Towards a characterization of universal categories

Algebra Universalis, 1972

We consider examples of the n-categories having the same n-graph.

Advances in Mathematics, 2014

We construct an embedding G : Graphs → Ab of the category of graphs into the category of abelian groups such that for X and Y in Graphs we have

2015

Abstract. Let X be a topological graph; i.e., a union of finitely many points and arcs, with arcs joined only at end points. If Y is any locally connected metrizable compactum that is co-elementarily equivalent to X, then Y is homeomorphic to X. In particular, X and Y are homeomorphic if some lattice base for one is elementarily equivalent to some lattice base for the other. 1. introduction This paper is about the model-theoretic topology of compact Hausdorff spaces— also referred to as compacta—and our aim is to show that any topological graph is categorical, relative to the class of compacta that are locally connected and metrizable. As the term categorical has a range of interpretations, we begin with a general description of how it is used here. Suppose K is a class of objects, together with two reflexive symmetric relations, one finer than the other. To keep things straight, call the finer relation indistinguishability (always an equivalence relation) and the coarser one simila...

Journal of The London Mathematical Society-second Series, 1971

In [1, 2] R. Rado proved several theorems on universal graphs. It is the purpose of this note to point out that the substance of these theorems, in fact strengthenings of them, can be obtained from general algebraic results of B. Jonsson or from related model theoretic results of M. Morley and R. Vaught. However, as we explain below, it seems to be the case that not all Rado's results can be obtained in this way, a fact which gives a little added interest to the comparison of results.

This old essay deals with a connection between a relatively recent (1940s and 1950s) field of mathematics, category theory, and a hitherto vague notion of philosophical logic usually associated with Plato, the self-predicative universal or concrete universal. While category theory can be quite forbidding to the non-specialist, simple examples can be used based on inclusion between sets. Given two sets A and B, consider the property of being a set X that is contained in A and is contained in B. In other words, the property is the property of being both a subset of A and a subset of B. In this example, the "participation" relation is the subset relation. There is a set, namely the intersection, meet, or overlap of A and B, denoted AmeetB, that has the property (so it is a "concrete" instance of the property), and it is universal in the sense that any other set has the property if and only if it participates in the universal example: concreteness: AmeetB is a subset of both A and B, and universality: X is a subset of AmeetB if and only if X is contained in both A and B. Thus the intersection AmeetB is the concrete universal for the property of being a subset of A and a subset of B. I argue that category theory is relevant to foundations as the theory of concrete universals. Category theory provides the framework to identify the concrete universals in mathematics, the concrete instances of a mathematical property that exemplify the property is such a perfect and paradigmatic way that all other instances have the property by virtue of participating in the concrete universal.

The Electronic Journal of Combinatorics - Electr. J. Comb., 2006

This is an account for the combinatorially minded reader of various categories of directed and undirected graphs, and their analogies with the category of sets. As an application, the endomorphisms of a graph are in this context not only composable, giving a monoid structure, but also have a notion of adjacency, so that the set of en- domorphisms is both a monoid and a graph. We extend Shrimpton's (unpublished) investigations on the morphism digraphs of reexiv e digraphs to the undirected case by using an equivalence between a category of reexiv e, undirected graphs and the category of reexiv e, directed graphs with reversal. In so doing, we emphasise a picture of the elements of an undirected graph, as involving two types of edges with a single vertex, namely 'bands' and 'loops'. Such edges are distinguished by the behaviour of morphisms with respect to these elements.

2015

In this paper we define a sequence of monads T(∞,n)(n ∈ N) on the category ∞-Gr of ∞-graphs. We conjecture that algebras for T(∞,0), which are defined in a purely algebraic setting, are models of∞-groupoids. More generally, we conjecture that T(∞,n)-algebras are models for (∞, n)-categories. We prove that our (∞, 0)-categories are bigroupoids when truncated at level 2. Introduction The notion of weak (∞, n)-category can be made precise in many ways depending on our approach to higher categories. Intuitively this is a weak∞-category such that all its cells of dimension greater than n are equivalences. Models of weak (∞, 1)-categories (case n = 1) are diverse: for example there are the quasicategories studied by Joyal and Tierney (see [24]), but also there are other models which have been studied like the Segal categories, the complete Segal spaces, the simplicial categories, the topological categories, the relative categories, and there are known to be equivalent (a survey of models ...

A set A of vertices of a graph G is called d-scattered in G if no two d-neighborhoods of (distinct) vertices of A intersect. In other words, A is d-scattered if no two distinct vertices of A have distance at most 2d. This notion was isolated in the context of finite model theory by Gurevich and recently it played a prominent role in the study of homomorphism preservation theorems for special classes of structures (such as minor closed families). This in turn led to the notions of wide, semiwide and quasi-wide classes of graphs. It has been proved previously that minor closed classes and classes of graphs with locally forbidden minors are examples of such classes and thus (relativised) homomorphism preservation theorem holds for them. In this paper we show that (more general) classes with bounded expansion and (newly defined) classes with bounded local expansion and even (very general) classes of nowhere dense graphs are quasi wide. This not only strictly generalizes the previous results and solves several open problems but it also provides new proofs. It appears that bounded expansion and nowhere dense classes are perhaps a proper setting for investigation of wide-type classes as in several instances we obtain a structural characterization. This also puts classes of bounded expansion in the new context and we are able to prove a trichotomy result which separates classes of graphs which are dense (somewhere dense), nowhere dense and finite. Our motivation stems from finite dualities. As a corollary we obtain that any homomorphism closed first order definable property restricted to a bounded expansion class is a duality.

SSRN Electronic Journal, 2000

Today it would be considered "bad Platonic metaphysics" to think that among all the concrete instances of a property there could be a universal instance so that all instances had the property by virtue of participating in that concrete universal. Yet there is a mathematical theory, category theory, dating from the mid-20 th century that shows how to precisely model concrete universals within the "Platonic Heaven" of mathematics. This paper, written for the philosophical logician, develops this category-theoretic treatment of concrete universals along with a new concept to abstractly model the functions of a brain.