Reconstructing the canonical extension in the stable setting

Since the category of models does not determine a first-order theory up to any typical notion of equivalence, e.g. Morita equivalence, it has been a persistent theme in categorical logic to identify what extra structure is needed to obtain a “reconstruction theorem”. Examples in the literature include ultrastructure à la Makkai [6], or the data of a topological category [1, 8].

In this talk, we will discuss a reconstruction problem for Coumans’ first-order canonical extension [2], which has seen a recent renewal of interest, for instance in [3] (alongside a resurgence in reconstruction theorems more generally, e.g. [7]). Coumans’ construction takes the coherent doctrine $D_{𝕋}$ associated with a theory $𝕋$, à la Lawvere [4], and applies the canonical extension ${(-)}^{\delta }$ from lattice theory; this can be understood as introducing a new predicate for each (model-theoretic) type of $𝕋$. Indeed, if $𝕋$ is a classical theory (and assuming the axiom of choice), $D_{𝕋}^{\delta }$ consists of the powerset of types in each context.

We will present the following:

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  Firstly, we give a streamlined proof that the complete points of the canonical extension $D_{𝕋}^{\delta }$ correspond precisely to the countably saturated models of $𝕋$ (which is only alluded to implicitly in previous literature, cf. [5, §1.4]);

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  Secondly, we show that if $𝕋$ is countably stable and in a countable language, the canonical extension $D_{𝕋}^{\delta }$ can be reconstructed from merely knowing the underlying sets of the countable saturated models; in other words, given countably stable theories ${𝕋}_1,{𝕋}_2$, there is a natural isomorphism $D_{{𝕋}_1}^{\delta }\cong D_{{𝕋}_2}^{\delta }$ if and only if the countable saturated models of ${𝕋}_1$ and ${𝕋}_2$ are equivalent as categories over $\mathrm{𝐒𝐞𝐭}$.

The above results will be the subject of a submission to a forthcoming volume of the Outstanding Contributions to Logic series in honour of Hilary Priestley.