@SaalHardali -- I suppose to be precise it's $\Sigma^\infty_+$ rather than $\Sigma^\infty$, right?

Does this happen when $I = BG$ where $G$ is a finite group?

@TimCampion Yeah that's right. What do you mean by I=BG? For any localization of prsheaves on it?

@SaalHardali Oh -- I guess I misread your question. I'm still wondering when $\Sigma^\infty_+$ is levelwise even without any localization -- is this always the case when there's no localization?

Er -- I suppose it is, simply because limits and colimits in functor categories are computed levelwise

@TimCampion yeah. I came up with this question while wondering about the stabilization of the category of $\infty$-operads. I wonder whether or not its the same as just complete segal spectra...

If the suspension is levelwise it is but I suspect this is not the case so i'm wondering whether there's a fix for this which does give some explicit description of this category.

You could obviously ask the same about categories but I came to this from contemplating how the homotopy theory of 1-coloured reduced operads looks like.

If anyone knows a paper which talks about the stabilization of the category of operads that would be nice.

@SaalHardali The stabilization of the ∞-category of ∞-categories is known: it's just the category of spectra. More generally the stabilization of $\mathrm{Cat}_{/C}$ is $\mathrm{Fun}(\mathrm{Tw}C,\mathrm{Sp})$ (ref)

Ah yes! I just recalled reading a paper that does this (a long time ago).

I suspect something similar is true for ∞-operads

In any case what I'm interested in the orthogonal case kind of

I mean the case where the operad has no non-identity 1-ary operations

and only 1 colour

The argument that $Stab(Cat) = Stab(Top) = Sp$ is not hard: any 1-fold loop object is already an $\infty$-groupoid. Then since $\infty$-groupoids are closed under finite limits, an infinite loop object will be the same as an infinite loopspace.

The same argument shows that $Stab(SymMonCat) = Stab(E_\infty(Top)) = Sp$

I'm not sure about general $\infty$-operads though

Actually -- $Opd \subseteq Fun(Fin_\ast,Top)$ is closed under limits. So an infinite loop object in $Opd$ is an operad all of whose spaces are infinite loop spaces, and all of whose structure maps are maps of infinite loop spaces. That is, it's an operad in spectra with respect to the direct sum monoidal structure.

Maybe some Eckman-Hilton argument can show that this is the same data as a spectrum.

@TimCampion This is exactly what I wanted to be true

Could you repeat the argument please? I'm not sure I got it

So we use the model of stabilization via infinite loop objects

Loops of operads will be computed by taking loops levelwise, when you think of an operad as a functor $Fin_\ast \to Top$

When you take loops levelwise, you end up with an operad whose spaces are loopspaces and whose structure maps are loop maps.

Iterating this, you see that an infinite loop object in operads will be an operad whose spaces are infinite loopspaces and whose structure maps are infinite loop maps.

(@SaalHardali Let me know where I stop making sense)

Then a spectrum object will be a sequence of these things, one a delooping of the previous one.

The structure maps are still maps $O(n) \times O(m_1) \times \dots \times O(m_n) \to O(m_1 + \dots + m_n)$

and now you observe that having such structure maps as infinite loop maps is the same as having an operad in infinite loop spaces with the cartesian monoidal structure

I suppose some care will be required to make this argument rigorous...

Here's a general question: Let $C$ be a cocartesian monoidal category. What is an operad in $C$ concretely?

@TimCampion Got it, thanks.

Isn't an operad in a cartesian category a çomplete dendroidal segal object?

I mean it works for spaces so why not?

@SaalHardali Ah, but I'm asking about operads in cocartesian monoidal categories!

The direct sum is both cartesian and cocartesian

But I think the fact that it is cocartesian will make the description of operad objects simplify considerably

@CharlesRezk Regarding your question from a few days ago, first, I assume you mean $\Sigma^\infty_+ \mathbb C \mathbb P^\infty$, rather than $\Sigma^\infty \mathbb C \mathbb P^\infty$ in order to get a ring spectrum. Second, I must be missing something because I don't see how this isn't exactly what Snaith's theorem talks about -- $\Sigma^\infty_+ \mathbb C \mathbb P^\infty[ \beta^{-1}] = KU$, right?

@TimCampion I suspect he means the free $E_∞$-algebra on $\mathbb{CP}^∞$, which has as an underlying spectrum $\bigoplus_n (\Sigma^∞\mathbb{CP}^∞)^{⊗n}_{h\Sigma n}$, not the suspension spectrum

@CharlesRezk was Charles' question to which I'm referring

@DenisNardin But the suspension spectrum $(\Sigma^\infty_+$) of an infinite loop space is the free $E_\infty$ ring on it -- right?

@TimCampion Well, there's a difference between the group algebra $\mathbb{Z}[G]$ and the free algebra on the group $\mathbb{Z}[t_g\mid g\in G]$, isn't it?

I.e. I don't think he is using the $E_∞$-structure on $\mathbb{CP}^∞$ to write down the definition

So you're thinking take the free $E_\infty$-ring spectrum on the suspension spectrum of $\mathbb C \mathbb P^\infty$?

Well, that's literally what $\mathbb{P}(\Sigma^∞\mathbb{CP}^∞)$ usually means

Right

That is exactly what Charles wrote

I guess I should read more carefully

@TimCampion Missed the "co-" (unsurprisingly). In any case the stabilization of operads is operads in spectra (with cartesian structure). Thanks for helping me with this ^^

As for operads in cocartesian category sounds like given a good definition of this you automatically get a good definition of co-operads by taking op everywhere. From what I recall co-operads are tricky to define...