What does it mean to fix a point in an affine space?

Question

In their book Metric Affine Geometry, Snapper and Troyer state on page 59:

It cannot be stressed enough that the affine space $X$ is not a vector space. Its points cannot be added and there is no way to multiply by scalars. No point in $X$ is preferred; they all play the same role. In particular, there is no point in $X$ which makes a better origin for a vector space than any other point.

The situation changes radically if we choose a point $c$ in $X$ and keep it fixed. It is now possible to make $X$ into a left vector space over $k$ by using the one-to-one mapping $f$ from $X$ onto $V$ defined by $f(x) = \overrightarrow{c,x}$ for each $x \in X$. All we do is carry the vector space structure of $V$ over to $X$ by means of the mapping $f$.

So, here's what I want to know:

(1) What does it mean to "fix" a point in an affine space?

(2) Why does fixing a point suddenly give us the ability to assign vector space structure to the affine space $X$?

(3) If an affine space isn't a vector space, what are the axioms we define it by then?

Answer 1

1. It simply means to pick a point $\bf c$ in the space. For any choice $\bf c$ there is a unique vector space structure on $X$ that is (a) compatible with the affine space structure of $X$ and (b) $\bf c$ is the zero vector for that vector space structure. The point (no pun intended) of an affine space vis-a-vis a vector space is simply that there isn't a canonical way to pick a $\bf c$ in the first place.

2. This was partially answered in (1): An affine space $X$ is a set endowed with a particular operation that satisfies the affine space axioms (this answers (3)). Together with a declaration of which element ${\bf c} \in X$ is the zero vector, you can canonically build operations $+, \cdot$ that together with the underlying set $X$ satisfy the vector space axioms, i.e., make $X$ a vector space.

Part of what makes this confusing is that by definition an affine structure on a set $X$ includes an underlying vector space $V$, along with and an addition [displacement] operation $+ : V \times X \to X$, usually denoted $+$, satisfying some natural axioms.

Here $V$ is an auxiliary object: It doesn't have much to do with the underlying set $X$ until one declares the operation $+$, but via that operation it tells us how the elements of $X$ are related. Here's a natural analogy:

• A vector space over a field $\mathbb{F}$ is a triple $(V, +, \cdot)$ where $V$ is a set, and $+:V \times V \to V$ and $\cdot: \mathbb{F} \times V \to V$ are operations satisfying some natural axioms.

• An affine space over a vector space $V$ is a pair $(X, +)$ where $X$ is a set, and $+:V \times X \to X$ is an operation satisfying some natural axioms.

The main difference between these examples, and probably another potential source of the above confusion, is that any vector space $(V, +, \cdot)$ automatically determines an affine structure $+$ on $V$: In this case, $V$ is the underlying set, $(V, +, \cdot)$ is the underlying vector space, and $+$ is just the vector space addition. Making the affine space $(V, +)$ from a vector space $(V, +, \cdot)$ just amounts to "forgetting where the zero vector is."