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6 fixed an ambiguity in the description of Chow(k)

I don't know any alternate geometric construction of the commutativity contraint, but there is a natural way to explain this, as a simple fact of homological algebra (and assuming very optimistic conjectures about mixed motives).

Let $DM_{gm}(k)$ be Voevodsky's triangulated category of geometric (=constructible) motives over $k$, with rationnal coefficients. For any (smooth) $k$-scheme, we have its homological motive $M(X)$ in $DM_{gm}(k)$ (this construction being functorial in $X$). Voevodsky proved that the full pseudo-abelian subcategory of $DM_{gm}(k)$ DM_{gm}(k)$, closed under arbitrary Tate twists, spanned by objects of shape$M(X)^\wedge=\underline{Hom}(M(X),\mathbf{Q})$for$X$smooth and projective is equivalent to the category$Chow(k)$of Chow motives (the cohomological version, constructed from rational equivalence on cycles). Now comes the conjectural part. Assume that there exists a motivic$t$-structure on$DM_{gm}(k)$; see this paper of Beilinson for what this means and for the link with the standard conjectures. Let us denote by$MM(k)$the heart of this$t$-structure. The objects of the category$MM(k)$are, by definition, mixed motives. This category is tannakian: for any prime number$\ell$, the$\ell$-adic realization functor defines a conservative$t$-exact and symmetric monoidal functor from$DM_{gm}(k)$to the bounded derived category of finite dimensional$\mathbf{Q}_\ell$-vector spaces (this is part of what it means to be a motivic$t$-structure). Call a mixed motive$M$pure of weight$n$if$M[-n]$is pure of weight zero in the sense of Bondarko (see this paper for the link between$t$-structures and weight structures): this simply means that$M[-n]$belongs to$Chow(k)$, seen as a subcategory of$DM_{gm}(k)$. For$X$smooth and projective over$k$, as$M(X)^\wedge$is pure of weight zero, for any integer$n$, the cohomology object$H^n(X)=\tau^{\leq n}\tau^{\geq n}(M(X)^\wedge)[n]$is pure of weight$n$. This implies that we have a (non canonical) isomorphism$M(X)^\wedge\simeq \oplus_n H^n(X)[-n]$(the so called Chow-Künneth decomposition'. Denote by$M(k)$the full subcategory of$MM(k)$which consists of finite sums of objects$M$of$MM(k)$which are pure of weight$n$for some integer$n$. We then have a natural functor $$X\mapsto \oplus_n H^n(X)$$ from the category of smooth and projective$k$-schemes to$M(k)$. This functor factors (by construction) through the category$Chow(k)$: $$Chow(k)\to M(k)$$ and factors through numerical equivalence (just because the motivic$t$-structure is compatible with the monoidal structure, using Poincaré duality in$DM_{gm}(k)$), inducing an equivalence of categories $$Chow(k)/\text{numerical equivalence} \simeq M(k)$$ Now, the point is that$M(k)$is a tannakian subcategory of$MM(k)$, and the modified commutativity constraint of the tensor product on the category of pure motives up to numerical equivalence is the one inherited from the natural monoidal structure of$M(k)$through the equivalence above. 5 corrected bad shift I don't know any alternate geometric construction of the commutativity contraint, but there is a natural way to explain this, as a simple fact of homological algebra (and assuming very optimistic conjectures about mixed motives). Let$DM_{gm}(k)$be Voevodsky's triangulated category of geometric (=constructible) motives over$k$, with rationnal coefficients. For any (smooth)$k$-scheme, we have its homological motive$M(X)$in$DM_{gm}(k)$(this construction being functorial in$X$). Voevodsky proved that the full subcategory of$DM_{gm}(k)$spanned by objects of shape$M(X)^\wedge=\underline{Hom}(M(X),\mathbf{Q})$for$X$smooth and projective is equivalent to the category$Chow(k)$of Chow motives (the cohomological version, constructed from rational equivalence on cycles). Now comes the conjectural part. Assume that there exists a motivic$t$-structure on$DM_{gm}(k)$; see this paper of Beilinson for what this means and for the link with the standard conjectures. Let us denote by$MM(k)$the heart of this$t$-structure. The objects of the category$MM(k)$are, by definition, mixed motives. This category is tannakian: for any prime number$\ell$, the$\ell$-adic realization functor defines a conservative$t$-exact and symmetric monoidal functor from$DM_{gm}(k)$to the bounded derived category of finite dimensional$\mathbf{Q}_\ell$-vector spaces (this is part of what it means to be a motivic$t$-structure). Call a mixed motive$M$pure of weight$n$if$M[-n]$is pure of weight zero in the sense of Bondarko (see this paper for the link between$t$-structures and weight structures): this simply means that$M[-n]$belongs to$Chow(k)$, seen as a subcategory of$DM_{gm}(k)$. For$X$smooth and projective over$k$, as$M(X)^\wedge$is pure of weight zero, for any integer$n$, the cohomology object$H^n(X)[-n]=\tau^{\leq H^n(X)=\tau^{\leq n}\tau^{\geq n}(M(X)^\wedge)$n}(M(X)^\wedge)[n]$ is pure of weight $n$. This implies that we have a (non canonical) isomorphism $M(X)^\wedge\simeq \oplus_n H^n(X)[-n]$ (the so called Chow-Künneth decomposition'.

