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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 $\mathcal{S}_*$)

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    $\begingroup$ How do you make sense of $E_\infty$-group in a non-cartesian monoidal ($\infty$-,but it is irrelevant for my question)category ? For $E_\infty$-monoids, this is a good question but I'm not sure there's a more satisfying answer than "they are $E_\infty$-monoids in that category"... Unlike for sets, the basepoint here need not be "added". Examples of such monoids are of course given by the multiplicative structure on $\Omega^\infty E$, for $E$ a commutative ring spectrum $\endgroup$ Aug 10, 2021 at 12:03
  • $\begingroup$ @MaximeRamzi About $\mathbb{E}_{\infty}$-groups, I've been thinking about this, though I'm not yet sure: in the $1$-categorical case, it makes sense to speak of "group objects in $\mathsf{Sets}_*$" (as a property of monoids in $\mathsf{Sets}_*$), even though it is non-Cartesian: by a result of Péroux and Shipley (Lemma 2.4 of arXiv:1708.02592), every comonoid in $(\mathsf{Sets}_*,\wedge,S^0)$ comes uniquely from a comonoid in $(\mathsf{Sets},\times,\mathrm{pt})$, freely adjoined with a basepoint. $\endgroup$
    – Emily
    Aug 10, 2021 at 15:14
  • $\begingroup$ So, any monoid $A$ in $\mathsf{Sets}_*$ can be made into a bimonoid in it in a unique way, and since antipodes are also unique if they exist, the statement that $A$ has a Hopf monoid structure becomes a property, rather than extra structure. So in this sense we may say that $A$ is a group object in $\mathsf{Sets}_*$ iff it is a Hopf monoid in $\mathsf{Sets}_*$. $\endgroup$
    – Emily
    Aug 10, 2021 at 15:15
  • $\begingroup$ I'm unsure if this is still true in the $\infty$-categorical case, though. (I asked this as a separate question here). If it doesn't, then we can also just consider $\mathbb{E}_{\infty}$-Hopf monoids in $\mathcal{S}_*$, rather than group objects there (though this would make things less interesting, i think :/) $\endgroup$
    – Emily
    Aug 10, 2021 at 15:18
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    $\begingroup$ @Emily Yes, the functor $Ω^∞$ is lax symmetric monoidal (since it is the right adjoint to the symmetric monoidal functor $Σ^∞$), and so it sends $\mathcal{O}$-algebras in $\operatorname{Sp}$ to $\mathcal{O}$-algebras in $\mathcal{S}_\ast$ for every $\infty$-operad $\mathcal{O}$. Note that this gives a negative answer to your question about the characterization of $E_\infty$-monoids in $\mathcal{S}_\ast$, since $\Omega^\infty E$ usually is not of the form $X_+$ for any $X$ (as all connected components are equivalent as spaces) $\endgroup$ Aug 10, 2021 at 15:23

2 Answers 2

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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.

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  • $\begingroup$ Thanks for fixing the tex, Sam $\endgroup$
    – Peter May
    May 22, 2022 at 22:26
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My apologies for answering my own question, as well as the many edits. I've found answers to all but one of the original five questions, both reproduced below.

Questions:

