Homotopy theory of finite total orders, trees and chicken feet

A transfer system is a graph on a lattice satisfying certain restriction and composition properties. They were first studied on the lattice of subgroups of a finite group in order to examine equivariant homotopy commutativity, which then unlocked a wealth of links to combinatorial methods. On a finite total order [n], transfer systems can be used to classify different homotopy theories on [n]. The talk will involve plenty of examples and not assume any background knowledge.