BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Manuel Rivera (Purdue University) DTSTART:20221017T140000Z DTEND:20221017T150000Z DTSTAMP:20260423T024537Z UID:MITTop/53 DESCRIPTION:Title: Simplicial coalgebras under three different notions of weak equivalence\nby Manuel Rivera (Purdue University) as part of MIT topology seminar\n\ n\nAbstract\n\\noindent Motivated by constructing algebraic models for hom otopy types\, I will discuss three different homotopy theories on the cate gory of simplicial cocommutative coalgebras corresponding to the following notions of weak equivalence:\n\n\\vspace{2ex}\n\n\\begin{itemize}\n\n\\it em 1. maps of simplicial coalgebras which become quasi-isomorphisms of dif ferential graded (dg) coalgebras after applying the normalized chains func tor\n\n\\item 2. maps of simplicial coalgebras which become quasi-isomorph isms of dg algebras after applying the normalized chains functor followed by the dg cobar construction\, and\n\n\\item 3. maps of simplicial coalgeb ras which become quasi-isomorphisms of dg algebras after applying a locali zed version of the dg cobar construction.\n\n\\end{itemize}\n\n\\vspace{2e x}\n\n\\noindent Notion (1) was used by Goerss to provide a fully-faithful model for spaces up to F-homology equivalence\, for a F an algebraically closed field. I will explain how (2)\, which is drawn from dg Koszul duali ty theory\, corresponds to a linearized version of the notion of categoric al equivalence between simplicial sets as used in the theory of quasi-cate gories. I will also explain how (3) leads to a fully-faithful model for th e homotopy theory of simplicial sets considered up to maps that induce iso morphisms on fundamental groups and on the F-homology of the universal cov ers\, for F an algebraically closed field. One of the key points is a sort of homological formulation of the fundamental group. This is based on joi nt work with G. Raptis and also on work with F. Wierstra and M. Zeinalian. \n LOCATION:https://researchseminars.org/talk/MITTop/53/ END:VEVENT END:VCALENDAR