Examples of good categories with bad objects being better
问题内容
There is a philosophy attributed to Grothendieck that it is better to have a good category (e.g. mapping objects, abelian category, etc) with bad objects than a bad category with nice objects.
What are some examples of this? Please also describe some ways these good categories have been helpful.
回答 (1)
Honestly, this happens just about any time you stick a smaller category into a bigger one. And as alluded to by Noah Schweber, this is not unique to category theory, and is ubiquitous across mathematics.
Simplest examples that comes to mind: embedding the category of (trivial) vector bundles into the category of coherent sheaves (the latter being abelian) on a noetherian scheme. Embedding the category of coherent sheaves into the category of quasi-coherent sheaves on a scheme say (the latter containing all product/coproducts, as opposed to just finite ones, and being better behaved for non-noetherian schemes, and for example allowing us to define a pushforward functor (for qcqs maps)), and embedding the category of quasi-coherent sheaves into the category of $\mathcal{O}_X$-modules (the latter having all injectives, which I believe is not true in general for $QCoh$ of a scheme, and thus allowing us to define sheaf cohomology).
Other ones that come to mind are from derived categories, for example, considering $D_{qc}(X)$, the who's objects are complexes of $\mathcal{O}_X$-modules with quasi-coherent cohomology sheaves, instead of $D(QCoh(X))$ (who's objects are complexes of quasi-coherent sheaves). While these two agree in many situations (e.g. $X$ is quasi-compact and separated scheme), the difference becomes quite clear when $X$ is allowed to be a stack - for example $X = BE$ for an elliptic curve $E$ (since $QCoh(BE) = Vect$). Then things like flat base-change don't work for the latter but they do for the former.
While we're mentioning stacks, embedding schemes into algebraic spaces allows us to take quotients by finite groups. Embedding schemes into stacks allows us to take quotients by (fppf) group schemes, and allows several moduli functors to be representable.
Another key one that comes to mind is, assuming $X$ is a qcs scheme (hence $D_{qc}(X) = D(QCoh(X))$), then embedding $D^b_{qc}(X)$ (objects are bounded complexes of $\mathcal{O}_X$-modules with quasi-coherent cohomology sheaves) into $D_{qc}(X)$, since the latter has all coproducts, and hence it makes sense to talk about compactness and compact generation. To my understanding, the "best" proof of Grothendieck duality uses these concepts.
It's also worth mentioning another source of examples is just about any time someone mentions $\infty$-categories, for example $\infty$-categorical enhancements of several derived categories because they contain limits/colimits, and importantly these limits/colimits are what we want them to be (which allows us to do e.g. descent/gluing for derived categories).