My Problem with 0

In my research recently, I’ve been debating between two setups for a category. My category is supposed to have, as objects, a finite set $s$ of spaces, a partition $\Lambda$ of $s$, and a map from the disjoint union of those spaces to a space $M$. I tend to bundle all of this information up into $\rho$ (for the finite set, it’s partition, and the collection of spaces) and $f$ (the map to $M$). In my situation, $M$ is a disjoint union of copies of $\mathbb{R}^n$. The spaces I have in $\rho$ have, for a while, been affine spaces. But there’s also always been a question about maybe having them be vector spaces. The difference, of course, is the existence of 0.

There are a few ways to think about affine spaces. The least precise is to say it is a vector space that forgot where its 0 is. With this idea, a pointed affine space is (essentially) a vector space. Every affine space has an underlying vector space, and given two points in the affine space, you can find their difference, which will be a vector in the underlying vector space. Since differences are defined in a vector space, every vector space is (essentially) an affine space whose underlying vector space is the one you started with.

Now, I have this collection of spaces (either vector or affine) and an affine map $f$ to $M$ – that is, the map is affine (linear after a linear translation) on each space in $\rho$. Since I have an equivalence relation $\Lambda$ (by abuse of notation) on my spaces in $\rho$, I can take the transitive closure of the image, $f(\Lambda)$, and get an equivalence relation on $M$. I then have this process in mind where I convert this equivalence relation on $M$ to one of a particularly nice form, which I have been calling ‘locally affine.’ For more, see my earlier writeup.

Part of the process to convert an equivalence relation to one that is locally affine involves looking at pairs of points that are parallel to some linear subspace of (a component of) $M$. For reasons that deserve to be called ‘continuity’, this is not a great procedure to do in $M$. If two points are parallel to some line, and you wiggle the two points a little, there’s no reason to assume they are still parallel. And that messes some things up (at least, can, and seems to with what I’ve been hoping to do), or, if nothing else, makes them uglier. So what I’ve been trying to accomplish, or approximate, is to do the similar operations on my original $\rho$ in my category of ‘abstract’ locally affine partitions. I would like to convert the original $\rho$ to something pretty similar, staying in the category I’m defining while not changing the (locally affine coarsening of the) image of the equivalence relation too much.

It’s tempting to assume that my spaces in $\rho$ are affine spaces, because a big part of making the locally affine partitions above is taking affine spans of things. Taking the affine span of a vector space would give me an affine space, but that would mean I’ve changed categories (from a category using vector spaces to a category using affine spaces). But taking the affine space of affine spaces causes no such problem. The problem with using affine spaces is that sometimes I also want to take linear spans, which is not defined for affine spaces. Taking linear spans only works if you have a 0. Grr. I’m going to have to start being more clever, or more careless. I wish I knew which.