I know that traditionally, an affine space is "what is left from a vector space, after removing the origin". Given any set $X$ and a ring $K$, we can consider the set of all formal affine linear combinations of X (affine here means the coefficients must add up to 1), as an affine space. (So there is a Set $\stackrel{F\left(X\right)} \longrightarrow$$ AffineLeftModules functor. What does the fact that the coefficients should add up to 1 has to do anything with the intuition about affine spaces? More generally, What can be said in general about a category, whose objects have "affine spaces"-like properties, in a meaningful way?
[Math] What does affinity means? (In categorical terms)
affine-geometrycategory-theory
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Equivalent categories are identical except that they might have different numbers of isomorphic "copies" of the same objects. One way of making this precise is as follows. Say a category $\mathcal{C}$ is skeletal if isomorphic objects of $\mathcal{C}$ are equal. Given any category $\mathcal{C}$, you can find an equivalent skeletal full subcategory (or "skeleton") $\mathcal{D}$ of $\mathcal{C}$ by just taking one representative of each isomorphism class of objects (the inclusion functor $\mathcal{D}\to\mathcal{C}$ is then an equivalence). Furthermore, if $\mathcal{C}$ and $\mathcal{D}$ are both skeletal, and $F:\mathcal{C}\to\mathcal{D}$ is an equivalence, then $F$ is actually an isomorphism (this is easy to see from the characterization of equivalences as fully faithful and essentially surjective; note, however, that an inverse of $F$ as an equivalence might not be an inverse of $F$ as an isomorphism). It follows that the skeleton of a category is unique up to isomorphism, and two categories are equivalent iff their skeleta are isomorphic.
Another way to make it precise is the following. Let $F:\mathcal{C}\to\mathcal{D}$ be an equivalence of categories. Then $F$ is naturally isomorphic to an isomorphism of categories $F':\mathcal{C}\to\mathcal{D}$ iff for each object $C$ in $\mathcal{C}$, the set of objects isomorphic to $C$ has the same cardinality as the set of objects isomorphic to $F(C)$. (Proof sketch: Fix representatives of each isomorphism class in $\mathcal{C}$ and $\mathcal{D}$, together with isomorphisms from each object to the representative and bijections between the corresponding classes in $\mathcal{C}$ and $\mathcal{D}$. Then modify $F$ so that it stays the same on the representatives, but on all other objects it is given by what it does on the representatives together with the data above.) So in this sense, if you know a category up to equivalence, the only thing you don't know about it is how many isomorphic copies of each object it has.
More importantly, pretty much every useful thing you can say about objects in a category is invariant under isomorphisms. So, you never care about the distinction between an object and some other object that is isomorphic to it (at least, when you have chosen a specific isomorphism between them). In the case of locally free sheaves and vector bundles, the point is not that locally free sheaves and vector bundles are literally in bijection with each other; rather, the point is that isomorphism classes of locally free sheaves are in bijection with isomorphism classes of vector bundles, in a way compatible with the maps between these objects. When you unravel what this compatibility should mean, you find that an equivalence of categories exactly gives you the compatibility you are looking for.
Consider a flat sheet. Given two points on the sheet, you can't add them, but you can describe how to go from the first point to the second point. And if you have instructions explaining how to go from point $A$ to point $B$, and further you have instructions for going from point $B$ to point $C$, then you can combine them into a single set of instructions for going from point $A$ to point $C$. And you can take those instructions (e.g., walk 10 feet north and 2 feet east) and apply them to any starting point to get a different ending point. If you say that you are going to measure everything with respect to a specific starting point $O$, every other point $X$ can be thought of as $O$ and the directions from $O$ to $X$.
An affine space is a generalization of this idea. You can't add points, but you can subtract them to get vectors, and once you fix a point to be your origin, you get a vector space. So one perspective is that an affine space is like a vector space where you haven't specified an origin. And given any vector space, you get an affine space by forgetting where the origin is, forgetting addition, but keeping subtraction and allowing yourself to add differences to points, so that you don't have $A+B$, but you do have $A+(B-C)$. Differences give directions, and you can add directions to points, but you can't add points to points.
We can generalize this idea further, which may shed some light: Let $G$ be a group and $X$ a space. An action of $G$ on $X$ is a map $G\times X\to X$ (written $g.x$) such that $g.(h.x)=(gh).x$ for every $g,h\in G, x\in X$. A $G$-torsor is a space $X$ with a group action such that for every $x,y\in X$, there is a unique $g\in G$ with $g.x=y$. $G$ is encoding the ways of getting from one point to another, there is a way to go from one point to any other point, the directions one can give can be applied to any starting point, and one can combine two sets of directions together by following the first set and then the second.
An affine space, is just a $G$-torsor where $G$ is a vector space.
The simplest example here is a point. The next simplest example is a line where we can say how far apart two points are, but no particular point is special. We can take a point and move left or right by one unit, but points cannot be added. So an affine line is like the number line, but we've forgotten where $0$ is.
Best Answer
An affine space over a field $k$ is a set equipped with a family of operations called affine linear combination, one for each tuple $c_1, ... c_n \in k$ that adds up to $1$, which is supposed to behave like the operation $(v_1, ... v_n) \mapsto c_1 v_1 + ... + c_n v_n$ does on a vector space. These operations satisfy various compatibility relations which guarantee that affine linear combinations compose the way they ought to. You can think of this as an object with less structure than a vector space in that in a vector space you would allow arbitrary tuples. More precisely, there is an obvious forgetful functor from vector spaces to affine spaces and you can think of the data it forgets as the origin.
The category of affine spaces is closely related to the category of heaps. It admits a forgetful functor to $\text{Set}$, and the functor you describe is its left adjoint.
The geometric intuition is that you can't add elements of an affine space, but you can take generalized averages; for example, over $\mathbb{R}$ you can find midpoints. See also convex space.
The algebraic intuition is that you start with a vector space but restrict yourself to taking as operations not all linear combinations, but only those which are covariant under translation (that is, under moving the origin) in the sense that
$$\text{operation}(v_1 + v, ... v_n + v) = \text{operation}(v_1, ... v_n) + v.$$
These are precisely the ones whose coefficients sum to $1$.