I've been thinking about Bertrand Toen's approach to studying the homotopy theory of schemes, and I've come across an inconsistency in my understanding of the subject that I was hoping somebody might be able to iron out for me.


If I take a field $k$ of characteristic $\neq\ell$, then if I have understood this


<http://www.aimath.org/WWN/motivesdessins/Toen.pdf>


correctly, there should be an $\ell$-adic schematic homotopy type $h(X)$ associated to every smooth and projective variety over $k$, such that the 'absolute' $\ell$-adic cohomology of $X$, which coefficients in a local system, can be calculated as an appropriate hom set inside the derived category of perfect complexes on $h(X)$. Specifically, I understood that the derived category of perfect complexes on $h(X)$ should be equivalent to the full subcategory of the usual $\ell$-adic derived category $D^b\_c(X\_{\mathrm{et}},\mathbb{Q}\_\ell)$ whose cohomology sheaves are lisse.


The reason this is confusing me is because I also thought that the schematic homotopy type associated to the point $\mathrm{Spec}(k)$ should be the classifying stack $BG\_k^\mathrm{alg}$ associated to the $\mathbb{Q}\_\ell$ pro-algebraic completion $G\_k^\mathrm{alg}$ of the absolute Galois group of $k$.


But the derived category of perfect complexes $D\_\mathrm{perf}(BG\_k^\mathrm{alg})$ on this classifying stack, as far as I can tell, should not be equivalent to $D^b\_c(\mathrm{Spec}(k)\_\mathrm{et},\mathbb{Q}\_\ell)$ - if I take some finite dimensional continuous $G\_k$-representation $V$ then the cohomology computed in the former category will be the algebraic group cohomology 
$$\mathrm{Hom}\_{D\_\mathrm{perf}(BG\_k^\mathrm{alg})}(\mathbb{Q}\_\ell,V[i])=H^i(G\_k^\mathrm{alg},V)$$
whereas the cohomology computed in the latter category will be continuous group cohomology 
$$\mathrm{Hom}\_{D^b\_\mathrm{lisse}(X\_\mathrm{et},\mathbb{Q}\_\ell)}(\mathbb{Q}\_\ell,V[i])=H^i\_\mathrm{cts}(G\_k,V).$$
These two won't coincide, and this suggests that there are no suitable subcategories of $D\_{\mathrm{perf}}(BG\_k^\mathrm{alg})$ and $D^b\_c(\mathrm{Spec}(k)\_\mathrm{et},\mathbb{Q}\_\ell)$ that will coincide. 


So what's going on? I've convinced myself that the two expectations
$$ D\_\mathrm{perf}(h(X))\cong D^b\_\mathrm{lisse}(X\_\mathrm{et},\mathbb{Q}\_\ell)$$
and 
$$h(\mathrm{Spec}(k))\cong BG\_k^\mathrm{alg} $$
are incompatible - so which one should I hold on to? Or have I missed a trick somewhere?


EDIT: Clarified what I mean by the 'cohomology' calculated in both categories.



Pro-algebraic versus continuous Galois cohomology, and schematic homotopy types
This answer is due to Jon Pridham.


While we might not expect $H^i(G\_k,V)=H^i(G\_k^\mathrm{alg},V)$ for every finite dimensional, continuous $G\_k$-representation $V$, there are certain results from the motivic theory that suggest that this might be true if $k$ is a number field (or local field of char $0$), and that the representation 'comes from geometry'.


So although we possible should not expect a schematic homotopy type $h(\mathrm{Spec}(k))$ which is a gerbe, and whose derived category of perfect complexes is equivalent to $D^b\_c(\mathrm{Spec}(k)\_\mathrm{et},\mathbb{Q}\_\ell)$, there may well be a gerbe whose derived category of perfect complexes is equivalent to the subcategory of $D^b\_c(\mathrm{Spec}(k)\_\mathrm{et},\mathbb{Q}\_\ell)$ generated by complexes of geometric origin, at least when $k$ is a number field.


Essentially, the existence of the motivic $t$-structure 'almost' implies that the motivic schematic homotopy type is a gerbe, from which it would follow that the $\ell$-adic schematic homotopy type associated to motivic representations (i.e. representations of geometric origin) is a gerbe.


