Title: Equivalence between combinatorial and topological dihomotopy
Speaker: Sanjeevi Krishnan (Ohio State University)
Abstract: Quillen equivalences between topological spaces and various presheaves formalize an equivalence between combinatorial and topological homotopy. This talk describes a recent extension of that Quillen equivalence for dihomotopy, homotopy respecting extra directionality on a space. On the combinatorial side, the fibrant objects model what might be called higher ordered groupoids, including nerves of preordered groups. On the topological side, the cofibrant objects include compact conal manifolds with locally constant, free, positive and generating cone fields. Neither model structure is cofibrantly generated in the usual sense, a seemingly unavoidable feature of dihomotopy. This talk describes the equivalent model structures; some of the recent theory of algebraic model structures [Riehl] and their underlying algebraic weak factorization systems [Gambino, Garner, Grandis] making the constructions possible; and sketches a simple application, a homological algebra for general monoids.
Seminar URL: https://www.asc.ohio-state.edu/fontes.17/homotopy_seminar/