## pre-Nobuo

Hi, we’re going to try and get things running around here.

Our objective is to have a glance at what algebraic geometry might look like if investigated using the functor-of-points approach. Essentially I’ll be trying to understand part of these notes.

Before getting all technical, let’s roughly outline our approach. The general idea is as follows: we want to start from a ‘simple’ category $\mathsf{C}$ and end with an ‘interesting’ category $\widetilde{\mathsf{C}}$. More precisely: start from a category $\mathsf{C};$ enlarge it to a category $\hat{\mathsf{C}};$ identify a subcategory of nice objects $\widetilde{\mathsf{C}},$ yielding inclusions

$\displaystyle \mathsf{C} \subseteq \widetilde{\mathsf{C}} \subseteq \hat{\mathsf{C}}.$

To understand what we are trying to do, let’s have a look at topological manifolds.

• Definition: A topological space ${X}$ is a manifold if it admits an open cover consisting of subsets homeomorphic to open subsets of ${\mathbb{R}^n}$ (where we let n vary).

With this broad definition a manifold is simply the glueing of some open subsets of ${{\mathbb R}^n}$ (along other open subsets). This can be made (categorically) more precise.
Let’s call ${\mathsf{C}}$ the category whose objects are open subsets of Euclidean spaces ${\mathbb{R}^n}$ (with ${n}$ variable), and whose morphisms are continuous maps. Given a manifold ${X,}$ let’s pick an open cover $\{U_i\}$, where $U_i \in \mathsf{C}$, and denote the intersections ${U_i \cap U_j}$ by $U_{ij}$. Then ${X}$ is actually the colimit of the diagram

$\displaystyle \coprod_{i,j} U_{ij} \rightrightarrows \coprod_k U_k.$

(To be pedantic we should have said that each ${U_i}$ is homeomorphic to an object in ${\mathsf{C}}$)
This colimit thing is just a fancy way of saying that maps

$\displaystyle f: X \rightarrow Y,$

from ${X}$ into a toplogical space ${Y,}$ are in bijective correspondence with collections of maps

$\displaystyle \{f_i: U_i \rightarrow Y \}$

such that restrictions coincide

$\displaystyle f_i\vert U_{ij} = f_j\vert U_{ji}.$

This colimit fact suggests that to define the category of manifolds we only need ${\mathsf{C},}$ and a glueing process.

Now the colimit above was taken in ${\mathsf{Top},}$ so the glueing process took place outside of ${\mathsf{C}.}$ (when discussing affine schemes, the algebraic analogue of ${\mathsf{C},}$ we shall see that trying to glue without enlarging the category yields the wrong answer)

So again, we started with ${\mathsf{C}}$, enlarged it (enormously) to ${\mathsf{Top}}$, and then selected the objects which could be obtained by glueing: i.e. manifolds.

The same thing is done in algebraic geometry: one starts with ${\mathsf{Aff},}$ the category affine schemes (or commutative rings if you prefer); enlarges it to ${\mathsf{Esp},}$ the category of locally ringed spaces; and finally one singles out ${\mathsf{Sch},}$ the category of schemes.

Of course one might find other (equivalent) ways to enlarge our category ${\mathsf{C}}$, and thus other ways of performing the glueing. One of these ways uses the Yoneda lemma (or functor of points).