We provide a combinatorial approach to studying the collection of N ∞operads in G-equivariant homotopy theory for G a finite cyclic group of prime power order. In particular, we show that for G = C p n the natural order on the collection of N ∞ -operads is in bijection with the poset structure of the (n + 1)-associahedron. We further provide a lower bound for the number of possible N ∞ -operads for any finite cyclic group G. As such, we have reduced an intricate problem in equivariant homotopy theory to a manageable combinatorial problem.
Given a suitable stable monoidal model category C and a specialization closed subset V of its Balmer spectrum one can produce a Tate square for decomposing objects into the part supported over V and the part supported over V c spliced with the Tate object. Using this one can show that C is Quillen equivalent to a model built from the data of local torsion objects, and the splicing data lies in a rather rich category. As an application, we promote the torsion model for the homotopy category of rational circle-equivariant spectra from [18] to a Quillen equivalence. In addition, a close analysis of the one step case highlights important features needed for general torsion models which we will return to in future work. Contents 10. The torsion model for rational T-spectra 28 References 34
We introduce two new topological invariants of a rigidly-compactly generated tensor-triangulated category and study their associated support theories and relation to existing technology. The first, the smashing spectrum, is produced by proving that the frame of smashing ideals is always spatial, and is equipped with a surjective morphism to the Balmer spectrum which detects the failure of the telescope conjecture. The second, the big spectrum, results from taking the entire collection of localizing ideals seriously and considering prime localizing ideals. Although there are, in principle, a proper class of localizing ideals, we are able to prove the existence of at least one big prime lying over every Balmer prime. We conclude with a pair of examples illustrating our constructions.
We utilise the theory of crossed simplicial groups to introduce a collection of local Quillen model structures on the category of simplicial presheaves with a compact planar Lie group action on a small Grothendieck site. As an application, we give a characterisation of equivariant cohomology theories on a site as derived mapping spaces in these model categories.
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