A left-exact localization of a category is a reflective subcategory such that the reflector preserves finite limits. There are several prominent examples of such localizations, such as sheafification, and localization of module categories. Is there a general theory of such localizations?
I don't have any particular type of result in mind, but given the prominence of the two examples I mentioned, it seems like the topic of left-exact localizations must have been studied for its own sake.
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. 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.
