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We identify the motivicKGL/2-local sphere as the fiber of\psi^{3}-1on(2,\eta)-completed HermitianK-theory, over any base scheme containing1/2. This is a motivic analogue of the classical resolution of theK(1)-local sphere, and extends to a description of theKGL/2-localization of an arbitrary motivic spectrum. Our proof relies on a novel conservativity argument that should be of broad utility in stable motivic homotopy theory. 

We rework and generalize equivariant infinite loop space theory, which shows how to construct $G$-spectra from $G$-spaces with suitable structure. There is a classical version which gives classical $\mathrm{\Omega <\#comment/>}$- $G$-spectra for any topological group $G$, but our focus is on the construction of genuine $\mathrm{\Omega <\#comment/>}$- $G$-spectra when $G$is finite. We also show what is and is not true when $G$is a compact Lie group. We give new information about the Segal and operadic equivariant infinite loop space machines, supplying many details that are missing from the literature, and we prove by direct comparison that the two machines give equivalent output when fed equivalent input. The proof of the corresponding nonequivariant uniqueness theorem, due to May and Thomason, works for classical $G$-spectra for general $G$but fails for genuine $G$-spectra. Even in the nonequivariant case, our comparison theorem is considerably more precise, giving an illuminating direct point-set level comparison. We have taken the opportunity to update this general area, equivariant and nonequivariant, giving many new proofs, filling in some gaps, and giving a number of corrections to results and proofs in the literature. 

We develop the theory of saturated transfer systems on modular lattices, ultimately producing a “matchstick game” that puts saturated transfer systems in bijection with certain structured subsets of covering relations. We also prove that Hill’s characteristic function χ for transfer systems on a lattice P surjects onto interior operators for P, and moreover, the fibers of χ have unique maxima which are exactly the saturated transfer systems. Lastly, after an interlude developing a recursion for transfer systems on certain combinations of bounded posets, we apply these results to determine the full lattice of transfer systems for rank two elementary abelian groups. 

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. 

We perform Hochschild homology calculations in the algebro-geometric setting of motives over algebraically closed fields. The homotopy ring of motivic Hochschild homology contains torsion classes that arise from the mod-p motivic Steenrod algebra and generating functions defined on the natural numbers with finite non-empty support. Under Betti realization, we recover Bökstedt’s calculation of the topological Hochschild homology of finite prime fields. 

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. 

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 ]. 

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.