We present a detailed synthetic overview of the utilisation of categorical techniques in the study of order structures together with their applications in operational quantum theory. First, after reviewing the notion of residuation and its implementation at the level of quantaloids we consider some standard universal constructions and the extension of adjunctions to weak morphisms. Second, we p...

Several results on the translation of factorization systems along adjunctions are proved. The first of these results is the answer to the question posed by G.Janelidze.

We investigate functorial relationships between the categories of theories in different institutions, namely adjunctions, as a means of translating between the different specification spaces that they provide. We show that there is a canonical way in which adjunctions between the categories of signatures can be lifted to the categories of theories. This lifting is associated with a duality betw...

This paper provides geometrical descriptions of the Frobenius monad freely generated by a single object. These descriptions are related to results connecting Frobenius algebras and topological quantum field theories. In these descriptions, which are based on coherence results for self-adjunctions (adjunctions where an endofunctor is adjoint to itself), ordinals in ε0 play a prominent role. The ...

Doctrines are categorical structures very apt to study logics of different nature within a unified environment: the 2-category Dtn doctrines. Modal interior operators characterised as particular adjoints in Dtn. We show that they can be constructed from comonads well adjunctions it, and two constructions compare. Finally we amount information lost passage comonad, or an adjunction, modal operat...

A simple model for the localization of category $\mathbf{CLoc}_2$ oriented and time-oriented globally hyperbolic conformal Lorentzian $2$-manifolds at all Cauchy morphisms is constructed. This provides an equivalent description $2$-dimensional algebraic quantum field theories (AQFTs) satisfying time-slice axiom in terms only two algebras, one Minkowski spacetime flat cylinder, together with a s...

In this paper we explain the relationship between Frobenius objects in monoidal categories and adjunctions in 2-categories. Specifically, we show that every Frobenius object in a monoidal category M arises from an ambijunction (simultaneous left and right adjoints) in some 2-category D into which M fully and faithfully embeds. Since a 2D topological quantum field theory is equivalent to a commu...

The Stone-Cech compactification of a space X is described by adjoining to X continuous images of the Stone-tech growths of a complementary pair of subspaces of X. The compactification of an example of Potoczny from [P] is described in detail. The Stone-Cech compactification of a completely regular space X is a compact Hausdorff space ßX in which X is dense and C*-embedded, i.e. every bounded re...