2023/08/19 by Niemiec, Piotr
#05C15 #2010: Primary 22F30 #Category Theory (math.CT) #FOS: Mathematics #General Topology (math.GN) #Secondary 05C63
paper · doi:10.48550/arxiv.2308.09986
The main aim of the paper is to study in greater detail absolutely homogeneous structures (that is, objects with the property that each partial isomorphism extends to a global automorphism), with special emphasis on metric spaces and (possibly infinite, full) graphs with edge-coloring. Besides, a general categorical approach to this concept is presented. The main achievement of the paper is the discovery of one-to-one correspondence between absolutely homogeneous objects and certain classes (that become sets when isomorphic objects are identified) of "finite" objects that satisfy a few quite general axioms (such as amalgamation and heredity). It is also introduced and discussed in detail the concept of products for graphs with edge-coloring (that produces an absolutely homogeneous graph provided all factors are so). Among the most significant results of the paper, it is worth mentioning a full classification (up to isometry) of all absolutely homogeneous ultrametric spaces as well as of all absolutely homogeneous graphs with edge-coloring in which all triangles are isosceles or in which all triangles are (precisely) tricolored.