2019/07/19 by Jayampathy Ratnayake, Ratnayake, Jayampathy, Annanthakrishna Manokaran +3
Computer Science · Mathematics · #Advanced Algebra and Logic #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic, programming, and type systems
paper · pdf · doi:10.48550/arxiv.1907.09933
openalex publication_date 2019/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper presents the Sierpinski Gasket ( mathbbS) as a final coalgebra\nobtained by Cauchy completing the initial algebra for an endofunctor on the\ncategory of tri-pointed one bounded metric spaces with continuous maps. It has\nbeen previously observed that mathbbS is bi-Lipschitz equivalent to the\ncoalgebra obtained by completing the initial algebra, where the latter was\nobserved to be final when morphisms are restricted to short maps. This raised\nthe question "Is mathbbS the final coalgebra in the Lipschitz setting?".\nThe results of this paper show that the natural setup is to consider all\ncontinuous functions. The description of the final coalgebra as the Cauchy\ncompletion of the initial algebra has been explicitly used to determine the\nmediating morphism from a given coalgebra to the the final coalgebra. This has\nbeen used to show that if the structure map of a coalgebra is continuous, then\nso is the mediating morphism. The description of mathbbS given here not\nonly generalizes previous observations, but also unifies classical descriptions\nof mathbbS. We also show, by means of an example, that mathbbS is not\nthe final coalgebra if we consider only Lipschitz maps.\n