indiscernible sequence?
Morley sequence?
Ramsey theorem?
Erdos-Rado theorem?
Ehrenfeucht-Fraïssé games (back-and-forth games)
Hrushovski construction?
generic predicate?
In any context it is interesting to ask when a contravariant hom-functor reflects isomorphisms, i.e. if homming into an object is enough to distinguish isomorphism classes. In a -categorical setting, the analogous question is if homming into an object reflects equivalences.
Conceptual completeness refers to the result of Makkai and Reyes that in the -category of pretoposes, the hom-2-functor reflects equivalences. More explicitly:
[Conceptual completeness] Let be a pretopos morphism. If the functor
is an equivalence of categories, then so is .
From a logical point of view, recall that pretoposes can be conceived as syntactic categories of coherent theories (up to elimination of imaginaries), so that a pretopos morphism corresponds to an interpretation of the theory into the theory , or equivalently to a model of the theory inside the category ; in particular, a pretopos morphism into is therefore a model of the source theory. In these terms, conceptual completeness states that if an interpretation between two coherent theories induces an equivalence between the categories of models , then and are bi-interpretable up to elimination of imaginaries, i.e. and are bi-interpretable.
Makkai duality states that factors through the -category of ultracategories (categories equipped with ultraproduct functors?) (which also contains ) and that is left-adjoint to .
Furthermore, the unit of this adjunction is an equivalence, so that a pretopos is equivalent to the category of ultrafunctors (ultraproduct-preserving functors) from to , i.e. any ultrafunctor is induced by taking points in models of some definable set .
This means: when viewed as a functor to the -category of ultracategories instead, creates equivalences also, so that the pretopos/theory can be reconstructed up to bi-interpretability from its ultracategory of models. This is what is known as strong conceptual completeness.
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 . 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.
(Maybe I’ll add some explicit computations later.)
Michael Makkai and Gonzalo Reyes, First order categorical logic: Model-theoretical methods in the theory of topoi and related categories, Springer-Verlag, 1977.
Michael Makkai, Strong Conceptual Completeness for First-Order Logic , APAL 40 (1988) pp.167-215. (freely available online)
Peter Johnstone, Sketches of an Elephant vol. II , Oxford UP 2002. (sec. D3.5, pp.931-939)
Jacob Lurie, Ultracategories, (pdf)
An approach which reframes conceptual completeness in terms of logical schemes is adopted in section 4.4 of
For the -analog of conceptual completeness, see section A.9 of
Last revised on July 17, 2025 at 09:05:37. See the history of this page for a list of all contributions to it.