Vector spaces with infinity The real numbers $\mathbb{R}$ can be identified with a one-dimensional real vector space. However in some contexts (e.g. convex analysis) it's useful to consider the reals augmented with positive and negative infinity, $\overline{\mathbb{R}} = \mathbb{R}\cup\{-\infty,\infty\}$.
It would be elegant and useful if there was a notion of "augmented vector space", meaning a vector space augmented with infinite values somehow, such that a one-dimensional augmented vector space could be identified with $\overline{\mathbb{R}}$ instead of $\mathbb{R}$.
According to Wikipedia, $\overline{\mathbb{R}}$ is not a field (at least with the particular definition that article uses), so we can't simply define a vector space over $\overline{\mathbb{R}}$, but perhaps there is some other way to do it. For example, one reasonable approach might be to identify infinite elements with rays and add those to the space.
Does this "augmented vector space" concept exist, and if so, what's it called?
I'm using the category theory tag because the ideal thing would be a category similar to $\mathbf{Vect_\mathbb{R}}$, but where the unit object is $\overline{\mathbb{R}}$ instead of $\mathbb{R}$. If the details have been worked out for something like that it would be great.
 A: There are ways to get at something similar, but the shortest answer is that there does not seem to be any convenient category of "convex things" such that maps with codomain $\bar{\mathbb R}$ coincide with convex functions in the ordinary sense. The problem is that convex functions aren't closed under composition. 
Since they don't compose, I think it's probably best to think of convex functions $X\to \bar{\mathbb R}$ as objects of a slice category of convex spaces over $\bar{\mathbb R}$. There is a nice finitarily monadic category of such spaces, essentially being sets equipped with combinations $tx+(1-t)y$ for all $t\in[0,1]$, whose morphisms are functions that strictly preserve the convex-linear combinations. Now, $\bar{\mathbb R}$ does not live in this category, but this is sensible, since there's no natural way to define convex-linear combinations such as $\frac 12 \infty +\frac 12(-\infty)$. For instance, you can easily prove that there's no such definition which continuously extends the convex combinations from $\mathbb R$. 
