Instances of "correcting" the compact objects of a category?

Question

I sense a familial resemblance between the following situations:

1. Equivariant homotopy theory: The $\infty$-category $Spaces^{BG}$ of Borel $G$-spaces fails to have a compact unit (= terminal object), even. So we pass to the bigger category $Spaces^{\mathcal O_G^{op}}$ of $G$-spaces (=presheaves on the orbit category).

2. Algebraic geometry: In $QCoh(X)$, the coherent sheaves often fail to be compact. So we pass to the larger category $IndCoh(X)$.

3. Chromatic homotopy theory: In the $K(n)$-local category of spectra $Spectra_{K(n)}$, the unit fails to be compact. I don't know what the fix is here -- maybe just pass to $Spectra_{E(n)}$?

4. $\infty$-Topos theory: In $Spaces$, the $\pi$-finite spaces (=coherent objects) fail to be compact. Again I'm not sure what the "fix" is -- is it talking about pro-$\pi$-finite spaces?

5. Animation: We start with some algebraic 1-category $\mathcal C$ like $Set$, $CRing$, or even $Top$, but we don't like the colimits there. So we extract the objects we do like -- the compact projectives $\mathcal C_{perf}$, and replace $\mathcal C$ with $P_\Sigma(\mathcal C_{perf}) = Fun^\times(\mathcal C_{perf}^{op},Spaces)$, the free completion of $\mathcal C_{perf}$ under sifted colimits.

Probably I could go on. The general pattern I'm driving at is something like "We have a category $\mathcal C$ which we like pretty well, but where the "nice, finite" objects are not compact. So we freely pass to a new category $\mathcal C'$ where those objects are compact."

Question 0: What are some other examples which might be added to the above list -- instances where one wants to "correct one's category to force certain objects to be more finite"?

Question 1: Is there something systematic to say about this process?

Question 2: If yes, then what? For instance, should the notion of "nice, finite" object be taken as "geometric input", or is there something formal to be said about some general intrinsic notion of finiteness which may be at odds with compactness?

Another thread linking many of the above examples is that often this move can be justified not just by looking at the category $\mathcal C$ in isolation, but by how $\mathcal C$ relates to other categories $\mathcal D$. Often this means that $\mathcal C$ really varies over some base (like $\mathcal C = C(G)$ for $G$ a group, $\mathcal C = \mathcal C(X)$ for $X$ a scheme, etc.), and often it means something like "the assignment $X \mapsto \mathcal C'(X)$ has better functoriality or 6-functor formalism - type properties than does $X \mapsto \mathcal C(X)$".

Question 3: Again, is there anything general to be said about this process, now from such a "global" perspective?

Question 4: If so, then what is the relationship between the "local" story from before and the "global" story here?

Note: I have some feeling that passing from $\mathcal C(X)$ to $\mathcal C'(X)$ can often be thought of as passing to some "compactification" of $X$, but trying to think generally about compactification seems to be a whole 'nother can of worms.

Answer 1

This is a fundamental part of the "functional analysis" of categories (thought of as analogues of topological vector spaces), and seems to come up everywhere you look at least in the setting of dg categories or stable $\infty$-categories [assumed to be presentable]. As you explain you often have several possible sources of finiteness - categorical (compactness), monoidal (dualizability), t-structure (coherence), finiteness with respect to a given functor - and you can choose to "complete your category for a different topology" by declaring a different class of small objects compact. e.g. a famous example fitting between your 1 and 2 is in modular representation theory, say for a finite group, where finite dimensional representations are not compact objects in modules for the group algebra, and this leads to the stable module category. I think most of these examples (again this applies at least to examples 1,2) can be thought of as finding a "decompletion" of your category -- using endomorphisms of the unit or more generally Hochschild cohomology you find your category lives over some parameter space ("support variety"), but only realizes complete objects along some subset. By enlarging the class of compact objects you're spreading the category over the entire space. (This theme of decompletion feels to me different than compactification but probably I'm missing the analogy.)

One satisfying "structural" source of examples is t-structures - this is explained beautifully in papers of Preygel and thoroughly in Appendix C of Lurie's Spectral Algebraic Geometry. Namely given a category with a suitable t-structure we can define a form of finiteness - coherent objects - and ind-complete with respect to this. Preygel calls this "regularizing" the t-structure, in reference to smoothness in algebraic geometry (Gaitsgory and collaborators call this "renormalization" which I find less evocative), while Lurie calls this "anticompletion", as it is opposite in a precise sense to completing the t-structure (and also is a "decompletion").

Another general (but maybe less intrinsic) way these examples arise is when you measure finiteness as seen through some given functor out of your category. This is the case for Koszul / bar-cobar dualities -- eg modules over an exterior algebra have a notion of finiteness given bounded coherent modules, which agrees with compactness as modules for the Koszul dual symmetric algebra (ie we measure the "size" of modules via Hom from the augmentation module), so by declaring these objects compact we get an equivalence with the symmetric algebra. Or in topology for a simply connected (or nilpotent) space you can similarly relate local systems (modules for chains on the based loop space) with module for cochains on the space, with different notions of finiteness.

Your "six-functor" examples I think have a similar flavor -- e.g. coherent sheaves provide a good notion of finiteness in particular since unlike perfects (the compact objects in QC) they're preserved by proper pushforwards (the augmentation for the exterior algebra can be thought of as arising this way), so ind-coherent sheaves are a natural setting for having !-pullbacks.

Finally a related source of examples in topology (again a form of Koszul duality or six-functor formalisms or decompleting BG, and appearing on the other side of geometric Langlands from IndCoh) is the notion(s) of "constructible" sheaves on a stack X. Namely we could define the large (:=presentable) category of sheaves on a stack as a limit of large categories of sheaves over schemes over X (or as a totalization for the Cech nerve of a cover) -- this is how D-modules on stacks naturally appear. OR we could consider small categories to be fundamental, like categories of constructible sheaves, and first define the small category on a stack as a limit of small categories in the same way. We then ind-complete to get a big category (so-called "renormalized" sheaves). This might be more natural eg in the l-adic sheaf setting.

The two versions differ exactly in the sense you ask. It's also an example of measuring smallness by a functor, in this case pullback to a given schematic cover. e.g. if your stack is a quotient of a space [scheme] by a group the difference amounts to Koszul duality, or the noncompactness of the unit on BG.(This is discussed for example in the papers of GKRV and AGKRRV [(Arinkin-)Gaitsgory-Kazhdan-(Raskin-)Rozenblyum-Varshavsky].)