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A slight variant $\tilde Mot_{num}(k,\mathbb{Q})$ of the category of pure motives $Mot_{num}(k,\mathbb{Q})$ is a Tannakian category equivalent to a category of representations of some algebraic group $GMot_k$ (the Motivic galois group):

$$ Mot_{num}(k,\mathbb{Q}) \simeq Rep(GMot_k) \,. $$

nLab

What it is known about the Derived category version (The derived category of motives) of this equivalence?

Equivalence between the sub-category of compact objects of (say for example) étale Motivic sheaves and the derived category of representations of finite dimension of (Ayoub) Motivic galois groups

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Let $k$ be a field and $\operatorname{DM}_{gm}(k)_{\mathbb Q}$ the ∞-category of rational geometric motives over $k$. A mixed Weil cohomology theory induces a symmetric monoidal exact functor

$$ R: \operatorname{DM}_{gm}(k)_{\mathbb Q} \to D_c(\mathbb Q), $$

where $D_c(\mathbb Q)$ are the compact=dualizable objects in the derived ∞-category of $\mathbb Q$. For abstract reasons, there exists a best possible commutative Hopf algebra $H$ in $D(\mathbb Q)$ such that, if $G=\operatorname{Spec}(H)$, then $R$ lifts to

$$ \tilde R: \operatorname{DM}_{gm}(k)_{\mathbb Q} \to \operatorname{Rep}_{D_c(\mathbb Q)}(G). $$

This general construction can be found in this paper by Iwanari. As you know, Ayoub has also constructed $H$, but only in the homotopy category of $D(\mathbb Q)$, which is not enough to define $\tilde R$.

Let me now assume that $k\subset \mathbb C$ and that $R$ is the Betti realization. Ayoub then proves that $H$ is connective, so that $\tau_{\leq 0}H$ is a Hopf algebra in rational vector spaces. If $G^{cl}=\operatorname{Spec}(\tau_{\leq 0}H)$, we then have a closed immersion $i\colon G^{cl}\hookrightarrow G$, whence

$$ i^*\circ \tilde R: \operatorname{DM}_{gm}(k)_{\mathbb Q} \to \operatorname{Rep}_{D_c(\mathbb Q)}(G^{cl}). $$

Choudhury and Gallauer Alves de Souza proved that the group scheme $G^{cl}$ is isomorphic to Nori's motivic Galois group.

The strongest conjecture one can make here is that $i^*\circ\tilde R$ is an equivalence of ∞-categories. This conjecture is equivalent to the conjunction of two conjectures: $\tilde R$ is an equivalence, and $G^{cl}=G$. It also clearly implies the conservativity conjecture (that $R$ is conservative) and the existence of the motivic $t$-structure. Conversely, the existence of a $t$-structure such that $R$ is $t$-exact implies that $G^{cl}=G$. The conservativity of $R$ together with the existence of a suitable $t$-structure should imply that $\tilde R$ is an equivalence, but I cannot find an applicable reference at the moment.

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    $\begingroup$ The functor $\tilde{R}$ has been also constructed by Pridham in arxiv.org/abs/1309.0637. $\endgroup$ – tttbase Dec 31 '16 at 21:40
  • $\begingroup$ @MarcHoyois Could you help me understand, or point to a reference, what does "geometric" mean for a motive. I familiar with the general construction of taking nisnevich sheaves and localizing by $A^1$-equivalences and stabilizing then rationalizing. What do I need to do from this point to get "geometric motives? Persumably those should correspond to the numerical equivalence relation, how is it encoded in this language? $\endgroup$ – Saal Hardali Dec 13 '18 at 6:55
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    $\begingroup$ @SaalHardali Geometric motives are the compact objects in the ∞-category of motives. $\endgroup$ – Marc Hoyois Jan 19 '19 at 7:40

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