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 (complete and cocomplete) 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 equivariant \(N_\infty\) operads has enabled recent advances. The talk will not assume prior knowledge of model category theory, though attendees interested in this context are invited to attend the pre-talk. Portions of this work are joint with Balchin, Franchere, Hafeez, MacBrough, Marcus, Osorno, Qin, Roitzheim, and Waugh.

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# Homotopical combinatorics

Kyle Ormsby, University of Washington / Reed College

Tuesday, October 25, 2022 - 3:30pm to 4:30pm

PDL C-401