arXiv subjects

Annanthakrishna Manokaran

Publications and source records attributed to Annanthakrishna Manokaran.

4 recordsLinked to original sources

arXiv2026

Presenting Point-Glued Self-Similar Objects as Final Coalgebras in Metric Spaces via Completion: A Categorical Hutchinson Theorem

We study self-similar spaces obtained by repeatedly gluing finitely many scaled copies of a pointed metric space along prescribed boundary points. A finite gluing specification defines endofunctors on suitable categories of pointed sets and pointed metric spaces. Under non-degeneracy and metric admissibility assumptions, the initial algebra of the set functor carries a canonical metric. Its Cauchy completion is a final coalgebra for the metric functor and, after forgetting the metric, also for the underlying set functor. We compare this construction with Iterated Function Systems. When the orbit algebra is injective and its comparison with the initial algebra is bi-Lipschitz, the attractor is isomorphic to the completed final coalgebra. This gives a categorical analogue of Hutchinson's theorem for overlaps captured by finitely many distinguished points. Examples include the unit interval, Cantor spaces, the Sierpiński gasket and $p$-adic integers.

math.CT↗

arXiv2021

Presenting the Sierpinski Gasket in Various Categories of Metric Spaces

This paper studies presentations of the Sierpinski gasket as a final coalgebra for functors on several categories of metric spaces with additional designated points. The three categories which we study differ on their morphisms: one uses short (non-expanding) maps, a second uses Lipschitz maps, and a third uses continuous maps. The functor in all cases is very similar to what we find in the standard presentation of the gasket as an attractor. We prove that the Sierpinski gasket itself is the final coalgebra of a naturally-occurring functor in the continuous setting. In the short setting, the final coalgebra exists but it is better described as the completion of the initial algebra, and this is not isomorphic to the Sierpinski gasket. In the Lipschitz setting, the final coalgebra does not exist. We determine the initial algebras in all three settings as well.

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arXiv2019

Sierpinski Gasket as a Final Coalgebra Obtained by Cauchy Completing the Initial Algebra

This paper presents the Sierpinski Gasket ($\mathbb{S}$) as a final coalgebra obtained by Cauchy completing the initial algebra for an endofunctor on the category of tri-pointed one bounded metric spaces with continuous maps. It has been previously observed that $\mathbb{S}$ is bi-Lipschitz equivalent to the coalgebra obtained by completing the initial algebra, where the latter was observed to be final when morphisms are restricted to short maps. This raised the question "Is $\mathbb{S}$ the final coalgebra in the Lipschitz setting?". The results of this paper show that the natural setup is to consider all continuous functions. The description of the final coalgebra as the Cauchy completion of the initial algebra has been explicitly used to determine the mediating morphism from a given coalgebra to the the final coalgebra. This has been used to show that if the structure map of a coalgebra is continuous, then so is the mediating morphism. The description of $\mathbb{S}$ given here not only generalizes previous observations, but also unifies classical descriptions of $\mathbb{S}$. We also show, by means of an example, that $\mathbb{S}$ is not the final coalgebra if we consider only Lipschitz maps.

math.CT↗

arXiv2017

On the Initial Algebra and Final Co-algebra of some Endofunctors on Categories of Pointed Metric Spaces

We consider two endofunctors of the form $~F:X\longrightarrow M\otimes X~$, where $~M~$ is a non degenerate module, related to the unit interval and the Sierpinski gasket, and their final co-algebras. The functors are defined on the categories of bi-pointed and tri-pointed metric spaces, with continuous maps, short maps or Lipschitz maps as the choice of morphisms. First we demonstrate that the final co-algebra for these endofunctors on the respective category of pointed metric spaces with the choice of continuous maps is the final co-algebra of that with short maps and after forgetting the metric structure is of that in the set setting. We use the fact that the final co-algebra can be obtained by a Cauchy completion process, to construct the mediating morphism from a co-algebra by means of the limit of a sequence obtained by iterating the co-algebra. We also show that the Sierpinski gasket $~(\mathbb{S},σ)~$ is not the final co-algebra for these endofunctors when the morphism are restricted to being Lipschitz maps.

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