While studying geometry, I came to know that a projective space (of dimension n associated to a vector space over a field K) can be seen as an affine space (of the same dimension and on the same field) extended with points at infinity. This extension perspective also pertains morphisms since, fixing an appropriate identification of the affine space into the projective one, affinities of the former can be viewed as special projectivities of the latter.
I am aware of the existence of category theory which can formalise similar interplays. Is there an interpretation of this case? More generally, I wonder to know (by willing scholars more advanced than me) if there exists an appropriate higher standpoint in which this relation can be interpreted.
Thank you.
Here's how I think rather concretely of projective versus affine spaces.
You start with the category of vector spaces, where the morphisms are linear isomorphisms.
The projective space associated with a vector space $V$ is the set $\newcommand\P{\mathbb{P}}$$\P V$ of $1$-dimensional subspaces. Projective transformations are maps naturally induced by linear isomorphisms. Any linear isomorphism $L: V \rightarrow W$ induces a map $$[L]: \P V \rightarrow \P W,$$ where for each $\ell \in \P V$, $$[L](\ell) = L(\ell).$$
To make this abstract, we can say that $\P$ is a projective space if there exists a vector space $V$ such that $\P = \P V$ and $F: \P \rightarrow \mathbb{Q}$ is a projective transformation if $\P = \P V$, $\mathbb Q = \P W$, and there exists a linear isomorphism $L: V \rightarrow W$ such that $F = [L]$.
This defines a functor from the category of vector spaces to the cateogry of projective spaces, where all morphisms are bijective.
An affine space arises as an affine hyperplane of a vector space. In other words, $\newcommand\A{\mathbb{A}}$$\A$ is an affine space if there exists a vector space $V$ and nonzero $\xi \in V^*$ such that $$\A = \xi^{-1}(1). $$ There is a natural injective map \begin{align*} \A &\rightarrow \P V\\ v &\mapsto [v], \end{align*} where $[v]$ is the $1$-dimensional subspace containing $v$. A map $\A_1 \rightarrow \A_2$ is an affine trasnformation if it is the restriction of a projective transformation $$[L]: \P V_1 \rightarrow \P V_2.$$
The complement $\P V\backslash \A$ is called the hyperplane at infinity. Given the definition above of an affine space, there is a unique way to extend $\A$ to an isomorphism class of projective spaces. That's what is meant by projectivizing $\A$ by adding the hyperplane at infinity. Also, any affine transformation has a unique extension to a projective transformation.