U. Haifa Topology & Geometry seminar: Sunday, Jan. 30, 2022. Speaker: Sanjeevi Krishnan (Ohio State University). “Title: A Kan Condition for Directed Homotopy”
Time: Sunday Jan 30th, 17:00 (Israel Time)
Location: On Zoom
Speaker: Sanjeevi Krishnan (Ohio State University)
Title: A Kan Condition for Directed Homotopy
The Kan condition guarantees that a simplicial set, a formal colimit of abstract simplices, models enough features of a topological space so that classical homotopy theory can be given a completely combinatorial description. There is a similar such Kan condition for cubical sets, formal colimits of abstract hypercubes. We present a generalization of the Kan condition suitable for directed homotopy theory, a homotopy theory for spaces with direction (like spacetimes). Concretely, we present a lifting condition on cubical sets, satisfied by all nerves of small categories, that yields a cubical approximation theorem for directed topology. Intuitively, this condition can be regarded as some kind of weak higher categorical structure on a cubical set. A distinguishing feature of the lifting condition is that it is an algebraic lifting condition [Garner, Grandis, Riehl] in the sense that the lifts have to be suitably natural. One simple application is a cochain-theoretic description of a directed $1$-cohomology theory, represented by directed classifying spaces of commutative monoids.
Meeting ID: 849 8759 1026