Snaith splitting for operads in spectra?

Let $$O$$ be an augmented operad. Then there is a functor $$J^O: Top_\ast \to Top_\ast$$ which takes $$X$$ to the free $$O$$-algebra on $$X$$ subject to the condition that the nullary operation coincides with the basepoint. The Snaith splitting tells us that $$\Sigma^\infty J^O(X) = \Sigma^\infty \vee_{n \geq 1} O(n)_+ \wedge_{\Sigma_n} X^{\wedge n}$$. Is there a version of this where $$X$$ is a pointed spectrum rather than a pointed space?

That is, let $$\mathbb S \to X$$ be a spectrum equipped with a map from the sphere spectrum. If $$O$$ is an augmented operad, there should be an $$O$$-algebra $$J^O(X)$$ such that the data of a map of $$O$$-algebra spectra $$J^O(X) \to A$$ is equivalent to a map of spectra $$X \to A$$ commuting with the maps from $$\mathbb S$$.

Question: Is there an easy formula for $$J^O(X)$$ when $$X$$ is a spectrum equipped with a map $$\mathbb S \to X$$? Can one at least say when $$J^O(X)$$ is nonzero?

Apparently such a formula will have to be a little more complicated than when $$X$$ is a space -- in particular, the dependence on the choice of map $$\mathbb S \to X$$ will be more subtle. For instance, if $$O$$ is an $$E_n$$ operad and $$\mathbb S \to X$$ is the zero map (or more generally smash-nilpotent), then $$J^O(X) = 0$$, rather than some large sum of wedges of $$X$$.

I think there's at least a colimit formula analogous to the James construction (which is, after all, the case $$O = E_1$$). For instance, when $$O = E_\infty$$, it should be the case that $$J^O(X)$$ is a certain colimit indexed over the category of finite sets and injections sending $$n \mapsto X^{\wedge n}$$. However, what I'd really like is a formula which (like the formula of the Snaith splitting) allows one to easily see that $$J^O(X) \neq 0$$ in reasonable cases, and so far this colimit formula has been a little too complicated for me to say this.

• You mean for O to be unital, not augmented, so that you’re asking about the left adjoint to the forgetful functor from O-algebras to E_0 algebras. I don’t think there is a splitting for this functor or any formula easier than the defining pushout of O-algebras (which can also be computed via a bar-like construction). It does come with a filtration whose associated graded is free on X/S, if that helps. (This gives you a sseq and you can try to see if 1 gets hit by a diffl). May 30, 2019 at 19:57

The best proofs of the `Snaith splitting' that you write down at the beginning, uses the $$J^{\mathcal O}$$ construction on spectra with units. Here is a complete proof:
Apply $$J^{\mathcal O}$$ to the zig-zag of equivalences of spectra (under and over $$S$$): $$\Sigma^{\infty} X_+ \xrightarrow{\sim} X \times S \xleftarrow{\sim} X \vee S.$$ On the left one gets $$\Sigma^{\infty}J^{\mathcal O}(X)_+$$ and on the right one gets precisely the wedge you wrote down (with an extra $$S$$ wedged on). QED