Let $\mathcal{S}$ be the $(\infty, 1)$-category of spaces. Then the compact objects of $\mathcal{S}$ are precisely the retracts of finite CW complexes. These are not the same as the finite CW complexes (i.e., the smallest subcategory of $\mathcal{S}$ containing $\ast$ and closed under finite homotopy colimits). Namely, given $X$ (let's say connected), the homology of the universal cover defines a perfect complex of $\mathbb{Z}[\pi_1 X]$-modules, which gives a class in $\widetilde{K}_0( \mathbb{Z}[\pi_1 X])$, which vanishes when $X$ is a finite CW complex. This class is the Wall finiteness obstruction; it is a classical result that this is the only obstruction for $X$ to be a finite CW complex. The reason for its existence is that taking the "image" of an idempotent is not a finite homotopy colimit (though it would be if one worked with $n$-categories for some $n$).
I'm curious about other examples of this type of finiteness obstruction. For instance, what happens if we do this in (for example) the $(\infty, 1)$-category of $E_\infty$-rings? The analogs of the finite CW complexes are the $E_\infty$-rings obtained from the free ones (free on a generator in some degree) via finite colimits, and the compact objects are these and the retracts. Is there an analog of the finiteness obstruction here? Other examples: algebras over some other operad, simplicial commutative rings, (pre)sheaves of spaces (resp. spectra), and equivariant spaces (resp. spectra). The only example that I am aware of considers the derived category of modules over a ring, where $\widetilde{K_0}$ measures the difference between "perfect complexes" and "those representable via finite complexes of frees."
