This paper develops the basics of the theory of involutive categories and shows that such categories provide the natural setting in which to describe involutive monoids. It is shown how categories of Eilenberg-Moore algebras of involutive monads are involutive, with conjugation for modules and vector spaces as special case. The core of the so-called Gelfand-Naimark-Segal (GNS) construction is identified as a bijective correspondence between states on involutive monoids and inner products. This correspondence exists in arbritrary involutive symmetric monoidal categories. 1

Abstract. This paper starts from the elementary observation that what is usually called a predicate lifting in coalgebraic modal logic is in fact an endomap of indexed categories. This leads to a systematic review of basic results in predicate logic for functors and monads, involving induction and coinduction principles for functors and compositional modal operators for monads. 1

This paper contributes a feedback operator, in the form of a monoidal trace, to the theory of coalgebraic, state-based modelling of components. The feedback operator on components is shown to satisfy the trace axioms of Joyal, Street and Verity. We employ McCurdy’s tube diagrams, an extension of standard string diagrams for monoidal categories, for representing and manipulating component diagrams. The microcosm principle then yields a canonical “inner” traced monoidal structure on the category of resumptions (elements of final coalgebras / components). This generalises an observation by Abramsky, Haghverdi and Scott.

The aim of this chapter is to construct new foundations for quantum logic and quantum spaces. This is accomplished by merging algebraic quantum theory and topos theory (encompassing the theory of locales or frames, of which toposes in a sense form the ultimate generalization). In a nutshell, the relation between these fields is as follows. First, our mathematical interpretation of Bohr’s ‘doctrine of classical concepts ’ is that the empirical content of a quantum theory described by a noncommutative (unital) C*-algebra A is contained in the family of its commutative (unital) C*-algebras, partially ordered by inclusion. Seen as a category, the ensuing poset C(A) canonically defines the topos [C(A), Set] of covariant functors from C(A) to the category Set of sets and functions. This topos contains the ‘Bohrification ’ A of A, defined as the tautological functor C ↦ → C, as an internal commutative C*-algebra. Second, according to the topos-valid Gelfand duality theorem of Banaschewski and Mulvey, A has a Gelfand spectrum Σ(A), which is a locale internal to the topos

Abstract. This paper contributes to the theory of coalgebraic, statebased modelling of components via two additions: a feedback operator in the form of a monoidal trace, and a three-dimensional string calculus for representing and manipulating composite component diagrams. The feedback operator on components is shown to satisfy the trace axioms by Joyal, Street and Verity. As a corollary, we appeal to the microcosm principle and derive a canonical traced monoidal structure on the category of resumptions. This generalises an observation by Abramsky, Haghverdi and Scott. 1