It is folklore that tree-width is monotone under taking subgraphs (i.e. injective graph homomorphisms) and contractions (certain kinds of surjective graph homomorphisms). However, although tree-width is obviously not monotone under any surjective graph homomorphism, it is not clear whether contractions are canonically the only class of surjections with respect to which it is monotone. Under the requirement that the decomposition shape must be preserved, we prove that this is indeed the case. Our results provide a framework for answering questions of this sort for many other kinds of combinatorial data structures (such as directed multigraphs, hypergraphs, Petri nets, circular port graphs, half-edge graphs, databases, simplicial sets etc.) for which natural analogues of tree decompositions can be defined. Furthermore and of independent interest, we prove these results by introducing the notion of a lasso, a generalization of contractions of graphs to arbitrary categories with pushouts of monomorphisms.
Earlier, it was shown that if a (locally small) total category 𝒦 is totally distributive then its wave semicategory 𝒦 is a taxon in the sense of Koslowski. And there is then an equivalence 𝒦→ ^≃imod(1,𝒦) , where the latter is the category of small interpolative modules from 1 to 𝒦 , again in the sense of Koslowski. Here we show that if a taxon T is what we call Freyd/Street-small then imod(1,T) is totally distributive.
We use the Chu construction to functorially build BV-categories from duoidal categories, demonstrating that candidate models of BV-logic can be cofreely constructed from a fragment of a model of Guglielmi’s sequencing operator. By using this construction to show that the strong Hyland envelope is a BV-category, we find a way to build a canonical model of spatio-temporal relationships between agents in spacetime from any symmetric monoidal category. The concrete physical interpretation of spacetime events in this model as intervention–context pairs resolves deficiencies in previous attempts to give a general categorical semantics to quantum supermaps.
In this paper, a topological representation for posets is presented by using irreducible Frink filters. Utilizing the topological representation, two particular Δ _1 -completions for posets are constructed. Furthermore, two categorical dualities are established for posets that differ essentially in their treatment of morphisms. The first, extending the Celani-González duality for meet-semilattices, interprets morphisms between topological spaces as certain relations, while keeping functions on the posets side. The second duality, in contrast, preserves the morphism structure on both sides as actual mappings, thereby yielding a fully functorial contravariant equivalence. Moreover, when restricted to the full subcategory of bounded distributive lattices, this second duality specializes to the classical Stone-type duality.
