Dear Andrej, I can offer a couple of thoughts, one simple and the other deeper but speculative. Note Andre's remark: it's not bounded linear orders that are classified by simplicial sets, but distinctly bounded linear orders (i.e. the bounds are distinct). The simple thought is that bounded linear orders are exactly lattices (necessarily distributive) satisfying the axiom x \/ y = x or x \/ y = y This gives a geometric theory Morita equivalent to that of bounded (distinctly bounded, if necessary) linear orders. The reason that's not completely uninteresting is because of the way the simplicial sets classify. You would expect (distinctly) bounded linear orders to be classified by the topos of _co_variant functors from the category of finite (distinctly) bounded linear orders to Set, and in fact this is the case. (Finite here = isomorphic to a finite cardinal.) But the category of finite distinctly bounded linear orders is dual to the simplicial category Delta, which gives the desired result. The duality is essentially a restriction of that between finite posets and finite distributive lattices, but the distributive lattices in this case are linear. (I discussed this issue in my paper "Strongly algebraic = SFP (topically)", where I developed some sufficient conditions for a classifying topos to be a presheaf topos, in situations like this where the finitely presentable models are finite.) Here are the deeper speculations. There's a series of dualities between discrete structures and Stone structures. (1) Discrete Boolean algebras dual to Stone objects (from the original Stone duality) (2) Discrete distributive lattices dual to Stone posets (from Priestley duality) (3) Discrete semilattices dual to Stone semilattices (4) Discrete posets dual to Stone distributive lattices (5) Discrete objects (sets) dual to Stone Boolean algebras These are all in "Stone Spaces" - sections II.4 and VI.3. These dualities become equivalences if you replace the structured Stone objects on the right by costructured Boolean algebras. I would conjecture that one can write down geometric theories for the costructured Boolean algebras on the right, Morita equivalent to those for the structured sets on the left. In fact, the last time I saw Japie Vermeulen he said he thought it was known, but he didn't show me any proof then and I haven't seen any since. (The context was that we were discussing the idea of representing a topos not, in the usual manner, by its category of discrete spaces - sheaves - but by its category of Stone spaces - dual to sheaves of Boolean algebras. A suitable constructive version of duality (5) would show that the sheaves can be recovered from the Stone space.) Since the distinctly bounded linear orders fit within the left hand side of the series of dualities, the conjecture would suggest there's a Morita equivalent theory of costructured Boolean algebras. All the best, Steve. On 31 Jul 2010, at 08:55, Andrej Bauer wrote:
The presheaf category of simplicial sets is the classifying topos for the theory L of a bounded linear order.
In general, there could be other theories which are "Morita equivalent" to L in the sense that their classifying toposes are equivalent to simplicial sets. Are any such known, preferably occurring in nature?
With kind regards,
Andrej
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