nLab
unbounded topos

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Topos Theory

topos theory

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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 SetSet, 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 SetSet. 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 FinSetFinSet as base topos.

An unbounded topos is a non-Grothendieck topos which does admit a geometric morphism to SetSet (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:Setp\colon \mathcal{E} \to Set is unbounded if pp is not bounded. Equivalently, \mathcal{E} (equipped with pp) is not equivalent to the category of sheaves on some small site (equipped with the usual geometric morphism to SetSet).

This means that for any set {A i} iI\{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:XYf,g\colon X \to Y of \mathcal{E} such that for all iIi\in I, and all t:A iXt\colon A_i \to X, we have ft=gtf\circ t = g\circ t, but fgf\neq g.

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

Examples

There are relatively few examples of unbounded toposes.

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

  • If KK is a topological group, the category Unif(K)Unif(K) of sets with a uniformly continuous KK-action is a SetSet-topos. In the case that KK has no smallest open subgroup, then Unif(K)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 MM on a Grothendieck topos whose functor part is not accessible is a cocomplete topos MCoalgMCoalg over SetSet which is not locally presentable hence not a Grothendieck topos.

    • As a particular example, one can take a left exact endofunctor FF on SetSet and form the corresponding comonad (X,Y)(X,Y×F(X))(X,Y) \mapsto (X,Y\times F(X)) on Set×SetSet\times Set and the topos of coalgebras for this (which is equivalent to the Artin gluing Gl(F)Gl(F) of FF). 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 setset of sets, when such a setting is properly defined, is not bounded over setset.

  • More coming soon!

References

The unbounded toposes GSetGSet, Unif(K)Unif(K) and MCoalgMCoalg are mentioned in B3.1.4 of

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

Last revised on November 2, 2015 at 20:27:43. See the history of this page for a list of all contributions to it.