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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.

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  • $\begingroup$ I'm not sure that all localisations occur as reflective subcategories. If you have a reflective subcategory then it can be considered as a localisation, but not the other way around. Example: Take the category of topological groupoids. This can be localised at the fully faithful essentially 'surjective' (actually need local sections, not just surjective) functors. The result is a category which is (equiv to) the Poincare category of the 2-category of topological stacks. This is larger than the category you started with, so isn't a reflective subcategory. $\endgroup$
    – David Roberts ♦
    Commented Nov 14, 2010 at 23:25
  • $\begingroup$ Sorry, yes, this is a good point. I guess I really had the idea of "Bousfield Localizations" in my head, where one finds the desired category inside a given model category. $\endgroup$
    – Eric Finster
    Commented Nov 14, 2010 at 23:28
  • $\begingroup$ I think this stems from Lurie's terminology. When he says localization, he means reflective localization. This is because he is building up to look at reflect (left-exact) localizations of $\infty$-presheaves to form $\infty$-topoi. But yes, there are many interesting localizations which are not reflective. (And there's coreflective ones, like compactly generated spaces sitting in $Top$). $\endgroup$
    – David Carchedi
    Commented Nov 15, 2010 at 1:33

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This may not be the sort of thing you're looking for (and it only answers your first question), but...

A category C is always both reflective and coreflective in the category $C^{\mathbf{2}}$ of arrows of C. You have the codomain and domain functors $c,d \colon C^{\mathbf{2}} \to C$ and the identity-morphism functor $i \colon C \to C^{\mathbf{2}}$, and adjunctions $c \dashv i \dashv d$, where the counit of the first and unit of the second are identities. For example, if $f \colon a \to b$ in $C$, then $\eta_f = (f,1_B) \colon f \to 1_B$, and so on.

This holds in any bicategory with enough comma objects, except with the word 'identities' replaced with 'isomorphisms'. It follows from quite general properties of the simplex 2-category, or from the fact that you can factor any adjunction (here the identity $1 \dashv 1$) into a reflection and a coreflection.

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