BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Chaitanya Leena Subramaniam (Université Paris Diderot) DTSTART:20210917T130000Z DTEND:20210917T140000Z DTSTAMP:20260423T004551Z UID:ACPMS/7 DESCRIPTION:Title: De pendent type theory and higher algebraic structures\nby Chaitanya Leen a Subramaniam (Université Paris Diderot) as part of Algebraic and Combina torial Perspectives in the Mathematical Sciences\n\n\nAbstract\nIn classic al universal algebra\, every family of algebraic structures (such as monoi ds\, groups\, rings\, modules\, small categories\, operads\, sheaves) can be classified by a syntactic (algebraic or essentially algebraic) equation al theory. A cornerstone of universal algebra is the equivalence between a lgebraic theories and finitary monads on the category of sets\, due to Law vere\, B\\'enabou and Linton. Higher algebraic structures (such as loop sp aces\, E-k spaces\, infinity-categories\, infinity-operads and their modul es and algebras\, stacks\, spectra) are algebraic structures up to homotop y in spaces ("spaces" = topological spaces\, simplicial sets or any other model of homotopy types). It is a long-standing presupposition among homot opy type theorists that the dependent types introduced by Martin-L\\"of ar e particularly well-suited to providing syntactic theories and a universal algebra for higher algebraic structures. In this talk\, we will see a (fe w) definition(s) of "dependently sorted/typed algebraic theory" and descri be a monad-theory equivalence strictly generalising that of Lawvere-Bénab ou-Linton. With respect to their Set-valued models\, dependently sorted al gebraic theories have the same expressive power as essentially algebraic t heories. However\, as we will see in this talk\, dependently sorted algebr aic theories have the advantage of having a good theory of models up-to-ho motopy in spaces\, which generalises the theory of homotopy-models of alge braic theories due to Schwede\, Badzioch\, Rezk and Bergner. We will see t hat many familiar algebraic structures (such as n-categories\, omega-categ ories\, coloured planar operads\, opetopic sets) are very naturally seen t o be models of dependently sorted algebraic theories. The crux of these re sults is a correspondence between the dependent sorts/types of any depende ntly sorted algebraic theory T\, and a certain "cellularity" underlying ev ery algebraic structure described by T (i.e. every T-model). The goal of t his talk will be to explain this correspondence between type dependency an d cellularity\, and why this cellularity marries well with homotopy theory .\n LOCATION:https://researchseminars.org/talk/ACPMS/7/ END:VEVENT END:VCALENDAR