Viewing Objects as Objects of a Reflective Subcategory with Bonus Structure

Question

A category $\mathcal{C}$ is called a reflective subcategory of $\mathcal{D}$ whenever the inclusion $i : \mathcal{C} \to \mathcal{D}$ admits a left adjoint (confusingly called $R$, the "reflector"). A key example is that of abelian groups as a subcategory of all groups. The inclusion admits a left adjoint (abelianization).

On the nlab page for reflective subcategories, they mention that we can view $R$ as a forgetful functor, and thus elements of $\mathcal{D}$ are objects of $\mathcal{C}$ with some "bonus structure".

The examples given are:

• $i$ from complete metric spaces to metric spaces, with the reflector given by completion. Then we can identify a metric space $M$ with its completion $\hat{M}$ equipped with a dense subset ($M \subseteq \hat{M}$).

• $i$ from fields to integral domains, with the reflector given by the field of fractions. Then the integral domain is just its field of fractions equipped with a numerator and denominator function.

It isn't hard to come up with other examples, which are (to me at least) more compelling. Notably

• $i$ from the category of undirected graphs to the category of directed graphs, where the reflector is the "symmetrize" operation. Then a directed graph is an undirected graph with a choice of (at least one) orientation for each edge.

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The question, then: Is this realization ever useful in practice? Or is it more useful for liberating one's mindset? It sounds like something which might provide a useful lens to view the world (particularly the example with graphs is similar to constructions which real combinatorialists already care about), but I cannot come up with any concrete examples of it being useful.

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Edit:

To clarify somewhat, thinking of an integral domain as a field equipped with numerator/denominator functions seems like it would encourage you to do field-y things with integral domains, knowing you can get the numerator and denominator back. This mindset might be useful in commutative algebra/algebraic geometry, where a common idea (from my limited experience) is to pass to the quotient field and then clear denominators at the end to get back to the integral domain you're actually interested in.

Are there other situations where this mindset aligns with a common proof idea? As a potential example (which perhaps exposes my lack of familiarity with basic analysis), are there theorems about metric spaces which one proves by passing to the completion, and then arguing your way back into the original space (which is dense in the completion)? Of course, if you have examples of this mindset in any reflective subcategory, I would love to hear them.

Thanks in advance!

Answer 1

I can't answer your particular question, so take it as a longer comment or side note.

Anyway, usually this 'liberated mindset' or 'useful len' will produce something new: at least simpler alternative proofs and often new theorems as well.

There are thousands of examples, but let me mention another one:

Viewing an adjunction $F\dashv G,\ F:\mathcal A\to\mathcal B$ as the collage $\mathcal F$ of a profunctor $\tilde F:\mathcal A^{op}\times\mathcal B\to\mathcal{Set}$ which is naturally isomorphic to $$F_*=\,\hom_{\mathcal B}(F-,\,-)\,\simeq\,\hom_{\mathcal A}(-,\,G-)\,=G^*$$ The collage of a profunctor $U:\mathcal A^{op}\times\mathcal B\to\mathcal{Set}$ is the category that extends the disjoint union of $\mathcal A$ and $\mathcal B$ by the elements of $U(a,b)$ as morphisms $a\to b$.

This brings in the theorem:

Every adjunction $F\dashv G$ factorizes as $F=R_{\mathcal B}I_{\mathcal A}$ and $G=C_{\mathcal A}I_{\mathcal B}$ where $I$ denotes full embedding and $R$ is a reflector, $C$ is a coreflector functor, $R_{\mathcal B}\dashv I_{\mathcal B}$ and $I_{\mathcal A}\dashv C_{\mathcal A}$.