Equivalent characterizations of the essential image of reflective subcategories

adjoint-functorscategory-theory

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.

Best Answer

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 $A$.

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