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).

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