Transfer systems and the combinatorics of model structures

Model structures underpin the modern enterprise of abstract homotopy theory and form presentations of (\infty,1)‑categories. Despite their fundamental nature, model structures have historically been studied en masse or applied in specific cases, and very little is known about the totality of model structures on a given (bicomplete) category. Homotopical combinatorics is an emerging field that remedies this situation by studying the enumerative combinatorics and structural properties of model structures on finite lattices. Specialized to a finite chain, we find rich connections with Catalan combinatorics, including (intervals in) the Tamari and Kreweras lattices. I will sketch homotopical combinatorics as it currently stands, including the surprising way in which the theory of transfer systems and equivariant N_\infty-operads has enabled recent advances.