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I feel that something deserves to be said about locally connected toposes regarded as classifying toposes. But I am not yet sure exactly what. Does anyone have an idea?
So if a topos $\mathcal{T}$ happens to be locally connected and at the same time we happen to regard it as a classifying topos for a geometric theory, then we should want to consider some additional properties or structrures on the algebras that it classifies.
For instance we might want to restrict attention to the algebras internal to other locally connected toposes such that these are classified by locally connected classifying geometric morphisms. Hence to consider classification not in the 2-category of toposes, but of locally connected toposes and locally connected geometric morphisms.
If so, that would mean to consider our topos as the classifying topos not just for a geometric theory, but for an infinitary first order theory. That is obvious enough, but is this also an interesting thing to say?
That’s an interesting question (sorry I’ve taken so long to reply to it). By general nonsense, any type of “theory” does have a classifying (= syntactic) category in the 2-category of categories in whatever doctrine it was expressed in – which for geometric theories we can take to be infinitary-pretoposes (with cocontinuous lex functors). By that approach, I think an infinitary first-order theory should have a classifying category in the 2-category of Heyting infinitary-pretoposes which are “locally-complete” in the sense of having all small intersections of subobjects (with cocontinuous Heyting functors that also preserve such intersections). Possibly there are some size issues that would have to be finessed.
Of course, any Grothendieck topos is a locally-complete Heyting infinitary-pretopos, as is any well-powered infinitary-pretopos by the adjoint functor theorem for posets. And any locally connected inverse-image functor is a locally-continuous cocontinuous Heyting functor. So we can regard any topos, whether or not it is itself locally connected (over Set), as classifying some infinitary first-order theory among locally-complete Heyting infinitary-pretoposes (with respect to locally-continuous cocontinuous Heyting functors), among which we include toposes.
I see two questions to ask, then, which I don’t know the answers to:
I presume that not every locally-complete cocontinuous Heyting functor between toposes is a locally connected inverse-image functor. So a topos which classifies a certain infinitary first-order theory, in the above sense, seemingly doesn’t necessarily classify the same theory with respect to locally-connected geometric morphisms out of other topoi. Maybe we need to allow the infinitary first-order theories to also include operations of infinite arity (so that the functors will be globally continuous and hence have left adjoints) and dependent types (to get the Beck-Chevalley condition for $\Pi$-functors and not just for $\forall$-functors).
If a given topos classifies some geometric theory as a topos, and also classifies some infinitary first-order theory as a locally-complete Heyting infinitary-pretopos, how are the two theories related?
Thanks for the reply. Interesting to hear that these questions are open.
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