next up previous
Next: What about triples? Up: Partial constructions Previous: Inference system interpretation

   
Topological interpretation

Tuples admit a topological interpretation:

How do morphisms come out?

Enriched with the following extra structure, they are covering systems, in the sense of MacLane and Meordijk (``Sheaves in Geometry and Logic'', pp. 524 - 527. This extra structure is a partial order $\leq$ representing inclusion between neighbourhoods. (This can be represented in various ways: see section 7.)

One can now define a notion of `point'. A point is what is common to a certain kind of set of neighbourhoods, satisfying two conditions. The first condition is that the set should be a filter: the set of neighbourhoods should be directed to the left and closed to the right (monotone) in $\leq$. (This means that for any pair of neighbourhoods there is another included in them both, and anything that includes a neighbourhood in the set is also in the set.) The other condition expresses the idea that a filter contains arbitrarily small neighbourhoods, and so `identifies a point'. The condition is that for any tree of basic coverings of any neighbourhood in the filter, the set of neighbourhoods at the leaves of that tree has non-empty intersection with the filter.

\begin{displaymath}\forall a : A, f : F(a), b : B^\ast(a) .
\exists c : C^\ast(a,b) . F(d^\ast(a,b,c))
\end{displaymath}

Warning: the following is probably erroneous. I am no longer sure one can identify points with functions $f : N \longrightarrow A$. For instance, if one takes A with two elements $a \leq 1$, and as basic coverings, only the singletons $\{ a \}$ covering a, and $\{ 1 \}$ covering 1, then a point is determined by the proposition which says it has a as an element. Hence the collection of points is like the collection of all propositions, and hence cannot be a set.

One may (at least sometimes) identify points with certain countable sequences $f : \mbox{$\mathit{Nat}$ }
\longrightarrow A$ of neighbourhoods, namely those satisfying the following conditions:


next up previous
Next: What about triples? Up: Partial constructions Previous: Inference system interpretation
Peter Hancock
1998-04-08