Commutativity of pairs of reflective localizations

Question

Suppose there are two classes of morphisms $w_1, w_2$ in $C$ and two two reflective localizations $L_1: C \overset{\rightarrow}{\hookleftarrow} C^\text{$w_1$-local}: i_1$ and $L_2: C \overset{\rightarrow}{\hookleftarrow} C^\text{$w_2$-local}: i_2$. I am wondering when $L_1$ restricts to an adjunction $$L_1: C^\text{$w_2$-local} \overset{\rightarrow}{\hookleftarrow} C^\text{$w_1$, $w_2$-local}.$$ I think this holds when if the endofunctors $i_1\circ L_1$ and $i_2 \circ L_2$ of $C$ commute.

Question: Is there an easily checkable criterion on $w_1$, $w_2$ to determine whether these endofunctors commute?

Another reasonable condition is to assume that for any morphism $f_1: X_1 \rightarrow Y_1$ in $w_1$, the objects $X_1$, $Y_1$ are $w_2$-local. In this case, $f_1: X_1 \rightarrow Y_1$ is a morphism in $C^\text{$w_2$-local}$, and hence we can form $(C^\text{$w_2$-local})^\text{$w_1$-local}$ (which I believe is equivalent to $C^\text{$w_1$, $w_2$-local}$) and produce the reflective localization $$ C^\text{$w_2$-local} \overset{\rightarrow}{\hookleftarrow} (C^\text{$w_2$-local})^\text{$w_1$-local} = C^\text{$w_1$, $w_2$-local}. $$ However, I am not sure that this localization is the restriction of $L_1$.

I also suspect that this problem might be a special case of a more general setting with pairs of adjoint functors that are not reflective localizations, i.e. a pair of adjoint functors whose composition endofunctors commute.

** I am thinking about this problem in the $\infty$-categorical setting but an answer in the setting of ordinary categories would also be helpful.

Answer 1

Here is one general result in terms of $w_1$ and $w_2$ (or their saturations).

Suppose that $C$ is presentable, and that for any morphism $f: X_2\rightarrow Y_2$ in $w_2$, the objects $X_2, Y_2$ are $\overline{w_1}$-colocal, meaning that $Hom(X_2, f)$ and $Hom(Y_2, f)$ are isomorphisms for any morphism $f$ in the saturation $\overline{w_1}$ of $w_1$. Then $i_1 \circ L_1$ preserves $w_2$-local objects.

To see this, recall that we need to prove that $i_1\circ L_1$ preserves $w_2$-local objects: if $c$ is $w_2$-local, then $i_1\circ L_1(c)$ is also $w_2$-local. In the presentable case, the unit $\eta_c: c\rightarrow i_1 \circ L_1(c)$ is in the saturation $\overline{w_1}$ of $w_1$.

For any morphism $f: X_2\rightarrow Y_2$ in $w_2$ there is a commutative diagram $\require{AMScd}$ \begin{CD} Hom(Y_2, c) @>\eta_c>>Hom(Y_2, i_1 \circ L_1(c))\\ @V f V V @VV f V\\ Hom(X_2, c) @>\eta_c>> Hom(X_2, i_1 \circ L_1(c)) \end{CD} Since $c$ is $w_2$-local by assumption, the left vertical map is an equivalence. Since $\eta_c$ is in $\overline{w_1}$ and $X_2$ and $Y_2$ are $\overline{w_1}$-colocal, the top and bottom maps are also equivalences. Hence, the right vertical map is also an equivalence and hence $i_1 \circ L_1(c)$ is a $w_2$-local object, as desired.

Ideally, one wouldn't need to form $\overline{w_1}$ but only consider $w_1$. It is true that if an object $c$ is $w_1$-local, then $c$ is $\overline{w_1}$-local. However, I am not sure about colocal (due to the mix-and-match of localizations and colocalizations).

Question: If $c$ is $w_1$-colocal, then is $c$ $\overline{w_1}$-colocal?