Ordered Linear Spaces I
I saw a really great talk today by Howard Barnum of the Los Alamos National Laboratory. It dovetails wonderfully with what I’ve been talking about here, and I think I can help him by bringing my categorical inclination to bear on his subjects. I’ll omit the motivation he was using because I can’t really explain the background, but it makes for a great example of a category.
We define the category of “ordered linear spaces” by starting with a totally ordered field . If you know what the real numbers are (I still haven’t defined them here) use them, but otherwise you can get away for now with rational numbers. We consider the category of finite-dimensional vector spaces over and -linear maps between them.
Now an ordered linear space is a finite-dimensional vector space equipped with a certain partial order, compatibly with the linear structure. We can do this by specifying a “cone” of vectors to consider as being bigger than . Then exactly when . We require that if then so is each with in the field . From this we can tell that if with , , and , then by the transitive property of partial orders. That is, the cone contains the line segment between any two of its points. In this situation we say it is a “convex set”. Finally, we require that we can find a positive basis of our vector space — one consisting of positive vectors. This is an ordered linear space, which is an object of . Because the order is specified by its cone, we often call such a space a “cone”.
A morphism in our category is just a linear function from one ordered linear space to another that preserves the partial order. That is, we call a linear function “positive” if whenever in then in . In other words, it sends the one cone into the other. An isomorphism is an isomorphism of vector spaces which identifies the two cones — the must be the “same shape”, up to a linear transformation. A subcone — the image of a monomorphism — works out to be exactly what it seems like it should be: a convex cone that fits inside another cone.
There’s a functor from the category of finite sets to . We start with a finite set and construct the free vector space on it. We define the cone to be all those vectors with all components positive. For reasons related to our motivation, we call these cones — and any cone equivalent to one of them — “classical”. The linear transformation induced by a function between finite sets is clearly positive, and so this is indeed a functor. It’s not hard to see that the image of any morphism from a classical cone is again classical, and thus the classical cones form a full subcategory of .
There’s a lot more to be said about these things, but I’ll leave it here for now.