4

The duality between the category Sets and the category CABool of complete atomic boolean algebras is an example of a general Stone-type duality. At the same time, Sets is a topos. Are there other known examples of Stone-type dualities involving toposes ? On the algebraic side of such a duality, I guess the "completeness" property is essential since its corresponds (on the topological side) to the property of being extremally-disconnected, something we would expect for the objects of a topos, i.e. behaving like "sets".

Sephi
  • 357
  • 3
    For any topos $\mathcal{E}$, the power-object functor $\mathcal{E}^\mathrm{op} \to \mathcal{E}$ is monadic. This is a direct generalisation of the equivalence between $\mathbf{Set}^\mathrm{op}$ and $\mathbf{CABA}$. – Zhen Lin Apr 02 '14 at 16:45
  • I know that fact, however I don't clearly see how the algebras for the induced monad correspond to complete atomic boolean algebras. That's why I thought of using Stone-type dualities to find other categories of lattice-like structures that could be dual to toposes. – Sephi Apr 03 '14 at 12:35
  • You can think of the algebras as some kind of "internal complete atomic Heyting algebra", whatever that might mean. They are certainly internal lattices and even internal frames. – Zhen Lin Apr 03 '14 at 12:37
  • 2
    @Zhen Lin: You can interpret the phrase "complete atomic Heyting algebra" just fine in the internal language of $\mathcal{E}$, and with this interpretation the equivalence holds. (Use an intuitionistically useful version of "atomic", as detailled at the nLab. If you want to define completeness by mandating that "for any set I, any I-indexed family of elements possesses a supremum" instead of mandating that "any subset possesses a supremum", you can do so using Mike Shulman's stack semantics. The two definitions turn out to be equivalent.) – Ingo Blechschmidt Apr 03 '14 at 19:03
  • @Ingo Blechschmidt: Do you mean that a complete atomic boolean algebra is, in the internal language of Sets, just a complete atomic Heyting algebra ? I mean, is the boolean character detected internally ? Besides, could you give more details on the "intuitionistically useful version of "atomic"" ? – Sephi Apr 04 '14 at 08:15
  • 2
    @Sephi: No: You can define an internal notion of a "complete atomic Boolean algebra", but the category of those objects will not be equivalent to $\mathcal{E}^\mathrm{op}$. (This is because power objects are not internal Boolean algebras, but internal Heyting algebras.) The correct definition of atomic to use it at the nLab article on atoms. It really is the usual definition, just phrased in a constructively appropriate way. – Ingo Blechschmidt Apr 04 '14 at 11:15
  • 1
    @Sephi: The internal language of $\mathbf{Set}$ coincides with the usual mathematical language. In particular, $\mathbf{Set}$ is a Boolean topos if and only if your metalogic is classical. By the way: To prove the equivalence of $\mathcal{E}^\mathrm{op}$ with the category of internal complete atomic Heyting algebras, you just have to repeat the usual constructive proof of the equivalence $\mathbf{Set}^\mathrm{op} \simeq \mathbf{CAHA}$ (mind the "H"!) in the internal language of $\mathcal{E}$. – Ingo Blechschmidt Apr 04 '14 at 11:18

1 Answers1

2

An atom in an internally complete Heyting algebra in an elementary topos is an element (in the Kripke-Joyal semantics) for which the downward segment is isomorphic to the subobject classifier. The details on this and how to establish the characterization of P(X) as being a complete atomic Heyting algebra is (theorem 6) in the second chapter of “Lattice Theoretic and Logical Aspects of Elementary Topoi”, Christian Juul Mikkelsen, March 1976. Various Publications Series No. 25, Matematisk Institut, Aarhus Universitet.

Chris
  • 21