We study diagrams associated with a finite simplicial complex K, in various algebraic and topological categories. We relate their colimits to familiar structures in algebra, combinatorics, geometry and topology. These include: right-angled Artin and Coxeter groups (and their complex analogues, which we call circulation groups); Stanley-Reisner algebras and coalgebras; Davis and Januszkiewicz’s ...

For deterministic systems, expressed as coalgebras over polynomial functors, every tree t (an element of the final coalgebra) turns out to represent a new coalgebra At. The universal property of these coalgebras, resembling freeness, is that for every state s of every system S there exists a unique coalgebra homomorphism from a unique At which takes the root of t to s. Moreover, the tree coalge...

Software architects such as Garlan and Katz promote the separation of computation from coordination. They encourage the study of connectors as first-class entities, and superposition of connectors onto components as a paradigm for component-oriented programming. We demonstrate that this is a good model for what visual programming tools like IBM’s VisualAge actually do. Moreover, Fiadeiro and Ma...

The group of compactly supported homeomorphisms on a Tychonoff space can be topologized in number ways, including as colimit homeomorphism groups with given compact support, or subgroup the its Stone-\v{C}ech compactification. A is said to have Compactly Supported Homeomorphism Property (CSHP) if these two topologies coincide. authors provide necessary and sufficient conditions for finite produ...

We define a construction operation on hypergraphs based on a colimit and show that its expressiveness is equal to the graph expressions of Bauderon and Courcelle. We also demonstrate that by closing a set of rewrite rules under graph construction we obtain a notion of rewriting equivalent to the double-pushout approach of Ehrig. The usefulness of our approach for the compositional modelling of ...

Using category theoretic notions, in particular diagrams and their colimits, we provide a common semantic backbone for various notions of modularity in structured ontologies, and outline a general approach for representing (heterogeneous) combinations of ontologies through interfaces of various kinds, based on the theory of institutions. This covers theory interpretations, (definitional) langua...

A fundamental construction in the study of stable homotopy is the free infinite loop space generated by a space X. This is the colimit QX = lim −→ ΩΣX. The i homotopy group of QX is canonically isomorphic to the i stable homotopy group of X. Thus, one may obtain stable information about X by obtaining topological results about QX. One such result is the Kahn-Priddy theorem [7]. In another direc...

For a broad collection of categories K, including all presheaf categories, the following statement is proved to be consistent: every left exact (i.e. finite-limits preserving) functor from K to Set is small, that is, a small colimit of representables. In contrast, for the (presheaf) category K = Alg(1, 1) of unary algebras we construct a functor from Alg(1, 1) to Set which preserves finite prod...

Let $(\mathcal C,\otimes,1)$ be an abelian symmetric monoidal category satisfying certain conditions and let $X$ a scheme over in the sense of Toen Vaquie. In this paper we show that when is quasi-compact semi-separated, any quasi-coherent sheaf on may expressed as directed colimit its finitely generated submodules. Thereafter, introduce notion field objects satisfy several properties similar t...

Several modularity concepts for ontologies have been studied in the literature. Can they be brought to a common basis? We propose to use the language of category theory, in particular diagrams and their colimits, for answering this question. We outline a general approach for representing combinations of logical theories, or ontologies, through interfaces of various kinds, based on diagrams and ...