Infinite dimensional spaces frequently appear in physics; there are several approaches to obtain a good categorical framework for this type of space, and cartesian closedness of some category, embedding smooth manifolds, is one of the most requested condition. In the first part of the paper, we start from the failures presented by the classical Banach manifolds approach and we will review the m...

We investigate the frame properties and closedness for the shift invariant space

A general result relating skew monoidal structures and monads is proved. This is applied to quantum categories and bialgebroids. Ordinary categories are monads in the bicategory whose morphisms are spans between sets. Quantum categories were originally defined as monoidal comonads on endomorphism objects in a particular monoidal bicategory M. Then they were shown also to be skew monoidal struct...

The definition and calculus of extraordinary natural transformations [EK] is extended to a context internal to any autonomous monoidal bicategory [DyS]. The original calculus is recaptured from the geometry [SV], [MT] of the monoidal bicategory V-Mod whose objects are categories enriched in a cocomplete symmetric monoidal category V and whose morphisms are modules. Mathematics Subject Classific...

Monoidal logics are logics that can be seen as specific instances of monoidal categories. They are constructed using specific rules and axiom schemata that allow to make explicit the monoidal structure of the logics. Among monoidal logics, we find Cartesian logics, which are instances of Cartesian categories. As it happens, many paradoxes in epistemic, deontic and action logics can be related t...

The equivalence between a monoidal category and a strict one has been proved by some authors such as Nguyen Duy Thuan [8], Christian Kassel [2], Peter Schauenburg [7]. In this paper, we show another proof of the problem by constructing a strict monoidal category M(C) consisting of M -functors and M morphisms of a category C and we prove C is equivalent to it. The proof is based on a basic chara...

Kornel Szlachányi [28] recently used the term skew-monoidal category for a particular laxi ed version of monoidal category. He showed that bialgebroids H with base ring R could be characterized in terms of skew-monoidal structures on the category of one-sided R-modules for which the lax unit was R itself. We de ne skew monoidales (or skew pseudo-monoids) in any monoidal bicategory M . These are...