  1. Has the notion of an $\mathbb{E}_{\infty}$-monoid in pointed spaces been studied before, particularly in the classical homotopy theory literature, and certainly under another name?
  2. A natural example of an $\mathbb{E}_\infty$-monoid in pointed spaces is $\Omega^\infty E$ for $E$ any $\mathbb{E}_{\infty}$-ring spectrum, as pointed by Maxime Ranzi in the comments. What are some other nice examples?
  3. The $\infty$-category of $\mathbb{E}_\infty$-spaces admits a tensor product $\otimes_\mathbb{F}$, having unit the geometric realisation $$|\mathrm{N}_{\bullet}(\mathbb{F})|\cong\coprod_{n=0}^{\infty}\mathbf{B}\Sigma_{n}$$ of the groupoid of finite sets and permutations $\mathbb{F}$. Is there such a tensor product in the $\infty$-category of $\mathbb{E}_{\infty}$-monoids in $\mathcal{S}_*$, and, if so, what is its monoidal unit?
  4. The $\infty$-category of $\mathbb{E}_{\infty}$-spaces admits a number of point-set models, such as special $\Gamma$-spaces, commutative monoids in $*$-modules, and commutative monoids in $\mathcal{I}$-spaces. Similarly, the $\infty$-category of spectra has $\mathbb{S}$-modules, symmetric spectra, and orthogonal spectra as point-set models (there's also very special $\Gamma$-spaces for connective spectra). Does the symmetric monoidal $\infty$-category $\mathsf{Mon}_{\mathbb{E}_{\infty}}(\mathcal{S}_*)$ also admit point-set models, in the sense of a presentation by a monoidal model category?

Answers:

