I'm reading Ayoub's paper [Motifs des varietes analytiques rigides][1], but I'm not quite familiar with motives. In this paper, he defines the category of motives to be $\mathbf{RigDM}^{\rm eff}_{\rm Nis}(k, \mathbb{Q}) := \mathbf{Ho}_{\rm Nis, \mathbb{B}^1}(\mathbf{Compl}(\mathbf{PreShv}(\text{SmRig}_k, \mathbb{Q})))$. 

As I get it, we consider the site of all smooth rigid spaces over $k$, then consider all complexes of presheaves over this site. Just like the construction of derived category, now we invert all quasi-isomorphisms as Nisnevich sheaves, and invert all $\mathbb{B}^1-$homotopy, namely morphisms like $\mathbb{Q}(\mathbb{B}^1\times X)[i]\rightarrow \mathbb{Q}(X)[i]$.

What I'm interested in is the Theorem 2.5.35, it says that this category can be compactly generated by those $\mathbb{Q}(X^{\rm an})[i]$ with $X$ being proper smooth $k$-schemes.

As I follow the proof, the first step is the fact that the category $\mathbf{Ho}_{\rm Nis}(\mathbf{Compl}(\mathbf{PreShv}(\text{SmRig}_k, \mathbb{Q})))$(So hence $\mathbf{RigDM}^{\rm eff}_{\rm Nis}(k, \mathbb{Q})$) is compactly generated by those $\mathbb{Q}(X^{\rm an})[i]$ with $X$ being quasi-compact $k$-schemes. Then he uses methods of algebraic geometry to reduce $X$ to be those proper smooth ones.

However, I don't quite follow this first step. In the category of modules, since we allow taking arbitrary direct sums, from $\mathbb{Q}$ we get all free modules. And taking direct summands is also allowed, so we have projective modules. Then by projective resolution, we get all modules. And in the derived category, all complex is quasi-isomorphic to its cohomology, so we get the whole derived category.

I guess for sheaves, it also follows this process to generate the category of motives. As I understand, the sheaf $\mathbb{Q}(X)$ that $X$ represent is just something as a constant sheaf on $X$. However, I don't quite understand how this goes, and I don't even know if there are enough projective sheaves for this topology...


  [1]: https://www.uni-due.de/~bm0032/PreprintsSPP/%5B10%5D%20Ayoub.pdf