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Abstract We initiate the study of model structures on (categories induced by) lattice posets, a subject we dub homotopical combinatorics . In the case of a finite total order [ n ], we enumerate all model structures, exhibiting a rich combinatorial structure encoded by Shapiro’s Catalan triangle. This is an application of previous work of the authors on the theory of $$N_\infty $$ N ∞ -operads for cyclic groups of prime power order, along with new structural insights concerning extending choices of certain model structures on subcategories of [ n ].
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This content will become publicly available on January 1, 2026
The combinatorics of $N_\infty$ operads for $C_{qp^n}$ and $D_{p^n}$
We provide a general recursive method for constructing transfer systems on finite lattices. Using this, we calculate the number of homotopically distinct $$N_{\infty} $$ operads for dihedral groups $$D_{p^n}$$, p>2 prime, and cyclic groups $$C_{qp^n}$$, $$p \neq q$$ prime. We then further display some of the beautiful combinatorics obtained by restricting to certain homotopically meaningful $$N_\infty$$ operads for these groups.
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- Award ID(s):
- 2204365
- PAR ID:
- 10634001
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Glasgow Mathematical Journal
- Volume:
- 67
- Issue:
- 1
- ISSN:
- 0017-0895
- Page Range / eLocation ID:
- 50-66
- Subject(s) / Keyword(s):
- homotopical combinatorics, equivariant homotopy, N-infinity operads
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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