One of the classical theorems of model theory is the Chang-Łos-Suszko preservation theorem that states that the theories formulated in FOL (first order logic) that are preserved under direct limits (generalized unions) are precisely those theories that are axiomatizable by a collection of $\Pi_2$-sentences, i.e., those of the form $\forall x_1, ...,\forall x_m \exists y_1,...,\exists y_n ~\delta$, where $\delta$ is quantifier-free (see, e.g., these slides of Benno van den Berg).

The above motivates the following question concerning inverse limits (of inverse systems). NB: category theorists commonly refer to inverse limits as limits and direct limits as colimits.

Question. Is there a characterization of FOL theories that are preserved under inverse limits?

One of the classical theorems of model theory is the Chang-Łos-Suszko preservation theorem that states that the theories formulated in FOL (first order logic) that are preserved under direct limits (generalized unions) are precisely those theories that are axiomatizable by a collection of $\Pi_2$-sentences, i.e., those of the form $\forall x_1 ...\forall x_m \exists y_1...\exists y_n ~\delta$, where $\delta$ is quantifier-free (see, e.g., these slides of Benno van den Berg).

The above motivates the following question concerning inverse limits (of inverse systems). NB: category theorists commonly refer to inverse limits as limits and direct limits as colimits.

Question. Is there a characterization of FOL theories that are preserved under inverse limits?

One of the classical theorems of model theory is the Chang-Łos-Suszko preservation theorem that states that the theories formulated in FOL (first order logic) that are preserved under direct limits (generalized unions) are precisely those theories that are axiomatizable by a collection of $\Pi_2$-sentences, i.e., those of the form $\forall x_1, ...,\forall x_m \exists y_1,...,\exists y_n ~\delta$, where $\delta$ is quantifier-free (see, e.g., these slides of Benno van den Berg).

The above motivates the following question concerning inverse limits (NB: category theorists commonly refer to inverse limits as limits and direct limits as colimits).

Question. Is there a characterization of FOL theories that are preserved under inverse limits?

One of the classical theorems of model theory is the Chang-Łos-Suszko preservation theorem that states that the theories formulated in FOL (first order logic) that are preserved under direct limits (generalized unions) are precisely those theories that are axiomatizable by a collection of $\Pi_2$-sentences, i.e., those of the form $\forall x_1, ...,\forall x_m \exists y_1,...,\exists y_n ~\delta$, where $\delta$ is quantifier-free (see, e.g., these slides of Benno van den Berg).

The above motivates the following question concerning inverse limits (of inverse systems). NB: category theorists commonly refer to inverse limits as limits and direct limits as colimits.

Question. Is there a characterization of FOL theories that are preserved under inverse limits?