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Creators/Authors contains: "Balchin, Scott"

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  1. ; ; (, Glasgow Mathematical Journal)
    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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  2. ; ; (, European Journal of Combinatorics)
    We investigate the rich combinatorial structure of premodel structures on finite lattices whose weak equivalences are closed under composition. We prove that there is a natural refinement of the inclusion order of weak factorization systems so that the intervals detect these composition closed premodel structures. In the case that the lattice in question is a finite total order, this natural order retrieves the Kreweras lattice of noncrossing partitions as a refinement of the Tamari lattice, and model structures can be identified with certain tricolored trees. 
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  3. ; ; ; (, Mathematische Zeitschrift)
    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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  4. ; ; (, Tunisian Journal of Mathematics)
    We isolate a class of groups — called lossless groups — for which homotopy classes of G-N∞ operads are in bijection with certain restricted transfer systems on the poset of conjugacy classes Sub(G)/G. 
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