Oxford Seminar / Homotopy coherent Bousfield–Kan
From a weird little-known construction in complex geometry we can notice a new type of subdivision of simplices: half cubical, half simplicial, and very useful for working with certain geometric objects. These subdivisions are called homotopy-coherent simplices (previously wiggly simplices).
In this talk I will give an introduction to the theory and methodology of homotopy limits of cosimplicial spaces through some tools of 2-category theory, explaining the important construction of Bousfield–Kan, and how this generalises to the homotopy coherent setting following joint work with Jason Brown and Cheyne Glass. I will also show lots of pictures of triangles. If times permits, I will introduce the notion of homotopy homotopy limit.
Prerequisites. The talk will use Quillen model categories and cosimplicial simplicial sets, but I will try to give good enough speedy overviews of both of these definitions. I will also use some definitions from enriched category theory (namely weighted (co)limits) but in a way that will likely make any Australians in the room rather upset.
Recommended pre-reading. The following definitions: nerve of a category, barycentric subdivision of a simplex, simplicial set.