$\DeclareMathOperator{\Sp}{\mathrm{Sp}}$I am taking a special case $\Sp$ here, mainly because it has nice categorical properties.


Let $R$ be an $E\_\infty$-ring spectrum. In [Higher Algebra](http://people.math.harvard.edu/%7Elurie/papers/HA.pdf), Lurie proves we have a forgetful functor (part of monadic adjunction)
$$ U\_R:\operatorname{Mod}\_R(\Sp) \rightarrow \Sp$$
where $\Sp$ is in the $\infty$-category of spectra.


$U\_R$ reflects equivalences. But **is $U\_R$ faithful** in the sense that that the induced map of
$$Map(x,y)\rightarrow Map(U\_Rx,U\_Ry)$$
mapping spaces is $-1$-truncated in the $\infty$-category of spaces. i.e. the homotopy fibers are [$-1$-truncated](https://ncatlab.org/nlab/show/n-truncated+object+of+an+%28infinity%2C1%29-category).




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One categorically, $U$ is faithful in many cases, i.e. if we replace $\Sp$ with $\mathrm{Ab}$.
Perhaps the answer is false in $\infty$-categories.
I'd like to understand what goes wrong. Some comments on the following would be helpful:


* A counter example where $U\_R$ is not faithful. (i.e. is it faithful when $R=H\Bbb Z$? )
* A brief/reference explanation for what accounts of this.



Is the forgetful functor $\mathrm{Mod}_R \mathrm{Sp} \rightarrow \mathrm{Sp}$ faithful?
$U\_R$ obviously preserves delooping, so if that were the case, because $\pi\_0 map(X,Y) = \pi\_1 map(X, \Sigma Y)$, you would also get an isomorphism on $\pi\_0$, so an equivalence of mapping spaces.


In other words, $U\_R$ is faithful if and only if it is fully faithful. But now for a map of ring spectra $R\to S$, the forgetful $Mod\_S \to Mod\_R$ is fully faithful if and only if $R\to S$ is an epimorphism of ring spectra (good examples are localizations - be careful that classical examples such as $R\to R/I$ for a usual ring $R$ tend to fail).


This is to say that "being an $S$-module" becomes a property of an $R$-module, rather than additional structure - so of course you can expect that to be very rare.


In your example of $H\mathbb Z$, it doesn't hold at all - you can for instance detect it on the level of the ring of stable cohomology operations of singular cohomology, which is bigger than just $\mathbb Z$ (look at the (co)homology of Eilenberg-MacLane spaces)


