Abstract
While higher groupoids have a natural model in spaces, higher categories have no such well-accepted model. This makes the question of correctness of a given definition of higher categories difficult to answer. We argue that the question has a simple answer "locally", namely, categories are locally modelled on so-called manifold diagrams. The corresponding "local model" for spaces/groupoids can be formulated in classical terms by a generalised Thom-Pontryagin construction. The idea of locally modelling higher categories by manifold diagrams (most prominently in the case of Gray-categories) is not new and has been proposed by multiple authors. However, the niceness of this manifold-based perspective on higher categories has been somewhat obfuscated by the complexity of manifold geometry in higher dimensions in the past. Our work gives a fully algebraic formulation of this manifold perspective. Interestingly, the model of higher categories that is based on this algebraic formulation is not fully weak: It is a generalisation of (unbiased) Gray-categories to higher dimensions. This is the starting point of a wealth of further research, which reaches from a (version of) Simpson's conjecture to presentations of the extended cobordism n-category and the homotopy and cobordism hypotheses.