From motives to comodules over the motivic Hopf algebra
type: Item
instance of: scholarly article
title: From motives to comodules over the motivic Hopf algebra
zbMATH Open document ID: 1359.14022
author: Joseph Ayoub
published in: Journal of Pure and Applied Algebra
publication date: 2017-03-17T00:00:00Z
review text: 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)…
reviewed by: Tom Bachmann
Mathematics Subject Classification ID: 14F42
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 18G30
Mathematics Subject Classification ID: 18G55
Mathematics Subject Classification ID: 19E15
zbMATH DE Number: 6695841
zbMATH Keywords: motivic hopf algebra
zbMATH Keywords: homotopy hopf algebra
zbMATH Keywords: motive
zbMATH Keywords: homotopy comodule
MaRDI profile type: Publication
full work available at URL: j.jpaa.2016.12.015
OpenAlex ID: W2581336165
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DOI: 10.1016/J.JPAA.2016.12.015
description: scientific article; zbMATH DE number 6695841