Higher and Higher
But before leaping ahead to dimension 4, recall that a while back we came across sesquicategories, which are 2-dimensional but without an interchange law. Leinster's book talks about a 2-operad for sesquicategories, which has sesquicategories for algebras. Sorry, perhaps I should have mentioned that categories can arise from operads just like algebras. On page 295, Leinster mentions an induced 3-operad, arising from the sesquicategory 2-operad. This is a 3-operad called Gray, because its algebras are Gray categories. We need to understand these, and there is a lot about them in Gurski's thesis!