TITLE: Is there a recognition principle for $\mathbb{E}_{\infty}$-spaces with zero?
QUESTION [6 upvotes]: A commutative monoid with zero is a commutative monoid $A$ together with an element $0_{A}$ such that $0_{A}a=a0_{A}=0_{A}$ for all $a\in A$. They are precisely the monoids (in the sense of monoidal categories) in the category of pointed sets, equipped with the smash product.
Passing to the derived world, commutative monoids get replaced by $\mathbb{E}_{\infty}$-monoids in spaces, while pointed sets get replaced by pointed spaces. So the natural analogue of a commutative monoid with zero in homotopy theory should be an $\mathbb{E}_{\infty}$-monoid in the symmetric monoidal $\infty$-category $(\mathcal{S}_*,\wedge,S^0)$ of pointed spaces, called an $\mathbb{E}_{\infty}$-space with zero.
Question. The May recognition principle states that a space is (weakly equivalent to) an infinite loop space iff it is a grouplike $\mathbb{E}_{\infty}$-monoid in spaces. Is there a recognition principle for $\mathbb{E}_{\infty}$-spaces with zero?
(Or of a subclass of them, such as some appropriate version of “grouplike” that works for the non-Cartesian monoidal $\infty$-category $\mathcal{S}_*$)

REPLY [2 votes]: I see no real point in considering $E_{\infty}$ spaces with $0$ except in the context of the multiplicative structure of an
$E_{\infty}$ ring space or, essentially equivalently, the multiplicative structure of an $E_{\infty}$ ring spectrum.   $E_{\infty}$ ring spaces or even just $E_{\infty}$ spaces with $0$,  come with two basepoints, say $0$ and $1$, of which only $0$ should be thought of as truly the basepoint.  $E_{\infty}$ spaces with zero then give the multiplicative structure of $E_{\infty}$ ring spaces, in which one has two interrelated $E_{\infty}$ space structures, one "additive'' and one "multiplicative'' but with $0$. The $0$ in the multiplicative structure allows one to form two monads in the same category, the category of based spaces, which are interrelated in the sense prescribed in a beautiful 1969 paper "Distributive laws" of Beck.   Beck's theory is recalled in section 15 of "The construction of $E_{\infty}$ ring spaces from bipermutative  categories".   That paper and its companions "What precisely are $E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra" and "What are $E_{\infty}$ ring spaces good for?''  date to 2009.  They modernize the theory first developed in the 1977 Springer volume "$E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra'' that has already been referred to.  It was first proven there that grouplike $E_{\infty}$ ring spaces, those for which $\pi_0$ is a group under addition and therefore a commutative ring, are equivalent to connective $E_{\infty}$ ring spectra.   This and the whole of infinite loop space theory, additive, multiplicative, and especially equivariant are being reworked in a paper provisionally titled ``Group completions and the homotopical monadicity theorem'', by Hana Jia Kong, Foling Zou, and myself.