One of the strong motivations for the development of higher category theory is the modelling of the building blocks of spaces, the n-types. The ‘homotopy hypothesis’ asserts that a good model of weak n-categories should satisfy the homotopy hypothesis in the weak n-groupoid case. In the first part of the talk I will give an overview of a class of models of weak n-category, called Segal-type models, which do satisfy the homotopy hypothesis. These models are based on the combinatorics of multi-simplicial sets.
In the second part of the talk I will consider the question of modelling stable n-types, that is n-types of spectra. I will illustrate how Segal-type models with appropriate extra structure can be used for this purpose. The second part of the talk is joint work with Lyne Moser, Viktoriya Ozornova, Maru Sarazola and Paula Verdugo.