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This is a quote from Mac Lane and Moerdijk's Sheaves in Geometry and Logic:

A classifying topos for $T$-models is then a topos $\mathcal B(T)$ over $\mathbf{Sets}$ with the property that for every cocomplete topos $\mathcal E$ there is an equivalence of categories $$c_\mathcal E\colon\mathrm{Mod}(\mathcal E, T)\to \mathrm{Hom}(\mathcal E, \mathcal B(T))$$ which is natural in $\mathcal E$.

Why do they say cocomplete topos and not Grothendieck topos?

It is well-known that a Grothendieck topos is the same as a cocomplete elementary topos with a small generating set. So it seems they want to emphasize that they don't assume the small generating set. But then, why do they need the axioms for an elementary topos? The subobject classifier isn't needed when interpreting geometric theories $T$.

Note that a category satisfying all Giraud axioms except the small generating set need not have a subobject classifier, i.e., need not be an elementary topos!

When working with classifying topoi I think there are two choices:

  1. One allows $T$ to be large. Then the "classifying categories" one considers should be categories satisfying all Giraud axioms except the small generating set.

  2. One assumes $T$ to be small. In this case the classifying category has a small set of generators, hence it is elementary by accident.

In both cases one doesn't get exactly cocomplete elementary toposes!

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You make good points, and I am inclined to agree that the definition is strange. But it works, because if you have a classifying topos that works for all infinitary pretoposes then it also works for all cocomplete toposes in particular. If I recall correctly the concept of pretopos is not used in MacLane–Moerdijk, so maybe they say "cocomplete topos" instead as a compromise.

Incidentally, I believe the classifying toposes that arise in practice do work for all infinitary pretoposes. One way of justifying this is to observe that every small set of objects in an infinitary pretopos is contained in a (not necessarily full) subcategory that is a Grothendieck topos and such that the inclusion preserves finite limits, small colimits, and is conservative. Every "inverse image functor" from a Grothendieck topos will factor through such a subcategory (but the subcategory will depend on the functor), so the category of "inverse image functors" will be a union of categories of geometric morphisms between Grothendieck toposes. Similarly, every model of the theory you want to classify will also be contained in such a subcategory, so the category of models will be a union of categories of models in Grothendieck toposes.

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  • $\begingroup$ Thanks! "But it works, because if you have a classifying topos that works for all infinitary pretoposes then it also works for all cocomplete toposes in particular." What about the converse: suppose that for all cocomplete elementary toposes $\mathcal E$, $\mathrm{Mod}(\mathcal E, T)\cong \mathrm{Hom}(\mathcal E, \mathcal F)$ (natural in $\mathcal E$). Why does it follow that for all infinitary pretoposes $\mathcal E$, $\mathrm{Mod}(\mathcal E, T)\cong \mathrm{Hom}(\mathcal E, \mathcal F)$ (natural in $\mathcal E$)? Here, $\mathcal F$ is a candidate for the classifying topos. $\endgroup$ Feb 5 at 16:24
  • $\begingroup$ I gave a heuristic explanation in the second paragraph. Have you read it? $\endgroup$
    – Zhen Lin
    Feb 5 at 16:26
  • $\begingroup$ I am struggling to understand it. Also, you talk about toposes that "arise in practice". My question is about the general case. $\endgroup$ Feb 5 at 16:27
  • $\begingroup$ The classifying toposes that arise in practice classify models of geometric theories that are generated by a small signature and a small set of axioms. Every Grothendieck topos can be considered a classifying topos of this form, so it is general enough in my opinion. $\endgroup$
    – Zhen Lin
    Feb 5 at 16:35

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