2021/10/13 by Victoria Noquez, Noquez, Victoria, Lawrence S. Moss +1
Arts and Humanities · #18B99 #28A80 #Category Theory (math.CT) #FOS: Mathematics #Musicology and Musical Analysis
paper · doi:10.48550/arxiv.2110.06404
openalex publication_date 2021/10/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.