Localizations are an extremely important part of modern homotopy theory. Both the category of spaces an spectra have a plethora of interesting localizations: at a fixed prime, rational, with respect to a given homology theory, etc.
I started wondering if it's possible to create "toy models" of some kind where the collection of possible localizations is somehow under control in a nice way. This would seem very difficult to accomplish using model categories, since a major theme of the model category literature is that there are almost always lots of localizations around. It's possible one could do something with $\infty$-categories, since the notion of localization makes sense there as well, but for now I'll just stick to ordinary categories for simplicity. In fact, localizations are just special reflective subcategories, so let's just think about those.
So suppose I have a category $\mathcal C$. Can I always find a new category which has $\mathcal C$ as a (proper, so that this is nontrivial) reflective subcategory? Yes, though I'm not very happy with my solution. Let $\mathcal C_0$ be the set of objects of $\mathcal C$ regarded as a discrete category. The inclusion $\mathcal C_0 \rightarrow \mathcal C$ is a diagram in $\mathcal Cat$, and it's easy to see that the Grothendieck construction on this diagram has $\mathcal C$ as a reflective subcategory, albeit in a rather trivial way. Are there other (possibly better) examples?
Now what if I have a collection of categories $\mathcal C_i$ for $i \in \mathcal I$. Is it possible to build a new category which contains each of the $\mathcal C_i$ as a reflective subcategory?
In general are there constructions which affect the collection of reflective subcategories in predictable ways? For example, it's easy to see that in the disjoint union $\mathcal C \coprod \mathcal D$ of two categories, the collection of reflective subcategories is just the product of those of $\mathcal C$ and those of $\mathcal D$. The cartesian product $\mathcal C \times \mathcal D$ seems a bit harder to understand.
I'd be interested to hear about homotopy theoretic variants as well. It just seemed easier to me to think about categories first.