Affine Map as a Morphism of Affine Vector Spaces

ORIGINAL ANSWER REPLACED BY: $\newcommand\vcal{\mathcal{V}} \newcommand\acal{\mathcal{A}} \newcommand\F{\mathbb{F}} $

Let $\vcal$ be the category of vector spaces over the field $\F$ of real or complex scalars. The morphisms are linear maps between vector spaces. Observe that using the standard topology of a real or complex vector space, a map $L: V \rightarrow W$ is linear if it is additive, i.e., for any $v_1,v_2 \in V$, \begin{align*} L(v_1+v_2) = L(v_1)+L(v_2). \end{align*}

An object in the category $\acal$ of affine spaces over $\F$ consists of a pair $(A,V)$, where $V$ is a vector space over $\F$, with an operation \begin{align*} A \times V &\rightarrow A\\ (a,v) &\mapsto a+v \end{align*} that satisfies the following properties:

1. For each $a \in A$, the map \begin{align*} V &\rightarrow A\\ v &\mapsto a+v \end{align*} is bijective. The inverse map is denoted by subtraction: \begin{align*} A &\rightarrow V\\ b &\mapsto b-a. \end{align*}
2. For each $v \in V$, the map \begin{align*} A &\rightarrow A\\ a &\mapsto a+v \end{align*} is bijective.
3. For any $a \in A$, $v, w \in V$, \begin{align*} a+(v+w) &= (a+v)+w.\\ \end{align*}

To define a morphism of $\acal$, first observe that given affine spaces $(A,V)$ and $(B,W)$ and a map $F: A \rightarrow B$, there is for each $a \in A$, an induced map \begin{align*} dF_a: V &\rightarrow W\\ v &\mapsto F(a+v)-F(a). \end{align*} Call $dF_a$ the differential of $F$ at $a$. A map $F: A \rightarrow B$ is a morphism of $\acal$ if for any $a \in A$ and $v \in V$, \begin{align*} F(a+v) &= F(a) + dF_a(v). \end{align*} From this it follows that if $v_1,v_2 \in V$, \begin{align*} dF_a(v_1+v_2) &= F(a+v_1)-F(a)+F(a+v_2)-F(a)\\ &= dF(v_1)+dF(v_2). \end{align*} It follows that $dF$ is linear. Moreover, if $a_1, a_2 \in A$, then \begin{align*} dF_{a_2}(v) &= F(a_2+v)-F(a_2)\\ &= F(a_1+(a_2-a_1)+v) - F(a_1+(a_2-a_1))\\ &= F(a_1) + dF_{a_1}((a_2-a_1)+v) - F(a_{1}) - dF_{a_1}(a_2-a_1)\\ &= dF_{a_1}(v). \end{align*} Therefore, the differential of $F$ does not depend on $a\in A$ and can be denoted simply by $dF$.

It follows that a map $F: A \rightarrow B$ is a morphism if and only if there is a linear map $L: V \rightarrow W$ such that for any $a \in A$, \begin{align*} F(a+v) = F(a)+L(v). \end{align*}

There are a couple ways to do this. Personally I find it unsatisfying to think of affine spaces in terms of their associated vector spaces: it is actually possible to give an axiomatization of affine spaces that does not require this. In the same way that a vector space over a field $K$ is a set equipped with operations that allow us to define arbitrary $K$-linear combinations

$$(v_1, \dots v_n) \mapsto \sum c_i v_i,$$

an affine space over a field $K$ can be defined as a set equipped with operations that allow us to define only affine $K$-linear combinations

$$(p_1, \dots p_n) \mapsto \sum c_i p_i, \sum c_i = 1.$$

In the vector space case, instead of talking about all linear combinations at once we conventionally reduce to talking only about addition and scalar multiplication, which can be used to build all linear combinations. In the affine case we can no longer do this. We can restrict our attention to binary affine combinations

$$(p_1, p_2) \mapsto c p_1 + (1 - c) p_2, c \in K$$

since arbitrary affine combinations can be built out of compositions of these. These parameterize the line containing $p_1$ and $p_2$ (whereas general affine combinations parameterize the hyperplane containing some set of points). But I don't think the resulting axioms look all that nice. It seems cleaner to me to use all affine combinations at once.

Formally, the collection of all affine combination operations organize into a gadget called a Lawvere theory, and we can define an affine space as what is called a model of this Lawvere theory, which means a set equipped with affine combination operations which satisfy exactly the relations that affine combinations do. (Vector spaces can be defined in the same style, using linear combinations.) Then a morphism of affine spaces is a function which respects all these affine combination operations. This may look like a lot of structure to carry around but in practice it just means we can do completely ordinary-looking calculations with affine combinations. For example it is straightforward to verify that any affine subspace of a vector space is an affine space in this sense.

Then you can prove as a theorem that every affine space is obtained from a vector space, rather than making it part of the definition, and also prove that morphisms of affine spaces correspond to affine maps in the usual sense. The details seem a bit tedious to spell out though.

A very similar idea can be used to define the notion of a convex space, where you only allow convex linear combinations.