nLab unbounded topos

Idea

Most geometric morphisms between toposes are bounded, and so most toposes can be considered as either a base topos - the universe of discourse - such as Set, or as a topos of internal sheaves over another topos (for example any Grothendieck topos).

While there are lots of examples of toposes which aren’t Grothendieck toposes, that is, bounded toposes over $Set$, for example the topos FinSet of finite sets, or realisability toposes, this is due to the fact they do not admit a geometric morphism to $Set$. Toposes of this form usually can be considered as a base topos/universe of discourse for non-standard versions of reasoning, for example finitism can be seen as taking the topos $FinSet$ as base topos.

An unbounded topos is a non-Grothendieck topos which does admit a geometric morphism to $Set$ (or some other specified base topos), but which isn’t equivalent to the category of sheaves on some small site.

Definition

A topos $\mathcal{E}$ equipped with a geometric morphism $p\colon \mathcal{E} \to Set$ is unbounded if $p$ is not bounded. Equivalently, $\mathcal{E}$ (equipped with $p$) is not equivalent to the category of sheaves on some small site (equipped with the usual geometric morphism to $Set$).

This means that for any set $\{A_i\}_{i\in I}$ of objects of $\mathcal{E}$, which would aspire to be a separating family for $\mathcal{E}$, there is a pair of morphisms $f,g\colon X \to Y$ of $\mathcal{E}$ such that for all $i\in I$, and all $t\colon A_i \to X$, we have $f\circ t = g\circ t$, but $f\neq g$.

More generally, we could work over a given base topos $\mathcal{S}$, and then an $\mathcal{S}$-topos $p\colon \mathcal{E} \to \mathcal{S}$ is unbounded if $p$ is not a bounded geometric morphism or equivalently it is not equivalent to the category of sheaves on an internal site in $\mathcal{S}$.

Examples

There are relatively few examples of unbounded toposes.

• Given a non-locally small groupoid $G$ with only a set of isomorphism classes of objects, the functor category $Cat(G,Set) =: GSet$ is a topos. It has a geometric morphism to $Set$, namely the global sections functor $\Gamma(X) = GSet(1,X)$, which is unbounded. $GSet$ is moreover cocomplete, Boolean and even locally small.

• If $K$ is a topological group, the category $Unif(K)$ of sets with a uniformly continuous $K$-action is a $Set$-topos. In the case that $K$ has no smallest open subgroup, then $Unif(K)$ is still Boolean and locally small, but is not cocomplete (it fails to have infinite coproducts), and so not a Grothendieck topos.

• The topos of coalgebras of a pullback-preserving comonad $M$ on a Grothendieck topos whose functor part is not accessible is a cocomplete topos $MCoalg$ over $Set$ which is not locally presentable hence not a Grothendieck topos.

  • As a particular example, one can take a left exact endofunctor $F$ on $Set$ and form the corresponding comonad $(X,Y) \mapsto (X,Y\times F(X))$ on $Set\times Set$ and the topos of coalgebras for this (which is equivalent to the Artin gluing $Gl(F)$ of $F$). In this case however it is not clear that there exist such endofunctors without assuming the existence of large cardinals (for instance the existence of a proper class of measurable cardinals is sufficient to give such an endofunctor).

    (DR: I think this result has since been improved, see the paper J. Adamek, V. Koubek and V. Trnkova, “How large are left exact functors?” in TAC. I will have to sort out whether what they are saying applies)

• Any topos that is not locally small, relative to some fixed topos $set$ of sets, when such a setting is properly defined, is not bounded over $set$.

• More coming soon!

Related topics

• internal sheaves
• internal site
• bounded geometric morphism

References

The unbounded toposes $GSet$, $Unif(K)$ and $MCoalg$ are mentioned in B3.1.4 of

• Peter Johnstone, Sketches of an Elephant

as being (at the time) essentially the only examples of unbounded toposes.