The following paper has been accepted at the 33rd International Conference on Concurrency Theory (CONCUR 2022), to be held in Warsaw next week: A Kleene Theorem for Higher-Dimensional Automata Uli Fahrenberg, Christian Johansen, Georg Struth, Krzysztof Ziemiański (LR(D)E, NTNU Gjøvik, U of Sheffield, Warsaw U) We prove a Kleene theorem for higher-dimensional automata (HDAs). It states that the languages they recognise are precisely the rational subsumption-closed sets of interval pomsets. The rational operations include a gluing composition, for which we equip pomsets with interfaces. For our proof, we introduce HDAs with interfaces as presheaves over labelled precube categories and use tools inspired by algebraic topology, such as cylinders and (co)fibrations. HDAs are a general model of non-interleaving concurrency, which subsumes many other models in this field. Interval orders are used as models for concurrent or distributed systems where events extend in time. Our tools and techniques may therefore yield templates for Kleene theorems in various models and applications.