Let be a category. Then presheaves on are

One can also look at presheaves of abelian groups/rings/modules etc., so then functor will factor through for example.

Let be a ring and be an -module. Then we had a functor

where

There exists a functor

Let be a presheaf on and be an object of . Then the natural map

is a bijection of sets. In particular, taking , we have

that is, is fully faithful.

In the above notation, we have

A morphism between presheaves and is a natural transformation :

The natural transformation consists of the data of

for every object satisfying the compatibility conditions:

We construct an explicit inverse. Consider for each object in the map

where we note that and . To see that this is an inverse, first denote the maps by

We have

Conversely, let . Then we have

But since is a natural transformation, it must satisfy

In other words, for all objects in and morphisms , we have

A presheaf is called representable if for some in , where is a natural isomorphism. If we consider applied to itself, we find that there is the identity morphism

We call the universal family. Dually, is corepresentable if for some in .

Let be a topological space. There is a fully faithful functor

i.e., the presheaf is . The Lemma (Yoneda)Lemma (Yoneda) tells us that an open subset is determined by which open sets it contains. Alternatively, we can replace the poset by a subposet , a basis of the topology, and we could say that the open subsets of a topology are determined by which basis elements a set contains.

Let . We provisionally call this the category of “affine schemes”. We denote a ring in by in . Then there is a fully faithful functor given by Lemma (Yoneda)Lemma (Yoneda)

Consider the following presheaf, the forgetful functor :

But now we have (as sets)

Therefore is representable by with universal family .

The Lemma (Yoneda)Lemma (Yoneda) says that a map of presheaves is the same as picking an element of :

Consider the curve defined by

If denotes the line , then meets all lines with multiplicity either or . The lines which don’t meet are .

Now consider

We write , where not all , for the equivalence class of in , and therefore

for . We can think of this as a line through the origin:

We see that is compact because it is the quotient of a sphere. We have a homeomorphism

We can now consider

The image of is not closed in , because is compact and is not. A coordinate for is equal to . If this point is to lie on , then we must have

This new equation is homogeneous. The vanishing locus in

is thus -invariant: for if then , and therefore

for all . We thus have

The extra points in that are not in are , since we must have the last coordinate be . We can also see that this implies because there is a sequence in

Hence, .

Now, line is in . We have

The factorisation shows that anything on the lines will meet the curve with degree .

Better: We go back to considering the equation . For , we have

where and . Hence, the lines which did not meet previously become

We “correct” the lines via a change of coordinates into the following:

Now, all lines meet with multiplicity .

Another way to view this is to look at the product

and consider the subspace called the incidence variety , defined as the locus

Then, we get a map from the composition

The fibre of this map tells us how many times our curve meets the line .
This map is “relatively compact”, and it as degree at one point. So each fibre has at least points. One can compute using a bit more algebra that every fibre has exactly points.