In rational homotopy theory, one can study the rational homotopy and cohomology categories via an algebraic structure, respectively the rational Lie Algebra model and the Sullivan cdga model.


If I am not wrong, over a prime there is a similar theorem for $\mathbb{F}\_p$ cohomology, that is Mandell's Theorem: under adequate assumptions, $\mathbb{F}\_p$ cohomology together with its $E\_{\infty}$ structure does a job analogous to Sullivan cdga in rational homotopy theory.


I am definitely not confident in this field, but I don't understand if $p$-homotopy groups $\mathbb{F}\_p \otimes \pi\_n$ (or some $p$-adic friend) are determined by Mandell's model or not, and what is the analog of the Lie model.


What I want in the end is a generalization of the notion of coformality to the $p$-adic world, which I recall: $X$ is said to be coformal if there exists a (zig-zag of) quasi-isomorphism connecting the rational homotopy groups of $\Omega X$ with the free Lie Algebra on $C^\* X$.



Analogues of Sullivan Theory at a prime for coformality
Mandell shows that, under some hypotheses, the $\mathbb{F}\_p$-cochains detect the $\mathbb{F}\_p$-homotopy type in the sense that there is such an equivalence $X \simeq Y$, if and only if, there is an equivalence of $E\_\infty$-algebras $C\_\*(X; \mathbb{F}\_p) \simeq C\_\*(X;\mathbb{F}\_p)$. This is different than saying that $\mathbb{F}\_p$-homotopy theory is equivalent to the homotopy theory of $E\_\infty$-algebras in $\mathbb{F}\_p$-cochain complexes because it says nothing about essential surjectivity or equivalence of mapping spaces. Nonetheless, it is still reasonable to call a space $p$-formal if there is an equivalence of $E\_\infty$ algebras $C\_\*(X;\mathbb{F}\_p) \simeq H\_\*(X;\mathbb{F}\_p)$, and this is a useful, if rarely satisfied, condition. As an aside, Mandell showed this result actually be improved to an equivalence of homotopy theories if one instead takes coefficients in $\bar{\mathbb{F}}\_p$.


Mandell also shows that our only guess for the Lie model of $\mathbb{F}\_p$-homotopy theory, the Koszul dual of $C\_\*(X;\mathbb{F}\_p)$ is actually contractible. I expect this result implies there is not an easy way to detect homotopy groups from the $E\_\infty$-algebra model.


All is not lost! The correct way to model $p$-torsion information with Lie algebras was demonstrated by Heuts. Using chromatic homotopy theory, one can construct the $v\_n$-localization of spaces and spectra. These localizations can see $p$-torsion information when $n>0$ and when $n=0$ it coincides with rationalization. Heuts showed that $v\_n$-local spaces are modeled by Lie algebras in $v\_n$-local spectra. I don't think there is a definition of coformality in this context, but I would be very interested to see one.


