塞皮罗维茨地毯作为最终余代数
The Sierpinski Carpet as a Final Coalgebra
AI总结：
本文通过构造正方形集的范畴和内函子，证明塞皮罗维茨地毯与最终余代数在双Lipschitz等价，并探讨了分形集与递归结构的关系。
摘要：
我们推进了将最终余代数作为数学中循环性来源与实数集分形集之间的联系的计划。特别是，我们关注塞皮罗维茨地毯，将其视为单位正方形的分形子集。我们构建了一个正方形集的范畴及其上的一个内函子，该函子对应于沿线段粘合正方形集的复制。我们证明了对于我们的函子，初始代数和最终余代数存在，并且最终余代数与塞皮罗维茨地毯在双Lipschitz等价。在此过程中，我们与诸如迭代构造初始代数作为余极限、核心递归代数以及哈钦森对分形集的经典处理等主题建立了联系。
英文摘要：
We advance the program of connections between final coalgebras as sources of circularity in mathematics and fractal sets of real numbers. In particular, we are interested in the Sierpinski carpet, taking it as a fractal subset of the unit square. We construct a category of square sets and an endofunctor on it which corresponds to the operation of gluing copies of a square set along segments. We show that the initial algebra and final coalgebra exist for our functor, and that the final coalgebra is bi-Lipschitz equivalent to the Sierpinski carpet. Along the way, we make connections to topics such as the iterative construction of initial algebras as colimits, corecursive algebras, and the classic treatment of fractal sets due to Hutchinson.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
