2012/11/21 by Arash Rastegar, Rastegar, Arash
Mathematics · #Advanced Topology and Set Theory #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1211.4968
openalex publication_date 2012/11/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We define the concept of self-similarity of an object by considering endomorphisms of the object as `similarity' maps. A variety of interesting examples of self-similar objects in geometry, algebra and arithmetic are introduced. Self-similar objects provide a framework in which, one can unite some results and conjectures in different mathematical frameworks. In some general situations, one can define a well-behaved notion of dimension for self-similar objects. Morphisms between self-similar objects are also defined and a categorical treatment of this concept is provided. We conclude by some philosophical remarks.