Reflective Localizations vs. categories of local objects

Question

Given a category $\mathcal{C}$ and a set (let's not bother with size issues here) $\mathcal{W} \subseteq \text{Mor}(\mathcal{C})$ we may form the category $\mathcal{C}[\mathcal{W}^{-1}]$ obtained by formally inverting all arrows belonging to $\mathcal{W}$. If we're lucky, then the localization functor $j: \mathcal{C} \longrightarrow \mathcal{C}[\mathcal{W}^{-1}]$ admits a fully faithful right adjoint $\iota: \mathcal{C}[\mathcal{W}^{-1}] \hookrightarrow \mathcal{C}$, in which case one speaks of a reflective localization.

The category $\mathcal{C}[\mathcal{W}^{-1}]$ viewed as a full subcategory of $\mathcal{C}$ via $\iota$ obviously consists of $\mathcal{W}$-local objects, i.e. for every $A \in \mathcal{C}[\mathcal{W}^{-1}]$ and every $f \in \mathcal{W}$ the map $\text{Hom}(f,A)$ is a bijection.

The nLab entry for "reflective localization" seems to claim that also the converse holds, i.e. that every $\mathcal{W}$-local object lies in the essential image of $\iota$, or equivalently, that for every $\mathcal{W}$-local $X$ the unit

$\eta(X): X \longrightarrow \iota(j(X))$

of the adjunction is an isomorphism. However I don't see how to prove that. By the triangle equation it follows (since $\iota$ is fully faithful) that $\eta(X)$ becomes an isomorphism after applying $j$, but I don't see how this helps in proving that $\eta(X)$ itself is an isomorphism.

So the question is, does this reverse implication hold at all?

Answer 1

To avoid confusing myself, I will write $L : \mathcal{C} \to \mathcal{C} [\mathcal{W}^{-1}]$ for the localising functor and $R : \mathcal{C} [\mathcal{W}^{-1}] \to \mathcal{C}$ for its right adjoint. (Note that $R$ is automatically fully faithful – the hard part is existence!)

As you say, for any object $Y$ in $\mathcal{C} [\mathcal{W}^{-1}]$, $R Y$ is automatically a $\mathcal{W}$-local object in $\mathcal{C}$: indeed, for any object $Z$ in $\mathcal{C}$, $Z$ is a $\mathcal{W}$-local object if and only if $\mathcal{C} (-, Z) : \mathcal{C}^\mathrm{op} \to \mathbf{Set}$ factors through $L : \mathcal{C} \to \mathcal{C} [\mathcal{W}^{-1}]$. So suppose $Z$ is a $\mathcal{W}$-local object. Since $\epsilon : L R \Rightarrow \mathrm{id}_{\mathcal{C} [\mathcal{W}^{-1}]}$ is a natural isomorphism, the triangle identities imply $L \eta : L \Rightarrow L R L$ is also a natural isomorphism. In particular, $$\mathcal{C} (\eta_Z, Z) : \mathcal{C} (R L Z, Z) \to \mathcal{C} (Z, Z)$$ is a bijection, so there is a unique morphism $\alpha : R L Z \to Z$ such that $\alpha \circ \eta_Z = \mathrm{id}_Z$. But $L \alpha = \epsilon_{L Z}$, so $$\eta_Z \circ \alpha = R L \alpha \circ \eta_{R L Z} = R \epsilon_{L Z} \circ \eta_{R L Z} = \mathrm{id}_{R L Z}$$ and therefore $\eta_Z : Z \to R L Z$ is indeed an isomorphism.

So the conclusion is that $\mathcal{C} [\mathcal{W}^{-1}]$ is equivalent to the full subcategory of $\mathcal{C}$ spanned by the $\mathcal{W}$-local objects.

Answer 2

Let $\iota: \mathcal{A}\subset \mathcal{C}$ a replete, full, reflexive subcategory with $F: \mathcal{C}\to \mathcal{A}$ left adjoint to $\iota$, and $\eta_X: X\to F(X)$ the canonical unity. The canonical counity $\epsilon$ is a isomorphism (because $\iota$ is full and faithfull) and $F(\eta_X)$ is a isomorphism (triangle identity).

Let $W:= \{ f \in |\mathcal{C}|_1\ |\ F(f)\ is\ Iso \}$ (in your post you can consider the saturation...).

Let $\mathcal{B}\subset \mathcal{C}$ the full subcategory of $W$-replete objects defined as in your post.

Claim: $\mathcal{A}= \mathcal{B}$: Let $A\in \mathcal{A}$ and $f: X\to Y$ by $F(f)\in Iso$, considering the square of $f, F(f), \eta_X, \eta_Y$, by universal property follow that $A\in \mathcal{B}$.

Let $B\in \mathcal{B}$, we have to show that $\eta_B\in Iso$, considering $\eta_B\in W$ follow a morphism $g: F(B)\to B$ with $1_B= g\circ \eta_B$, and from $(\eta_B\circ g)\circ \eta_B=1\circ \eta_B $ follow that (universal property): $\eta_B\circ g=1$.

Edit:

COnsidering the case $\mathcal{A}= \mathcal{C}[\mathcal{W}^{-1}]$ for some class of morphisms $\mathcal{W}$, and suppose that we have a full immersion $\iota: \mathcal{C}[\mathcal{W}^{-1}]\to \mathcal{C}$ left adjoint of the natural functor $F: \mathcal{C}\to \mathcal{C}[\mathcal{W}^{-1}]$, and let $W$ defined as above, then $W$ is the saturation of $\mathcal{W}$ and $\mathcal{C}[\mathcal{W}^{-1}]\cong \mathcal{C}[W^{-1}]$.

THen the $W$-local objects are the $\mathcal{W}$-local objects, and this is a replete class.

Edit ABout the second question, after the Zhen Lin counterexample, I can say only this: From [P], T.13.11, p.98, given an adjuntion $<\iota, F>: \mathcal{C}\to\mathcal{A}$ with $\iota$ full and faithfull we have that $\mathcal{A}$ is equivalent to $\mathcal{C}[W^{-1}]$ where $W$ is the class of morphisms $f$ such that $F(f)$ is a isomorphism, and $W$ is also the class of morphism's $w$ such that $\mathcal{C}(w, A)$ is a isomorphism (bijection) for any $A\in\mathcal{A}$. Then if $\mathcal{A}\subset \mathcal{C}$ is the full subcategory of the $\mathcal{W}$-stable objects for some class of morphisms $\mathcal{W}$, then $W$ is the "Galois-closure" of $\mathcal{W}$.

Furthermore, about the existence of the adjoint $\iota$: given a class lef-calculable of morphisms $\mathcal{W}$ of a category $\mathcal{C}$, let $F: \mathcal{C}\to \mathcal{C}[\mathcal{W}^{-1}]$ the natural functor. From [P] 15.3 follow that $F$ has a right adjoint (and it is full and faithfull) IFF for any $X\in \mathcal{C}$ there exist a morphism $u_X: X\to X'$ where $X'$ is $\mathcal{W}$-stable and $u_X$ belong to the saturation $W$ of $\mathcal{W}$ IFF for any $X$ the category $X\downarrow W$ (the full subcategory of $X\downarrow\mathcal{C}$ of the $W$'s with $X$ as domain) has a final object (I seem that the proff work without the left-calculable hypothesis).

Biblio: [P]: Theory of Categories , Nicolae Popescu & Liliana Popescu