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Motivic slices and coloured operads. (English) Zbl 1258.18012
The main purpose of this paper is to determine the algebraic structure on the slice filtration of algebras in the motivic stable homotopy category of F. Morel and V. Voevodsky [Publ. Math., Inst. Hautes Étud. Sci. 90, 45–143 (1999; Zbl 0983.14007)].
The authors first give an axiomatic definition of slice filtrations in the homotopy category of a stable combinatorial simplicial symmetric monoidal model category \(\mathcal M\). They consider a nested sequence of subcategories \(\mathcal C_i\), \(i\in\mathbb Z\), of the homotopy category \(\mathcal C = \mathbf{Ho}(\mathcal M)\) so that each \(\mathcal C_i\) is generated by homotopy colimits and extensions of a set of objects in \(\mathcal M\). The authors moreover assume that the tensor unit lies in \(\mathcal C_0\) and that the tensor product maps \(\mathcal C_i\times\mathcal C_j\) into \(\mathcal C_{i+j}\). The slice functor determined by this data assigns a sequence of objects \(s_i X\in\mathbf{Ho}(\mathcal M)\) fitting in exact triangles \(c_{i+1} X\rightarrow c_{i} X\rightarrow s_{i} X\rightarrow c_{i+1} X[1]\) to any \(X\in\mathbf{Ho}(\mathcal M)\), where \(c_i\) refers to the colocalization functor associated to the category \(\mathcal C_i\). The authors prove that the slice filtration of an \(A_{\infty}\)-algebra (respectively, \(E_{\infty}\)-algebra) in \(\mathcal M\) has a underlying \(A_{\infty}\)-algebra (respectively, \(E_{\infty}\)-algebra) structure, and establish a similar statement for modules over \(A_{\infty}\)-algebras (respectively, \(E_{\infty}\)-algebras). The arguments rely on an analysis of the homotopy invariance of (colored) endomorphism operads under the application of localization functors in the category \(\mathcal M\).
Let \(S\) be a separable noetherian base scheme. The methods of the authors apply to Voevodsky slice filtration of the motivic stable homotopy category \(\mathbf{SH}(S)\), and where we take the category generated by the \(i\)th Tate suspensions \(\Sigma_T^i X\) of effective motives \(X\in\mathbf{SH}(S)^{\mathrm{eff}}\) to define \(\mathcal C_i\). The authors examine the case of the algebraic K-theory and hermitian K-theory spectra, and of the algebraic cobordism spectrum, for which we have an explicit description of the slice filtration in terms of motivic Eilenberg-MacLane spectra. The authors pose that the \(E_{\infty}\)-algebra structure obtained on the slice filtration of these motivic spectra can be deduced from the natural \(E_{\infty}\)-algebra structure associated to the motivic Eilenberg-MacLane spectra.

18G55 Nonabelian homotopical algebra (MSC2010)
14F42 Motivic cohomology; motivic homotopy theory
55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
18D50 Operads (MSC2010)
14F35 Homotopy theory and fundamental groups in algebraic geometry
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