From motives to comodules over the motivic Hopf algebra
The article under review complements other articles by the same author studying the motivic Hopf algebra. In this article the motivic Hopf algebra is upgraded to a ``homotopy Hopf algebra'', and a comparison functor from triangulated motives to the homotopy category of homotopy comodules over the motivic Hopf algebra is constructed. This is essentially aiming towards a derived version of the (conjectural) Tannakian reconstruction theory for ordinary motives. To recall the machinery, let \(F: \mathcal{C} \leftrightarrows \mathcal{D}: U\) be an adjunction of symmetric monoidal categories, with \(F\) symmetric monoidal (and so \(U\) lax monoidal). Then \(UF\) is a monad on \(\mathcal{C}\) and hence for every \(X \in \mathcal{C}\) there is an associated cosimplicial object \((UF)^\bullet X\). If \(X\) is a commutative monoid in \(\mathcal{C}\) then \((UF)^\bullet X\) is a cosimplicial commutative monoid in \(\mathcal{C}\), since \(UF\) is lax monoidal. Since any object of \(\mathcal{C}\) is canonically a module over the monoidal unit \(\mathbb{1}\), for every \(X \in \mathcal{C}\) we obtain a cosimplicial module \((UF)^\bullet X\) over the cosimplicial commutative monoid \((UF)^\bullet \mathbb{1}\). If \(\Gamma: \mathcal{C} \to \mathcal{E}\) is a further lax symmetric monoidal functor, then we can consider the composite \[ \Gamma (UF)^\bullet: \mathcal{C} \to \Gamma (UF)^\bullet \mathbb{1}\text{-Mod}. \] In very good cases, one may hope for this functor to be an equivalence, thus recovering \(\mathcal{C}\). Moreover, recall that the category of Hopf algebras in \(\mathcal{E}\) is equivalent to the category of cosimplicial commutative algebras in \(\mathcal{E}\) satisfying certain additional properties. Thus in good cases the cosimplicial commutative algebra \(\Gamma(UF)^\bullet \mathbb{1}\) determines a Hopf algebra \(H(\mathcal{C}, F, \Gamma)\), and the reconstruction functor takes the more familiar form \[ \Gamma (UF)^\bullet: \mathcal{C} \to H(\mathcal{C}, F, \Gamma)\text{-coMod}. \] As a final twist, if the category \(\mathcal{E}\) has a notion of weak equivalence, then one may weaken the properties on a cosimplicial algebra in \(\mathcal{E}\) which characterise a Hopf algebra suitably to obtain a notion of a \textit{homotopy Hopf algebra}. Sometimes \(\Gamma (UF)^\bullet \mathbb{1}\) may turn out to be a homotopy Hopf algebra, in which case we denote it by \(\underline{H}(\mathcal{C}, F, \Gamma)\). Then one may consider also a homotopical version of modules over a homotopy Hopf algebra, and hence ``derive everything in sight'' to obtain a third reconstruction functor \[ R \Gamma (UF)^\bullet: Ho(\mathcal{C}) \to Ho(\underline{H}(\mathcal{C}, F, \Gamma)\text{-hocoMod}). \] The article under review carries out this program for \(\mathcal{C} = DM(k, \Lambda)\) where the field \(k\) is provided with a complex embedding \(\sigma\) and \(\Lambda\) is a \(\mathbb{Q}\)-algebra. The functor \(F\) is given by Betti realisation \(Bti^*\), and \(\Gamma\) is global sections, in other words \(\mathcal{D} = \mathcal{E} = D(\Lambda)\). This way the author constructs a homotopy Hopf algebra \(\underline{H}_{mot}(k, \sigma, \Lambda) = \underline{H}(DM(k, \Lambda), Bti^*, \Gamma)\) enhancing the motivic Hopf algebra constructed in earlier papers, and a triangulated comparison functor \[ DM(k, \Lambda) \to Ho(\underline{H}_{mot}(k, \sigma, \Lambda)\text{-hocoMod}). \] In order to carry out the above program, the author has to employ many long and technical arguments, including establishing appropriate model structures for homotopy comodules and providing a new symmetric monoidal model for passing to spectra over a \(\mathbb{Q}\)-linear category.
scientific article; zbMATH DE number 6695841