Wow... didn't realize that the \infty-operadic Grothendieck construction wasn't really written down anywhere except for Gijs's work using dendroidal sets...

@TylerLawson regarding your answer to my question, I suspect the functor Ass^⊗-->Cat_Δ should only be lax monoidal? Does that seem right?

I was thinking of the functor that quite literally sends <n> to M^n (plus-or-minus taking cofibrant-fibrant objects inside M)

I think that's at least strong monoidal

Oh, maybe I misinterpreted, it's probably a lax functor / pseudofunctor (there's a natural iso from F(g) . F(f) to F(gf))

Let a commutative group G act on a scheme X. We can then form the quotient stack [X/G] and there is a very nice description of the inertia stack in this situation. Is there a similar nice description of the inertia group of [X/G] when G is a commutative group scheme? That is, when G is not necessarily the associated sheaf of a constant group functor?

A sanity check : G is a finite group, and assume M is a module over the constant Mackey functor of integers, or equivalently M is a cohomological Mackey functor in the sense of Thevenaz-Webb, could we build a Moore spectrum of M in the sense that the integer-graded homology is concentrated in degree 0 and is M?

The homology has coefficient the constant Mackey functor Z

no - for most groups, there's no Moore spectrum for the constant Mackey functor

in fact, for any nontrivial group

as shown in Appendix B of web.math.rochester.edu/people/faculty/doug/otherpapers/…

this was shown to me by @Tyler

oh, I think I might have misinterpreted the question - you're talking about homology with coefficients in the constant Mackey functor, so the sphere is a Moore spectrum from the constant Mackey functor in this sense

I strongly suspect the answer is still no

I don't think the constant Mackey functor has well-enough behaved homological algebra for this kind of thing to hold

@SaulGlasman Yeah, after thinking for a while I believe the answer shall be negative

I need a resolution of length 3 or more in general to resolve a Z module, and it seems that there are a lot of obstructions of recovering a G-CW-complex out of it