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The intention of the paper is to show the applicability of the general categorical framework of open maps to the setting of two models - higher dimensional automata (HDA) and timed higher dimensional automata (THDA) - in order to transfer general concepts of equivalences to the models. First, we define categories of the models under consideration, whose morphisms are to be thought of as simulations. Then, accompanying (sub)categories of observations are chosen relative to which the corresponding notions of open maps are developed. Finally, we use the open maps framework to obtain two abstract bisimulations which are established to coincide with hereditary history preserving bisimulations on HDA and THDA, respectively.