nLab conceptual completeness

Contents

model theory

Dimension, ranks, forking

• forking and dividing?

• Morley rank?

• Shelah 2-rank?

• Lascar U-rank?

• Vapnik–Chervonenkis dimension?

Contents

Idea

In any context it is interesting to ask when a contravariant hom-functor $\operatorname{Hom}(-,X)$ reflects isomorphisms, i.e. if homming into an object is enough to distinguish isomorphism classes. In a $2$-categorical setting, the analogous question is if homming into an object reflects equivalences.

Definition

Conceptual completeness refers to the result of Makkai and Reyes that in the $2$-category $\mathbf{Pretop}$ of pretoposes, homming into the pretopos Set $\operatorname{Hom}_{\mathbf{Pretop}}(-,\mathbf{Set}) : \mathbf{Pretop} \to \mathbf{Cat}$ reflects equivalences.

That is, if $f : T_1 \to T_2$ is a pretopos morphism and $- \circ f : \operatorname{Hom}_{\mathbf{Pretop}}(T_2, \mathbf{Set}) \to \operatorname{Hom}_{\mathbf{Pretop}}(T_1,\mathbf{Set})$ is an equivalence, then $f$ was also. Pretopos morphisms are just interpretations of theories; a pretopos morphism into Set is an interpretion of the source theory into the internal logic of Set, and is therefore a model.

Since pretoposes are just effectivizations of syntactic categories of first-order theories, this means: if a functor between the categories of models of $T_1$ and $T_2$ is an equivalence and is induced by an interpretation $T_1 \to T_2$, then that interpretation must be part of a bi-interpretation, i.e. $T_1$ and $T_2$ are equivalent as syntactic categories via pretopos morphisms.

Makkai duality states that $\operatorname{Hom}_{\mathbf{Pretop}}(-,X)$ factors through the $2$-category of ultracategories (categories equipped with ultraproduct functors?) (which also contains $\mathbf{Set}$) and that $\operatorname{Hom}_{\mathbf{Pretop}}(-,\mathbf{Set})$ is left-adjoint to $\mathbf{Hom}_{\mathbf{Ult}}(-,\mathbf{Set})$.

Furthermore, the unit of this adjunction is an equivalence, so that a pretopos $T$ is equivalent to the category of ultrafunctors (ultraproduct-preserving functors) from $\mathbf{Mod}(T)$ to $\mathbf{Set}$, i.e. any ultrafunctor $\mathbf{Mod}(T) \to \mathbf{Set}$ is induced by taking points in models of some definable set $X \in T$.

This means: when viewed as a functor to the $2$-category of ultracategories instead, $\operatorname{Hom}_{\mathbf{Pretop}}(-,\mathbf{Set})$ creates equivalences also, so that the pretopos/theory $T$ can be reconstructed up to bi-interpretability from its ultracategory of models. This is what is known as strong conceptual completeness.

Remarks

In his AMS monograph on duality and definability in first-order logic, Makkai refined the above reconstruction result to work with just the (ultra) core of the ultracategory of models of $T$. Awodey and his students replace the ultracategory structure on this groupoid with a related topology instead; this is the “spectral groupoid” which forms the basis of the logical scheme approach.

Examples

(Maybe I’ll add some explicit computations later.)

References

An approach which reframes conceptual completeness in terms of logical schemes is adopted in section 4.4 of

For the $(\infty, 1)$-analog of conceptual completeness, see section A.9 of

Last revised on January 8, 2019 at 08:52:52. See the history of this page for a list of all contributions to it.