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 
and end with an ‘interesting’ category 
. More precisely: start from a category 
enlarge it to a category 
identify a subcategory of nice objects 
yielding inclusions
To understand what we are trying to do, let’s have a look at topological manifolds.
- Definition: A topological space
is a manifold if it admits an open cover consisting of subsets homeomorphic to open subsets of 
(where we let n vary).
With this broad definition a manifold is simply the glueing of some open subsets of 
(along other open subsets). This can be made (categorically) more precise.
Let’s call 
the category whose objects are open subsets of Euclidean spaces (with 
variable), and whose morphisms are continuous maps. Given a manifold 
let’s pick an open cover 
, where 
, and denote the intersections 
by 
. Then is actually the colimit of the diagram
(To be pedantic we should have said that each 
is homeomorphic to an object in )
This colimit thing is just a fancy way of saying that maps
from into a toplogical space 
are in bijective correspondence with collections of maps
such that restrictions coincide
This colimit fact suggests that to define the category of manifolds we only need 
and a glueing process.
Now the colimit above was taken in 
so the glueing process took place outside of 
(when discussing affine schemes, the algebraic analogue of we shall see that trying to glue without enlarging the category yields the wrong answer)
So again, we started with , enlarged it (enormously) to 
, 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 
the category affine schemes (or commutative rings if you prefer); enlarges it to 
the category of locally ringed spaces; and finally one singles out 
the category of schemes.
Of course one might find other (equivalent) ways to enlarge our category , and thus other ways of performing the glueing. One of these ways uses the Yoneda lemma (or functor of points).