  1. Yes. These are called “$\mathbb{E}_{\infty}$-spaces with zero” or “based $\mathbb{E}_{\infty}$-spaces” in Rognes, Topological logarithmic structures, Remark 6.12. There are also other names depending on which model (if any) one chooses for them, which include “$\mathcal{L}_0$-spaces” and “commutative based $\mathcal{I}$–space monoids” in loc. cit, as well as “$\mathcal{O}$-spaces with zero” or “$\mathcal{O}_0$-spaces” in May–Quinn–Ray–Tornehave's, $\mathrm{E}_\infty$ Ring Spaces and $\mathrm{E}_\infty$ Ring Spectra, Section IV.1, for $\mathcal{O}$ an $\mathbb{E}_{\infty}$-operad.
  2. Geometric realisations of nerves of symmetric monoidal categories provide examples of $\mathbb{E}_{\infty}$-spaces. Similarly, geometric realisations of nerves of “symmetric monoidal categories with zero” give another class of examples of $\mathbb{E}_{\infty}$-spaces with zero. Indeed:
    • Define a monoidal category with zero to be an $\mathbb{E}_{1}$-monoid in $(\mathrm{N}^\mathsf{D}_{\bullet}(\mathsf{Cats}_{\mathsf{2},*}),\wedge,\mathsf{pt}\coprod\mathsf{pt})$, where $\mathsf{Cats}_{\mathsf{2},*}$ is the $2$-category of small (weakly-)pointed categories;
    • Concretely, such an object corresponds to a monoidal category $(\mathcal{C},\otimes,\mathbf{1}_{\mathcal{C}})$ equipped with an object $\mathbf{0}_{\mathcal{C}}$ and natural families of isomorphisms \begin{align*} \delta^{\ell,\mathbf{0}}_{A} &\colon \mathbf{0}_{\mathcal{C}}\otimes A \longrightarrow \mathbf{0}_{\mathcal{C}},\\ \delta^{r,\mathbf{0}}_{A} &\colon A\otimes \mathbf{0}_{\mathcal{C}} \longrightarrow \mathbf{0}_{\mathcal{C}}, \end{align*} called the left and right annihilators of $\mathcal{C}$, satisfying certain coherence conditions.
    • Similarly, braided and symmetric monoidal categories with zero are $\mathbb{E}_{2}$- and $\mathbb{E}_{3}$($=\mathbb{E}_{4}=\cdots=\mathbb{E}_{\infty}$-)monoids in $(\mathrm{N}^\mathsf{D}_{\bullet}(\mathsf{Cats}_{\mathsf{2},*}),\wedge,\mathsf{pt}\coprod\mathsf{pt})$. They are described as above, but there are a few extra coherence conditions combining braidings and left/right annihilators.
    • For example, any (braided, symmetric) bimonoidal category gives rise to such a (braided, symmetric) monoidal category with zero by forgetting the additive monoidal structure.
    • By the symmetric monoidality of nerves and geometric realisations, it follows that, for any symmetric monoidal category with zero $\mathcal{C}$, the space $|\mathrm{N}_{\bullet}(\mathcal{C})|$ is an $\mathbb{E}_{\infty}$-space with zero.
    • So for instance each of the examples here give also examples of $\mathbb{E}_{\infty}$-spaces with zero, and there are of course many more.
  3. Yes, the $\infty$-category $\mathsf{Mon}_{\mathbb{E}_{\infty}}(\mathcal{S}_*)$ has a canonical monoidal structure obtained by applying Gepner–Groth–Nikolaus, Theorem 5.1 to $\mathcal{C}=\mathcal{S}_*$. By the same result, the unit for this tensor product is the free $\mathbb{E}_{\infty}$-space with zero on $S^0$. While the free $\mathbb{E}_{\infty}$-space on $*$ is computed to be \begin{align*} \coprod_{n=0}^{\infty}*^{\times n}_{\mathsf{h}\Sigma_{n}} &\simeq \coprod_{n=0}^{\infty}\mathbf{E}\Sigma_{n}\times_{\Sigma_{n}}* \\ &\simeq \coprod_{n=0}^{\infty}\mathbf{B}\Sigma_{n}\\ &\cong |\mathrm{N}_{\bullet}(\mathbb{F})|, \end{align*} the classifying space of the groupoid of finite sets and permutations $\mathbb{F}$, we see that the free $\mathbb{E}_{\infty}$-space with zero on $S^0$ is given by \begin{align*} \bigvee_{n=0}^{\infty}((S^{0})^{\wedge n})_{\mathsf{h}\Sigma_{n}} &\simeq \bigvee_{n=0}^{\infty}(S^{0})_{\mathsf{h}\Sigma_{n}}\\ &\simeq \bigvee_{n=0}^{\infty}\mathbf{E}\Sigma_{n,+}\times_{\Sigma_{n}}S^0,\\ &\simeq \bigvee_{n=0}^{\infty}(\mathbf{B}\Sigma_{n})_+\\ &\simeq \left(\coprod_{n=0}^{\infty}\mathbf{B}\Sigma_{n}\right)_+\\ &\simeq |\mathrm{N}_{\bullet}(\mathbb{F})|_+\\ &\simeq |\mathrm{N}_{\bullet}(\mathbb{F}_+)|, \end{align*} where $\mathbb{F}_+$ is the symmetric monoidal category with zero obtained by freely adjoining an absorbing element $-\infty$ to $\mathbb{F}$ and extending the rest of the symmetric monoidal category structure of $\mathbb{F}$ to $\mathbb{F}_+$ suitably (in particular by defining $\langle n\rangle\oplus-\infty\overset{\mathrm{def}}{=}-\infty$ for all $\langle n\rangle\in\mathrm{Obj}(\mathbb{F}_+)$).
  4. We can just consider pointed versions of models for $\mathbb{E}_{\infty}$-spaces:
    • A model for $\mathbb{E}_{\infty}$-spaces is given by special $\Gamma$-spaces, which are certain pointed functors $E\colon(\Gamma^{\mathsf{op}},\langle0\rangle)\to(\mathsf{Sets},*)$. We can (probably) obtain a model for $\mathbb{E}_{\infty}$-spaces with zero by considering instead special pointed functors $E\colon(\Gamma^{\mathsf{op}},\langle0\rangle)\to(\mathsf{Sets}_*,S^0)$.
    • A model for $\mathbb{E}_{\infty}$-spaces is given by commutative monoids in $\mathcal{I}$-spaces. Similarly, commutative monoids in pointed $\mathcal{I}$-spaces give a model for $\mathbb{E}_\infty$-spaces with zero. This is observed in Rognes, Topological logarithmic structures, Remark 6.12.
    • A model for $\mathbb{E}_{\infty}$-spaces is given by commutative monoids in $*$-modules. Similarly to the case of $\mathcal{I}$-spaces above, commutative monoids in “pointed $*$-modules” form a model for $\mathbb{E}_\infty$-spaces with zero as well.
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