When is an elementary subclass reflective?

Question

Consider an elementary class, $K$, of some $\mathcal{L}$-theory, $T$ equipped with the usual $\mathcal{L}$-structure homomorphisms. (Not elementary embeddings, which elementary classes are more frequently equipped with.) Suppose we have $K' \subseteq K$, the elementary class of models of $T'$. When is $K'$ a reflective subcategory of $K$, i.e. when does the inclusion functor $\iota:K'\to K$ admit a left adjoint functor?

For example, I was considering monoids as a subcategory of groups. Here the theory of groups is given by $T' = T\cup \{\forall x \exists y\;xy=1\}$, where T is the theory of monoids. As far as I can tell though, the similarly structured sentence $\forall x\exists y\; x\sim y$ does not form a reflective subcategory for graphs, so presumably this cannot be characterized in terms of quantifier complexity, i.e. just being $\Pi_n$ or $\Sigma_n$.

Answer 1

This is a great question, which I will only partially address. A complete, general, answer to the question goes beyond the energy I am happy to put into MathOverflow.

Def. Let $T \subset T'$ be first order theories (having models with at least two elements and quotients of definable equivalence relations), so that the forgetful functor $U: \text{Mod}(T') \to \text{Mod}(T)$ yields the right adjoint of a reflection $L \dashv U$. I will say that $L$ is reasonable if it preserves ultraproducts. I will call a reasonable reflection the reflections of this kind.

Prop. Every reasonable reflection is induced adding formulas of the following form $$\forall \bar x \phi(\bar x) \Rightarrow \exists ! \bar y \psi(\bar y) \wedge \bar y = f(\bar x).$$

Sketch of proof. Consider the canonical structure of ultracategory (à la Makkai) on the categories of models. Recall that by studying their associated pretopos of coordinates, and because we assumed it has models with at least two elements and quotients of definable equivalence relations, we obtain the following reconstruction result dues to conceptual completeness. $$ \text{Ult}(\text{Mod}(T), \text{Set}) \simeq \text{Syn}(T).$$

Thus, the reasonable (!) reflection at the level of categories of models, induced a reflection at the level of syntactic categories. Reflections correspond to orthogonality formulas à la Adamek-Rosicky (see Rem 5.6 in Locally presentable and accessible categories).