background for subsequent work: in [Batanin-Cisinski-Weber, 2013] the Gray tensor pro...

Joyal and Street note in their paper on braided monoidal categories [9] that the 2–category V –Cat of categories enriched over a braided monoidal category V is not itself braided in any way that is based upon the braiding of V . The exception that they mention is the case in which V is symmetric, which leads to V –Cat being symmetric as well. The symmetry in V –Cat is based upon the symmetry of...

We introduce -algebras and -algebras as semantic domains for data re nement of imperative programming languages. The functorial semantics of -calculus is given by using the adjunction between the category of -algebras and the category of small locally ordered categories. We de ne the notion of upward and downward simulation between the interpretations of atomic commands, and re nements between ...

We devise a generic framework where a weakest precondition semantics, in the form of indexed posets, is derived from a monad whose Kleisli category is enriched by posets. It is inspired by Jacobs’ recent identification of a categorical structure that is common in various predicate transformers, but adds generality in the following aspects: 1) different notions of modality (such as “may” vs. “mu...

Joyal and Street note in their paper on braided monoidal categories [10] that the 2–category V–Cat of categories enriched over a braided monoidal category V is not itself braided in any way that is based upon the braiding of V. What is meant by “based upon” here will be made more clear in the present paper. The exception that they mention is the case in which V is symmetric, which leads to V–Ca...

Under a minimum of assumptions, we develop in generality the basic theory of universal algebra in a symmetric monoidal closed category V with respect to a specified system of arities j : J ↪→ V . Lawvere’s notion of algebraic theory generalizes to this context, resulting in the notion of single-sorted V -enriched J -cotensor theory, or J -theory for short. For suitable choices of V and J , such...

We show that the category of rational G-spectra for a torus G is Quillen equivalent to an explicit small and practical algebraic model, thereby providing a universal de Rham model for rational G-equivariant cohomology theories. The result builds on the Adams spectral sequence of [12], the enriched Morita equivalences of [23], and the functors of [27] making rational spectra algebraic.

Directed Algebraic Topology is a recent field, deeply linked with Category Theory. A 'directed space' has directed homotopies (generally non reversible), directed homology groups (enriched with a preorder) and fundamental n-categories (replacing the fundamental n-groupoids of the classical case). Applications have been mostly developed in the theory of concurrency. Unexpected links with noncomm...

We extend the theory of Quillen adjunctions by combining ideas of homotopical algebra and of enriched category theory. Our results describe how the formulas for homotopy colimits of Bousfield and Kan arise from general formulas describing the derived functor of the weighted colimit functor.