Takeuchi’s famous free Hopf algebra construction is analyzed from a categorical point of view, and so is the construction of the Hopf envelope of a bialgebra. Both constructions this way appear as compositions of well known and natural constructions. This way certain partially wrong perceptions of these constructions are clarified and their mutual relation is made precise. The construction of H...

Several attempts have been made of extending to graph grammars the unfolding semantics originally developed by Winskel for (safe) Petri nets, but only partial results were obtained. In this paper we fully extend Winskel’s approach to single-pushout grammars providing them with a categorical concurrent semantics expressed as a coreflection between the category of (semi-weighted) graph grammars a...

Classically, a Tychonoff space is called strongly 0-dimensional if its Stone-Čech compactification is 0-dimensional, and given the familiar relationship between spaces and frames it is then natural to call a completely regular frame strongly 0-dimensional if its compact completely regular coreflection is 0-dimensional (meaning: is generated by its complemented elements). Indeed, it is then seen...

We generalize the unfolding semantics, previously developed for concrete formalisms such as Petri nets and graph grammars, to the abstract setting of (single pushout) rewriting over adhesive categories. The unfolding construction is characterized as a coreflection, i.e. the unfolding functor arises as the right adjoint to the embedding of the category of occurrence grammars into the category of...

It is shown that for any commutative unital ring R the category HopfR of R–Hopf algebras is locally presentable and a coreflective subcategory of the category BialgR of R–bialgebras, admitting cofree Hopf algebras over arbitrary R–algebras. The proofs are based on an explicit analysis of the construction of colimits of Hopf algebras, which generalizes an observation of Takeuchi. Essentially be ...

We prove that in the category of achimedean lattice-ordered groups with weak unit there is no homomorphism-closed monoreflection strictly between the strongest essential monoreflection (the so-called “closure under countable composition”) and the strongest monoreflection (the epicompletion). It follows that in the category of regular σframes, the only non-trivial monoreflective subcategory that...

Classically, a Tychonoff space is called strongly 0-dimensional if its Stone-Čech compactification is 0-dimensional, and given the familiar relationship between spaces and frames it is then natural to call a completely regular frame strongly 0-dimensional if its compact completely regular coreflection is 0-dimensional (meaning: is generated by its complemented elements). Indeed, it is then seen...

We introduce the notion of strong concatenable process for Petri nets as the least refinement of non-sequential (concatenable) processes which can be expressed abstractly by means of a functor Q[ ] from the category of Petri nets to an appropriate category of symmetric strict monoidal categories with free non-commutative monoids of objects, in the precise sense that, for each net N , the strong...

This paper looks at the design process of the WamBam; a selfcontained electronic hand-drum meant for music therapy sessions with severely intellectually disabled clients. Using coreflection with four musical therapists and literature research, design guidelines related to this specific user-group and context are formed. This leads to a concept of which the most relevant aspects are discussed, b...

We study structures called d-frames which were developed by the last two authors for a bitopological treatment of Stone duality. These structures consist of a pair of frames thought of as the opens of two topologies, together with two relations which serve as abstractions of disjointness and covering of the space. With these relations, the topological separation axioms regularity and normality ...