Hyperimaginaries and Exactness of the Pro-Completion
Given a complete first-order theory $T$, we characterise elimination of hyperimaginaries in $T$ in terms of the exactness of the pro-completion $\textrm{Pro}(\textrm{def}(T))$ of the syntactic category $\textrm{def}(T)$ of $T$, thus extending Makkai's result connecting elimination of imaginaries with the exactness of $\textrm{def}(T)$. Likewise, we characterise the heq construction in terms of the exact completion of $\textrm{Pro}(\textrm{def}(T))$.
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