12:30 AM
@TylerLawson Thank you, sir!

10 hours later…
10:11 AM
@CharlesRezk $\underline{\mathsf{Spaces}}_G$ is a contravariant functor from the orbit category to $\mathsf{Cat}_{\infty}$ (see the definition I gave above). The object E_G\Gamma is an object in $\mathsf{Fun}_G(B_G\Gamma, \underline{\mathsf{Spaces}}_G)$
or equivalently in G-spaces over B_G\Gamma, just like the case when G is the trivial group. As you say, that functor category is equivalent to a full subcategory of G\time\Gamma spaces (which I think I tried to specify above with the alternate description of the functor category as presheaves on a subcategory of the orbit category for that products group).
Everything I said generalizes to the case of a G-groupoid, which is a presheaf of groupoids on the orbit category, in place of the special case of B_G\Gamma
And yes any group-like (coherent) monoid in Spaces_G would yield an example of this by taking classifying spaces pointwise (for the given functor on the orbit category).

9 hours later…
7:03 PM
@DylanWilson OK, I guess I understand. I'd say things like this. Let $\mathcal{X}$ be an infinity-topos, and $B$ an object of it. Then $B$ can be thought of as an "internal $\infty$-groupoid" of $\mathcal{X}$, by abstract abstract nonsense. The infinity-topos has an internal model of itself (up to size issues), an $\infty$-category object in $\mathcal{X}$ I'll call $\Omega$. ...
...Then there's an $\infty$-category $[B,\Omega]$ of "internal functors" and we have $[B,\Omega]=\mathcal{X}_{/B}$, basically formally using descent.
Then the diagonal map $B\to B\times B$, viewed as a morphism in $\mathcal{X}_{/B}$, corresponds to a natural transformation of functors in $[B,\Omega]$.
Taking $B=B_G\Gamma$ gives what you say, although some features of this are not formal consequences of the formalism I described: e.g., that $(\mathrm{Spaces}_G)_{B_G\Gamma}$ is equivalent to presheaves on certain $G\times \Gamma$-orbits.
Given $G$ and $\Gamma$, there is a kind of candidate for $B_G\Gamma$, which I'll call $B_G^{\mathrm{borel}} \Gamma$. It is defined as $G$-space as $\mathrm{Map}(EG, B\Gamma)$. In some cases (e.g., if $\Gamma$ is finite or a torus) it is the same as $B_G\Gamma$.
If not, then $(\mathrm{Spaces}_G)_{/B_G^{\mathrm{borel}}\Gamma}$ is equivalent to presheaves on something complicated and not describable using Lie groups, probably.

3 hours later…
9:58 PM
There's a map $(S^1)^{\times n} \to S^n$ from the $n$-torus to the $n$-sphere which collapses all but the top cell. Does this map split stably?

Yes

Cool! Is that easy to see?
Is it something as general as "the top cell of an orientable manifold always splits off stably"?

It's because $\Sigma(X\times Y) \approx \Sigma X \vee \Sigma Y \vee \Sigma(X\wedge Y)$.

Oh, duh!
Thanks
The context is that I think I have an alternate construction of Rognes' $S^{ad G}$ (that's the thing such that $S^{ad G} \wedge G = G^\vee$ that I was talking about the other day). This "potentially phony" $S^{ad G}$ has the property that it splits off of $G$ -- I just want to verify that this is not crazy.