@TimCampion Doesn't the Nielsen-Schreir theorem rule this out? A subgroup of F_2 of index k is free on k+1 generators (true for infinite k as well, then k+1=k). If the inclusion is an iso on H_1, then k+1=2, and thus the subgroup is all of F_2.

@OmarAntolín-Camarena I'm not sure that applies to infinite index subgroups

Happy Anniversary!

Let $X$ be a derived stack, $QC(X)$ the stable category of quasi-coherent sheaves on $X$, and $HH(QC(X))$ the stable category $QC(X) \otimes_{QC(X) \otimes QC(X)^{op}} QC(X)$. Is it true that endomorphisms of the unit object in $HH(QC(X))$ recovers $THH(X)$?

at least for perfect stacks, yes: see ben-zvi--francis--nadler. there they show that what you call HH(QC(X)) is QCoh(LX) where LX = X x_{X x X} X, and if i understand correctly then THH(X) is just the ring of functions of LX

any qc scheme with affine diagonal is perfect, for instance

So I'm actually trying to see why, for a perfect stack, THH(X) is computed by O(LX), which BZFN don't seem to show explicitly. Or maybe I've just missed something in my reading?

what's your definition of THH(X)

THH(Perf_X) as given by the realization of the cyclic bar construction

the equivalence between QCoh(LX) and HH(QCoh(X)) passes through the bar construction

that's how you define the tensor product of stable presentable oo-categories

see theorem 4.7 of bzfn

Gotcha - thanks!