# Equivalent characterizations of the essential image of reflective subcategories

I'm currently trying to prove the third item of the following exercise from Category Theory in Context,

Exercise 4.5.vii. Consider a reflective subcategory inclusion $$D \hookrightarrow C$$ with reflector $$L: C \to D$$.

1. Show that $$\eta L = L\eta$$, and that these natural transformations are isomorphisms.
2. Show that an object $$c \in C$$ is in the essential image of the inclusion $$D\hookrightarrow C$$, meaning that it is isomorphic to an object in the subcategory $$D$$, if and only if $$\eta_c$$ is an isomorphism.
3. Show that the essential image of $$D$$ consists of those objects $$c$$ that are local for the class of morphisms that are inverted by $$L$$. That is, $$c$$ is in the essential image if and only if the pre-composition functions $$C(b, c) \xrightarrow{f^*} C(a, c)$$ are isomorphisms for all maps $$f : a \to b$$ in $$C$$ for which $$Lf$$ is an isomorphism in $$D$$. This explains why the reflector is also referred to as “localization.”

Here $$\eta$$ is the unit of the adjunction. I have managed to prove the first two items. However, reflective subcategories were just briefly introduced and so I am not sure how to relate item $$(3)$$ with $$(2)$$, if this is the right path to begin with. Any hints? I would also appreciate if someone can shed some light on the term localization and why $$(3)$$ 'explains' this nomenclature.

Hint: First prove that $$iLc$$ satisfies the condition in 3, where $$i:D\to C$$ is the inclusion. Then use apply 2 to conclude.
Regarding "localization" the point is that $$L$$ is characterized by the arrows it inverts, so the language is being imported from commutative algebra. Specifically, the motivating situation is that of $$R[S^{-1}]$$-modules, where $$S$$ is a multiplicative set in a commutative ring $$R$$. Every $$R[S^{-1}]$$ is functorially an $$R$$-module by restricting the scalars, and this functor is fully faithful, with the left adjoint $$R[S^{-1}]\otimes_R (-)$$. So this is a reflective subcategory. Furthermore an $$R$$-module admits an $$R[S^{-1}]$$-action if and only if it is local for those $$R$$-module maps inverted by tensoring with $$R[S^{-1}]$$, as in Riehl's point 3. The classical algebraic framework would reduce those maps to the maps $$s:R\to R$$ determined by elements of $$S$$, while the general categorical framework instead asks for locality with respect to all maps $$\eta_A:A\to A[S^{-1}]$$, for $$A$$ an $$R$$-module. We can bridge these frameworks by observing that locality with respect to each $$s$$ is equivalent to locality with respect to the single unit map $$R\to R[S^{-1}]$$, which implies locality with respect to every $$\eta_A$$ by considering the action of $$L$$ on a presentation of $$A$$.
• Thanks for your reply! I think I've got your hint, but still can't manage to finish this off. If $c$ is on the essential image via $w : c \simeq Ld$, then $Ld$ verifies the condition in $(3)$ and thus so does $c$ via factoring $f^*$ through $f^* : C(b,Ld) \to C(a,Ld), w_* : C(a,c) \to C(a,Ld), w_* : C(b,c) \to C(b,Ld)$. – qualcuno Feb 18 '19 at 4:19
• (cont.) However the converse is still unclear to me: if $c$ verifies the condition on $(3)$, we can apply this to $\eta_c$ (as $(1)$ guarantees that $L\eta_c$ is an iso). I see how this shows that $\eta_c$ is a split epi: if $a$ is the inverse of $\eta_c^*$, then $a(1_c)\eta_c =( \eta_c^* a)(1_c) = 1_c$. However, can we assert that this arrow is actually an iso? Having $Lc \simeq c$ for any $c$ in the essential image seems too strong, but I may have bad intuition on this. – qualcuno Feb 18 '19 at 4:19
• @GuidoA. Actually, $Lc$ is indeed isomorphic to $c$ in that case. Given an iso $a:c\to Lc'$, we have a naturality square involving $a,La,$ and $\eta_{Lc'}$, all of which are iso, as well as $\eta_c$. So the latter is an iso as well. – Kevin Arlin Feb 18 '19 at 21:08