Let $X$ be an $E_\infty$-space (not necessarily grouplike). Let $x \in \pi_0 X$ be an element; say that $x$ is **strictly commutative** if there is a map of $E_\infty$-spaces $\mathbb{Z}_{\geq 0} \to X$ that takes $1 \mapsto x$. (The terminology is abusive, as for an element to be strictly commutative is extra data than a condition.)

There is also a natural space of strictly commutative elements in $X$, given by the (derived) mapping space (in the homotopy theory of $E_\infty$-spaces) $\hom(\mathbb{Z}_{\geq 0}, X)$. I do not know of a simple presentation of $\mathbb{Z}_{\geq 0}$ as an $E_\infty$-space (the free $E_\infty$-space on one object is $\bigsqcup_{n \geq 0} B \Sigma_n$), so I am not sure how to write this space down in terms of $X$. If $X$ is grouplike, so that it can be identified with a connective spectrum, then this is the mapping space in spectra $\hom( H \mathbb{Z}, X)$.

What are examples of strictly commutative elements? For instance, I am interested in the following example: given an $E_\infty$-ring $R$, what is the space of strictly commutative elements in the infinite loop space $\Omega^\infty R$ with *multiplicative structure*? (Equivalently, what is the space of maps $S^0[\mathbb{Z}_{\geq 0}] \to R$ in $E_\infty$-rings?) One reason is that the $E_\infty$-ring $S^0[\mathbb{Z}_{\geq 0}]$ is easier to compute with than the free $E_\infty$-ring on a generator in degree zero, but seems to be less nice formally, and I'd like to know conditions under which an element in $\pi_0 R$ can be hit by a map from the monoid algebra.