The Categories of Graphs

In traditional studies of graph theory, the graphs allow only one edge to be incident to any two vertices, not necessarily distinct, and the graph morphisms must map edges to edges and vertices to vertices while preserving incidence. We refer to these restricted morphisms as strict morphisms. We relax the conditions on the graphs by allowing any number of edges to be incident to any two vertices, as well as relaxing the condition on graph morphisms by allowing edges to be mapped to vertices, provided that incidence is still preserved. We call the broader category of these graphs and these morphisms the Category of Conceptual Graphs and Graph Morphisms, denoted Grphs. We then define four other concrete categories of graphs created by combinations of restrictions of the graph morphisms as well as restrictions on the allowed graphs. We determine the categorial structure of these six categories of graphs by characterizing common categorially defined structures and properties and by characterizing six special types of monomorphisms, and dually six special types of epimorphisms. We also establish the Fundamental Morphism Theorem in two of the categories of graphs. We then provide an Elementary Theory for five categories of graphs, producing a list of first-order axioms that, when taken with the higher-order axiom of the existence of small products and coproducts, characterizes these five categories of graphs. We also provide a result toward Hedetniemi\u27s conjecture that arose from the study of the categories of graphs

Generalised morphisms of k-graphs: k-morphs

In a number of recent papers, (k+l)-graphs have been constructed from k-graphs by inserting new edges in the last l dimensions. These constructions have been motivated by C*-algebraic considerations, so they have not been treated systematically at the level of higher-rank graphs themselves. Here we introduce k-morphs, which provide a systematic unifying framework for these various constructions. We think of k-morphs as the analogue, at the level of k-graphs, of C*-correspondences between C*-algebras. To make this analogy explicit, we introduce a category whose objects are k-graphs and whose morphisms are isomorphism classes of k-morphs. We show how to extend the assignment \Lambda \mapsto C*(\Lambda) to a functor from this category to the category whose objects are C*-algebras and whose morphisms are isomorphism classes of C*-correspondences.Comment: 27 pages, four pictures drawn with Tikz. Version 2: title changed and numerous minor corrections and improvements. This version to appear in Trans. Amer. Math. So

Generalized Typed Attributed Graph Transformation Systems based on Morphisms Changing Type Graphs and Data Signature

Our aim is to extend the framework of typed attributed graphs in [1] to generalized typed attributed graphs. They are based on generalized attributed graph morphisms, short GAG-morphisms, which allow to change the type graph, data signature, and domain. This allows to formulate type hierarchies and views of visual languages defined by GAG-morphisms between type graphs, short GATG-morphisms. In order to study interaction and integration of views, restriction of views along type hierarchies, restriction and integration of consistent view models and reflection of behaviour between different typed attributed graph transformation systems we present suitable conditions for the construction of pushouts and pullbacks, and special van Kampen properties in the category GAGraphs of generalized attributed graphs. Moreover, we show that (GAGraphs,M) and (GAGraphsATG,M) are adhesive HLR categories for the class M of injective, persistent, and signature preserving morphisms

Operads, configuration spaces and quantization

We review several well-known operads of compactified configuration spaces and construct several new such operads, C, in the category of smooth manifolds with corners whose complexes of fundamental chains give us (i) the 2-coloured operad of A-infinity algebras and their homotopy morphisms, (ii) the 2-coloured operad of L-infinity algebras and their homotopy morphisms, and (iii) the 4-coloured operad of open-closed homotopy algebras and their homotopy morphisms. Two gadgets - a (coloured) operad of Feynman graphs and a de Rham field theory on C - are introduced and used to construct quantized representations of the (fundamental) chain operad of C which are given by Feynman type sums over graphs and depend on choices of propagators.Comment: 58 page

Topos-like Properties in Two Categories of Graphs and Graph-like Features in an Abstract Category

In the study of the Category of Graphs, the usual notion of a graph is that of a simple graph with at most one loop on any vertex, and the usual notion of a graph homomorphism is a mapping of graphs that sends vertices to vertices, edges to edges, and preserves incidence of the mapped vertices and edges. A more general view is to create a category of graphs that allows graphs to have multiple edges between two vertices and multiple loops at a vertex, coupled with a more general graph homomorphism that allows edges to be mapped to vertices as long as that map still preserves incidence. This more general category of graphs is named the Category of Conceptual Graphs. We investigate topos and topos-like properties of two subcategories of the Category of Conceptual Graphs. The first subcategory is the Category of Simple Loopless Graphs with Strict Morphisms in which the graphs are simple and loopless and the incidence preserving morphisms are restricted to sending edges to edges, and the second subcategory is the Category of Simple Graphs with Strict Morphisms where at most one loop is allowed on a vertex. We also define graph objects that are their graph equivalents when viewed in any of the graph categories, and mimic their graph equivalents when they are in other categories. We conclude by investigating the possible reflective and corefective aspects of our two subcategories of graphs