Anhinga anhinga playground

Let's look at the bijection mentioned in the previous post more closely. A model is "a map from the syntactic theory to a semantic space". In particular, given a Lindenbaum algebra L(T) we can build a "generic model" of T called M_T in L(T) (by mapping the propositional variables to their classes of equivalences in L(T) ).

Now, for a homomorphism f : L(T) → A, the corresponding model of the theory T in A is a map f o M_T (mapping the propositional variables to the elements of A).

Now, if we look at the post of June 9, we'll see a duality between generalized points and models. "Generic point" and "generic model" correspond to each other, and global points 1 → X correspond to standard models in the 2-valued Boolean algebra 2 = {false, true}, that is to the homomorphisms L(T) → 2.

One of the main ideas in the Vickers' text is that one can think about all kinds of spaces as spaces of models (one takes a space and starts thinking about its points as models of a logical theory of an appropriate type, usually more complex than classical propositional logic), and that in a large variety of logical situations one can take all models and form a space from them.

I've completed the first reading of Steve Vickers' rather brilliant text, "Locales and toposes as spaces", from here

http://www.cs.bham.ac.uk/~sjv/

The problem with all these categorical games is that one can understand every neat trick separately, but there are just too many of them, and it's difficult to hold the resulting picture together. So I think I'll try to write them down, one trick at a time, and this might help me.

Trick of the day: points as arrows. If a category has a terminal object 1, then (global) points of object X are (defined as) arrows 1 → X. However, in many cases there are not enough global points, so for any object A people define "points of X at stage A" as arrows A → X. In particular, the "generic point" of X is simply the identity arrow, id : X → X.