My understanding is that one purpose of stable motivic homotopy theory is to emulate classical stable homotopy theory. In particular, we would like Weil cohomology theories to be representable by motivic spectra. Can someone clarify this and/or give references for the construction of the spectrum representing, say, ladic cohomology?

9$\begingroup$ Good question. But even more than you expect: Not only Weyl cohomology theories are representable, there is even a notion of mixed Weil cohomologies and yes, they are representable! Check CisinskiDeglise paper here arxiv.org/abs/0712.3291 $\endgroup$– TintinJun 20, 2016 at 17:00
1 Answer
Let me elaborate the comment to be more precise.
Question:Can someone clarify if Weil cohomologies are representable and give a reference?
Answer: Yes, but it requires a hypothesis: the cohomology has to be defined as a sheaf cohomology. In the paper Mixed Weil cohomologies CisinskiDéglise (CD) defined the natural notion of mixed Weil cohomology (which restricted to the pure case is a Weil cohomology). Provided that your Weil cohomology is defined as a sheaf cohomology with the natural properties one may expect (check CD's notion of mixed Weil theory for the precise definition) CD constructed a spectrum representing your cohomology. At the end of the paper you can check how do CD work out the main examples.
Question: Can someone give a reference for the construction of the spectrum representing $l$adic cohomology?
Answer: There is no reference for this, to my knowledge [Edit: check Marc Hoyois comment from below. It seems there is such spectrum]. To apply above results the cohomology has to be a sheaf cohomology, as we said above. To my knowledge $l$adic cohomology is not a sheaf cohomology.
Nevertheless, there is something. Let $k$ be a countable perfect fiedl and $X$ be a smooth $k$scheme. Denote $\bar X=X\otimes \bar k$ for $\bar k$ a separable closure of $k$. Deligne defined a sheaf computing the $\mathbb{Q}_l$cohomology $H(\bar X,\mathbb{Q}_l)$ of $\bar X$. This cohomology is representable in the stable homotopy category (check the end of CD paper).
Remark: The result of CD has been extended to apply to other cohomologies with twists. The first result I know is from B. Drew's thesis in 2.1.8 for absolute Hodge cohomology, but the reference requires a lot of investment from the reader. The concrete case of real Deligne cohomology was worked out with analogue ideas to that of CD by HolmstromScholbach here in section 3 and a general result was then obtained by DégliseMazzari here (check the introduction and 1.4.10).

5$\begingroup$ If I recall correctly, your formula for ladic cohomology is valid when X is of finite type over an algebraically closed or finite field, but not in general (there's a $lim^1$ group that also contributes). In fact there is a motivic spectrum representing ladic cohomology over an arbitrary base: it is simply the homotopy limit of the motivic spectra representing etale cohomology with coefficients in $\mathbb Z/l^k(*)$. (I don't know a reference for this.) $\endgroup$ Jul 6, 2016 at 14:39

1$\begingroup$ Thank you very much for the correction, Marc. I think it is better that I let someone else talk on $l$adic cohomology. I edited the post. One question about taking the homotopy limit of the motivic spectra representing étale cohomology with coefficients in $\mathbb{Z}/l^k(*)$. Shouldn't that spectrum represent the cohomology defined by the sheaf $\mathbb{Z}_l$? $\endgroup$– TintinJul 6, 2016 at 14:59

5$\begingroup$ No, the terms of the limit spectrum are limits of etale EilenbergMac Lane sheaves $K(\mathbb Z/l^k(n), 2n)$ and you cannot commute the limit and $K$. That's of course the same reason ladic cohomology isn't etale cohomology with coefficients in $\mathbb Z_l$, which as far as I know is not even $\mathbb A^1$invariant. $\endgroup$ Jul 6, 2016 at 15:29

1$\begingroup$ now by the work of Bhatt and Scholze we can compute ladic cohomology as cohomology on some site(proetale site) $\endgroup$– aliFeb 11, 2021 at 18:14