General Theory of Left-Exact Localization?

Just to tie this one up, the cited result in Borceux (Prop 5.6.1) says the following:

Proposition: Let $\mathcal C$ be a finitely-complete category and let $L \mathcal C$ be a reflective subcategory. Let $\mathcal W$ be the class of morphisms inverted by the reflector $r: \mathcal C \to L\mathcal C$. Then $r$ is left exact if and only if $\mathcal W$ is stable under base change.

This continues to hold in the $\infty$-categorical context. Here is a proof adapted from Borceux which works $\infty$-categorically.

Proof: The "only if" direction is clear; we prove "if". First, since $i$ preserves terminal objects we have that the terminal object of $L\mathcal C$ is the terminal object of $\mathcal C$ and in particular is $\mathcal W$-local, so $L$ preserves terminal objects.

So it will suffice to show that $L$ preserves pullbacks. To this end, it is sufficient to show that if we have a natural transformation from the pullback square on the left to the one on the right below, and if the components $B \xrightarrow \sim B'$, $C \xrightarrow \sim C'$, and $D \xrightarrow \sim D'$ are all in $\mathcal W$, then so is the map $A \to A'$.

$\require{AMScd} \begin{CD} A @>>> B\\ @VVV @VVV\\ C @>>> D \end{CD} \qquad \Rightarrow \qquad\begin{CD} A' @>>> B'\\ @VVV @VVV\\ C' @>>> D' \end{CD}$

By pullback-stability and 2/3, the map $B \times_{D'} C \xrightarrow \sim A'$ is in $\mathcal W$. So by 2/3 it will suffice to show that the map $A \to B \times_{D'} C$ is in $\mathcal W$. By pullback-stability and 2/3, the map $D \xrightarrow \sim D \times_{D'} D$ is in $\mathcal W$, so it will suffice by pullback-stability to show that the following two squares are pullbacks:

$\begin{CD} A @>>> B @>>> D \\ @VVV @VVV @VV^\sim V \\ B \times_{D'} C @>>> B \times_{D'} D @>>> D \times_{D'} D \end{CD}$

This can be seen using the following two diagrams:

$\begin{CD} A @>>> B \\ @VVV @VVV \\ B \times_{D'} C @>>> B \times_{D'} D @>>> B \\ @VVV @VVV @VVV \\ C @>>> D @>>> D' \end{CD} \quad \begin{CD} B @>>> D \\ @VVV @VVV \\ B \times_{D'} D @>>> D \times_{D'} D @>>> D \\ @VVV @VVV @VVV \\ B @>>> D @>>> D' \end{CD} $

In each case we argue that the bottom-right square and the composite of the lower two squares is a pullback, so the bottom-left square is a pullback. Since the composite of the left two squares is also a pullback, it results that the top-left square is a pullback.

This is my bibliography on the subject. Indeed Bourceux was quite relevant in this topic.

• Borceux, Sheaves of algebras for a commutative theory, Ann. Soc. Sci. Bruxelles Sér. I 95 (1981), no. 1, 3–19
• Borceux and Kelly, On locales of localizations, J. Pure Appl. Algebra, Volume 46, Issue 1, 1987, Pages 1-34.
• Borceux and Veit, On the Left Exactness of Orthogonal Reflections J. Pure Appl. Algebra, 49 (1987), pp. 33-42.
• Borceux, Subobject Classifier for Algebraic Structures. Subobject classifier for algebraic structures J. Algebra, 112 (1988), pp. 306-314.
• Veit, Sheaves, localizations, and unstable extensions: Some counterexamples. J. Pure Appl. Algebra, Volume 140, Issue 2, July 1991, Pages 370-391.
• Borceux and Quinteiro. A theory of enriched sheaves. Cahiers de Topologie et Géométrie Différentielle Catégoriques 37.2 (1996): 145-162.
• Garner and Lack, Lex Colimits. J. Pure Appl. Algebra. Volume 216, Issue 6, June 2012, Pages 1372-1396.