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Recall that an elementary topos is a cartesian closed category with finite limits and a subobject classifier. A Grothendieck topos is a category equivalent to the category of sheaves on a site.

Are there examples of (co)complete elementary topoi that are not Grothendieck? On the "Cocomplete" side of things, such a category cannot be accessible, since a locally presentable elementary topos is automatically Grothendieck.

Karthik Yegnesh
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A classical example is $G$-$Set$ for a large group $G$. That this is a cocomplete elementary topos is not hard to see. Limits and colimits are formed at the underlying set level, and exponentials $Y^X$ are formed as usual as the set of functions $f: X \to Y$ with the $G$-action $(g, f) \mapsto g f$ defined by $g f: x \mapsto g f(g^{-1} x)$. The subobject classifier is the 2-element set with trivial $G$-action (again, as usual). Thus $E = G$-$Set$ is a cocomplete (elementary) topos.

It remains to see $E$ is not Grothendieck. If it were, then every continuous functor such as the underlying-set functor $U: E \to Set$ would have a left adjoint $F$, by the special adjoint functor theorem (the hypotheses for the SAFT are satisfied: a Grothendieck $E$ is well-powered and complete and has a cogenerator $\Omega^c$ where $c$ is the coproduct of associated sheaves of objects in a small site presentation). In particular, we would have $\hom_E(F(1), -) \cong U$, but the only candidate for the representing object $F(1)$ would be $G$, which is ruled out by largeness.

Edit: Adam Epstein pointed out in a comment that the intuition in the last sentence, regarding the only candidate for the representing object, is not correct for some $G$ such as a large simple group. The following addendum patches up this oversight.

Let $G$ be a large free group, say the union of the diagram of free groups $F(\alpha)$ obtained by applying the free group functor to von Neumann cardinals $\alpha$ and initial segment inclusions between them. Then the topos $Set^G$ of $G$-sets is not Grothendieck.

Supposing it is, then the underlying-set functor $U: Set^G \to Set$ is representable, by the special adjoint functor theorem. Hence $U \cong \hom(X, -)$ for some $G$-set $X$. We will show $U$ has a proper class of non-isomorphic representable subfunctors, i.e., the object $X$ has a proper class of quotients, which is impossible in a Grothendieck topos (or even in a locally small topos -- quotients of $X$ are in bijection with equivalence relations on $X$, forming a subcollection of a hom-set $[X \times X, \Omega]$).

To begin with, for the class of canonical inclusions $i_\alpha: F(\alpha) \to G$ we may uniformly exhibit retractions $r_\alpha: G \to F(\alpha)$ (retractions in the sense of group homomorphisms). We may then view $F(\alpha)$ as a $G$-set with action $G \times F_\alpha \to F_\alpha$ taking $(g, x)$ to $r_\alpha(g) \cdot x$, and the representable functor $\hom(F(\alpha), -)$ is naturally a subfunctor of $U$. Indeed, a map $f: F(\alpha) \to A$ is uniquely determined by the value $a = f(1)$ at the identity element $1 \in F(\alpha)$, as the intertwiner condition yields $f(r_\alpha(g)) = g a$, whence $f(x) = i_\alpha(x)a$ for general $x \in F(\alpha)$. Denoting the idempotent map $i_\alpha r_\alpha: G \to G$ by $p_\alpha$, the subfunctor inclusion is given componentwise by

$$\hom(F(\alpha), A) \cong \{a \in A: \forall_{g \in G}\; g a = p_\alpha(g) a\} \hookrightarrow U(A)$$

and all these subfunctors are non-isomorphic, for the simple crude reason that the $F(\alpha)$ have generally different cardinalities as sets (as soon as $\alpha$ is in the uncountable range).

Todd Trimble
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    Incidentally, this $G$-$Set$ is complete and well-powered (clearly). So the obstruction to using SAFT is solely in the lack of a cogenerator. Also, as an aside: cocomplete toposes $E$ are automatically complete (continued next comment). – Todd Trimble Dec 29 '16 at 23:44
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    For example, let's construct the product of a family ${B_i}_{i \in I}$. By cocompleteness, we have a map in $E$:

    $$p: \sum_{i \in I} B_i \to \Delta I,$$

    where $\Delta I = \sum_{i \in I} 1$, sending the summand $B_i$ to $i$ (the $i^{th}$ copy of $1$). Then

    $$\prod_{i \in I} B_i = \Pi_! p$$

    is the product, where $!$ is the unique map $\Delta I \to 1$. Indeed, maps $X \to \Pi_! p$ are in natural bijection with maps $!^\ast X \to p$ in $E/\Delta(I) \simeq E^I$ (this equivalence holds by infinite extensivity), i.e., with an $I$-indexed family of maps $X \to B_i$, as required.

     – Todd Trimble Dec 29 '16 at 23:44
  • Interesting! The converse is also true right? It seems to follow immediately from Pare's theorem. – Karthik Yegnesh Dec 30 '16 at 02:53
  • Karthik, yes that's quite right; well-observed. – Todd Trimble Dec 30 '16 at 05:12
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    Another example is the topos obtained by gluing two Grothendieck toposes along an inaccessible left exact functor. Unfortunately one can't make this very explicit, since such functors only exist under hypotheses involving large cardinals. – Mike Shulman Dec 30 '16 at 11:34
  • @MikeShulman Nevertheless, that's pretty interesting! I don't think I would have come up with that one. – Todd Trimble Dec 30 '16 at 14:40
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    @ToddTrimble there's somewhere in the Elephant that he lists known examples of unbounded geometric morphisms to Set, which include all cocomplete locally small toposes. One is an inaccessible gluing, one is actions of a large group (or more generally presheaves on a large category with essentially small slices), and one is uniformly continuous actions of a topological group (which is not cocomplete). – Mike Shulman Dec 31 '16 at 21:27
  • @MikeShulman Lucky for me, I asked my wife for the Elephant recently (as a belated Christmas gift), and so I'll be able to share in that wisdom soon. :-) – Todd Trimble Dec 31 '16 at 22:04
  • @ToddTrimble Hooray! Sorry for not being more explicit; my copy of the Elephant is at my office, and I'm not. – Mike Shulman Jan 01 '17 at 12:01
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    Something I always wonder about in this connection is what precisely is meant by a large group. None of the cited references, including the Elephant, seem to get into the matter, though. But note that the action of a large simple group on a set is necessarily trivial. Various examples can be defined, and proved simple, well within ZF - for example, the alternating group on a proper class, PSL_3 of a large field. Perhaps this is just too well-known for comment? – Adam Epstein Jun 07 '18 at 06:48
  • @AdamEpstein I think this gets down to "what do we mean by a (proper) class?" If we work in ZFC say, then we can think of a formula with a free variable (such as $x \notin x$) as giving a class, in other words identify classes with syntactic items, and work with formulas metamathematically. One may go on to define a category of classes and class functions, and show that this category has nice properties, such as having finite products. Once we have that, we can talk about a group object in that category. – Todd Trimble Jun 07 '18 at 10:33
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    @ToddTrimble Isn't it more than that? Those examples I gave are group objects in the category of classes and class functions. Yet (by simplicity) any action by one of these large groups $G$ on a set is trivial, hence $G$-$Set$ is equivalent (indeed isomorphic) to $Set$, whence a Grothendieck topos. – Adam Epstein Jun 07 '18 at 11:07
  • @AdamEpstein Sorry; I missed what you were driving at the first time. Thanks for pointing this out. Let me patch up my answer a little later. – Todd Trimble Jun 07 '18 at 11:43