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$\renewcommand{\S}{\mathcal{S}}\newcommand{\l}{\langle}\newcommand{\r}{\rangle}\newcommand{\op}{\mathsf{op}}\newcommand{\fin}{\mathrm{fin}}$Recently, I've noticed that the definitions of special $\Gamma$-spaces and spectra are quite close in spirit:

  • $\Gamma$-spaces are pointed functors $X\colon(\Gamma^\op,\l0\r)\to(\mathcal{S},*)$ from Segal's category to the category $\mathcal{S}$ of spaces. Moreover, we call $X$ (see Definition 7.1 here)

    1. special if, for each $\l n\r,\l m\r\in\mathrm{Obj}(\Gamma)$, the map $$X_{\l n\r\vee\l m\r}\to X_{\l n\r}\times X_{\l m\r}$$ induced by the inert surjections $\l n\r\vee\l m\r\to\l n\r$ and $\l n\r\vee\l m\r\to\l m\r$ is a weak equivalence.
    2. very special if it is special and equivalently
      • $\pi_0(X_{\l1\r})$ is a group.
      • The map $$X_{\l 2\r}\to X_{\l1\r}\times X_{\l1\r}$$ induced by the total map $\l2\r\to\l1\r$ and one of the two inert surjections $\l2\r\to\l1\r$ is a weak equivalence.

  • Spectra are reduced excisive functors $E\colon\mathcal{S}^\fin_*\to\S$ of $\infty$-categories, where

    1. $E$ is excisive if it sends pushouts to pullbacks.
    2. $E$ is reduced if $E(*)\simeq *$;

In particular, very special $\Gamma$-spaces are equivalent to connective spectra. In a separate question, I've asked about whether it's possible to view nonconnectivity as arising from enlarging Segal's category $\Gamma^{\mathsf{op}}\overset{\mathrm{def}}{=}\mathsf{Sk}(\mathsf{FinSets}_*)$ of finite pointed sets into the $\infty$-category $\S^\fin_*$ of finite pointed spaces.

From a different side of this comparison, however, I was also thinking about how we may compare the excision and "special" conditions to each other: indeed, the former implies the latter, and this makes spectra into intrinsically grouplike notions. Because of this and other properties, spectra are regarded as the analogue of $\mathsf{Ab}$ in higher algebra, and the connective ones recover precisely the $\mathbb{E}_\infty$-group objects in spaces.

Question. Is there a known suitable weakening of the excision condition, making it into a kind of "semi-excision condition", in such a way that reduced semi-excisive functors $\S^{\fin}_*\to\S$ ("semispectra") include the $\mathbb{E}_{\infty}$-monoids in spaces as precisely the "connective semispectra"?

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    $\begingroup$ I don't know the answer to your question, but be careful about point 1 for $\Gamma$-spaces : a $\Gamma$-space is not defined by sending coproducts to products; in fact that would not make sense (it is covariant from finite pointed sets to spaces). The Segal condition is slightly more subtle (in particular it is not a "Lawvere theory"-type definition and that is very important) $\endgroup$
    – Maxime Ramzi
    Commented Aug 19, 2021 at 9:25
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    $\begingroup$ @MaximeRamzi It makes sense here because $\mathrm{Fin}_*$ has a zero object, so one has projections from the coproduct. $\endgroup$
    – Marc Hoyois
    Commented Aug 19, 2021 at 16:21
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    $\begingroup$ You could relax the excision condition to only hold up to group completion, with Segal-excision giving the pointwise $E_\infty$-monoid structure. A connective such functor (i.e., one that preserves $n$-connective spaces for all $n$) would then just be an $E_\infty$-monoid (equipped with a connective delooping of its group completion, but that is unique). $\endgroup$
    – Marc Hoyois
    Commented Aug 19, 2021 at 16:23
  • $\begingroup$ Thanks Maxime and Marc! I've updated the question to be more precise. $\endgroup$
    – Emily
    Commented Aug 20, 2021 at 22:32

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