Higher and Higher

Thanks to n-Category Cafe for pointing out the availability of Gurski’s thesis on tricategories, probably destined to become a useful reference. In reply to John Baez, Tod Trimble (well known for his handwritten diagrams for tetracategories) immediately commented: if your work takes you beyond weak 3-categories and into the land of weak 4-categories, then you’ll especially need to read this!

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!

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