{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,9,6]],"date-time":"2024-09-06T07:13:57Z","timestamp":1725606837616},"reference-count":0,"publisher":"EasyChair","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:p>This paper is a contribution to the presentation of fractal sets in terms of final coalgebras.<\/jats:p><jats:p>The first result on this topic was Freyd's Theorem: the unit interval [0,1] is the final coalgebra of<\/jats:p><jats:p>a certain functor on the category of bipointed sets.  Leinster 2011 offers<\/jats:p><jats:p>a sweeping generalization of this result.   He is able to represent many of what would be intuitively<\/jats:p><jats:p>called \"self-similar\" spaces using (a) bimodules (also called profunctors or distributors),<\/jats:p><jats:p>(b) an examination of non-degeneracy conditions on functors of various sorts; (c) a construction of<\/jats:p><jats:p>final coalgebras for the types of functors of interest using a notion of resolution.    In addition to the<\/jats:p><jats:p>characterization of fractals sets as sets, his seminal paper also characterizes them as topological spaces.<\/jats:p><jats:p>Our major contribution is to suggest that in many cases of interest, point (c) above on resolutions<\/jats:p><jats:p>is not needed in the construction of final coalgebras.   Instead, one may obtain a number of spaces of<\/jats:p><jats:p>interest as  the Cauchy completion of  an initial algebra,<\/jats:p><jats:p>and this initial algebra is the set of points in a colimit of an omega-sequence of finite metric spaces.<\/jats:p><jats:p>This generalizes Hutchinson's 1981 characterization of fractal attractors  as<\/jats:p><jats:p>closures of the orbits of the critical points. In addition to simplifying the overall machinery, it also presents a metric space which is ``computationally related'' to the overall fractal.   For example, when applied to Freyd's construction, our method yields the metric space.<\/jats:p><jats:p>of dyadic rational numbers in [0,1].<\/jats:p><jats:p>Our second contribution is not completed at this time, but it is a set of results on \\emph{metric space}<\/jats:p><jats:p>characterizations of final coalgebras.   This point was raised as an open issue in Hasuo, Jacobs, and Niqui 2010,<\/jats:p><jats:p>and our interest in  quotient metrics comes from their paper.     So in terms of (a)--(c) above, our work<\/jats:p><jats:p>develops  (a) and (b) in metric settings while dropping (c).<\/jats:p>","DOI":"10.29007\/pw5g","type":"proceedings-article","created":{"date-parts":[[2018,1,23]],"date-time":"2018-01-23T17:59:06Z","timestamp":1516730346000},"page":"158-152","source":"Crossref","is-referenced-by-count":1,"title":["Fractal Sets as  Final Coalgebras Obtained by Completing an Initial Algebra"],"prefix":"10.29007","volume":"25","author":[{"given":"Larry","family":"Moss","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jayampathy","family":"Ratnayake","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Robert","family":"Rose","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"11545","event":{"name":"TACL 2013. Sixth International Conference on Topology, Algebra and Categories in Logic"},"container-title":["EPiC Series in Computing"],"original-title":[],"deposited":{"date-parts":[[2018,1,23]],"date-time":"2018-01-23T18:01:25Z","timestamp":1516730485000},"score":1,"resource":{"primary":{"URL":"https:\/\/easychair.org\/publications\/paper\/L2H5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[null]]},"references-count":0,"URL":"https:\/\/doi.org\/10.29007\/pw5g","relation":{},"ISSN":["2398-7340"],"issn-type":[{"type":"print","value":"2398-7340"}],"subject":[]}}