Denote by $M(k)$ the full subcategory of $MM(k)$ which consists of finite sums of objects $M$ of $MM(k)$ which are pure of weight $n$ for some integer $n$. We then have a natural functor $$X\mapsto \oplus_n H^n(X)$$ from the category of smooth and projective $k$-schemes to $M(k)$. This functor factors (by construction) through the category $Chow(k)$: $$Chow(k)\to M(k)$$ and factors through numerical equivalence (just because the motivic $t$-structure is compatible with the monoidal structure, using Poincaré duality in $DM_{gm}(k)$), inducing an equivalence of categories $$Chow(k)/\text{numerical equivalence} \simeq M(k)$$ Now, the point is that $M(k)$ is a tannakian subcategory of $MM(k)$, and the modified commutativity constraint of the tensor product on the category of pure motives up to numerical equivalence is the one inherited from the natural monoidal structure of $M(k)$ through the equivalence above.

4 made purity easier to understand

I don't know any alternate geometric construction of the commutativity contraint, but there is a natural way to explain this, as a simple fact of homological algebra (and assuming very optimistic conjectures about mixed motives).

Let $DM_{gm}(k)$ be Voevodsky's triangulated category of geometric (=constructible) motives over $k$, with rationnal coefficients. For any (smooth) $k$-scheme, we have its homological motive $M(X)$ in $DM_{gm}(k)$ (this construction being functorial in $X$). Voevodsky proved that the full subcategory of $DM_{gm}(k)$ spanned by objects of shape $M(X)^\wedge=\underline{Hom}(M(X),\mathbf{Q})$ for $X$ smooth and projective is equivalent to the category $Chow(k)$ of Chow motives (the cohomological version, constructed from rational equivalence on cycles).

Now comes the conjectural part. Assume that there exists a motivic $t$-structure on $DM_{gm}(k)$; see this paper of Beilinson for what this means and for the link with the standard conjectures. Let us denote by $MM(k)$ the heart of this $t$-structure. The objects of the category $MM(k)$ are, by definition, mixed motives. This category is tannakian: for any prime number $\ell$, the $\ell$-adic realization functor defines a conservative $t$-exact and symmetric monoidal functor from $DM_{gm}(k)$ to the bounded derived category of finite dimensional $\mathbf{Q}_\ell$-vector spaces (this is part of what it means to be a motivic $t$-structure).

Call a mixed motive $M$ pure of weight $n$ if $M[-n]$ is pure of weight zero in the sense of Bondarko (see this paper for the link between $t$-structures and weight structures). : this simply means that $M[-n]$ belongs to $Chow(k)$, seen as a subcategory of $DM_{gm}(k)$. For $X$ smooth and projective over $k$, as $M(X)^\wedge$ is pure of weight zero, for any integer $n$, the cohomology object $H^n(X)[-n]=\tau^{\leq n}\tau^{\geq n}(M(X)^\wedge)$ is pure of weight $n$. This implies that we have a (non canonical) isomorphism $M(X)^\wedge\simeq \oplus_n H^n(X)[-n]$ (the so called `Chow-Künneth decomposition'.

Denote by $M(k)$ the full subcategory of $MM(k)$ which consists of finite sums of objects $M$ of $MM(k)$ which are pure of weight $n$ for some integer $n$. We then have a natural functor $$X\mapsto \oplus_n H^n(X)$$ from the category of smooth and projective $k$-schemes to $M(k)$. This functor factors (by construction) through the category $Chow(k)$: $$Chow(k)\to M(k)$$ and factors through numerical equivalence (just because the motivic $t$-structure is compatible with the monoidal structure, using Poincaré duality in $DM_{gm}(k)$), inducing an equivalence of categories $$Chow(k)/\text{numerical equivalence} \simeq M(k)$$ Now, the point is that $M(k)$ is a tannakian subcategory of $MM(k)$, and the modified commutativity constraint of the tensor product on the category of pure motives up to numerical equivalence is the one inherited from the natural monoidal structure of $M(k)$ through the equivalence above.

3 corrected spelling
2 corrected a little mistake
1