BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Victoria Noquez (Indiana University) DTSTART:20201001T180000Z DTEND:20201001T190000Z DTSTAMP:20260423T021310Z UID:OLS/24 DESCRIPTION:Title: The Sierpinski Carpet as a Final Coalgebra Obtained by Completing an Initial Algebra\nby Victoria Noquez (Indiana University) as part of Online log ic seminar\n\n\nAbstract\nThe background for this work includes Freyd's Th eorem\, in which the unit interval is viewed as a final coalgebra of a cer tain endofunctor in the category of bipointed sets. Leinster generalized t his to a broad class of self-similar spaces in categories of sets\, also c haracterizing them as topological spaces. Bhattacharya\, Moss\, Ratnayake\ , and Rose went in a different direction\, working in categories of metric spaces\, obtaining the unit interval and the Sierpinski Gasket as a final colagebras in the categories of bipointed and tripointed metric spaces re spectively. To achieve this they used a Cauchy completion of an initial al gebra to obtain the required final coalgebra. In their examples\, the iter ations of the fractals can be viewed as gluing together a finite number of scaled copies of some set at some finite set of points (e.g. corners of t riangles). Here we will expand these ideas to apply to a broader class of fractals\, in which copies of some set are glued along segments (e.g. side s of a square). We use the method of completing an initial algebra to obta in the Sierpinski Carpet as a final coalgebra in a category of metric spac es\, and note the required adaptations to this approach\, most notably tha t we no longer get the initial algebra as the colimit of a countable seque nce of metric spaces. We will explore some ways in which these results may be further generalized to a broader class of fractals. Joint work with La rry Moss.\n LOCATION:https://researchseminars.org/talk/OLS/24/ END:VEVENT END:VCALENDAR