## Nobuo

In the previous post we started with (the category of open subsets of ‘s), embedded it inside and formed topological manifolds via gluing. (not much of a surprise there…)

We mentioned that there is another way to enlarge using the Yoneda lemma. Let’s (briefly) see how.

Let’s start from the easiest case: take to be a set. Pick any set with one element , in other words a *final* object in .

We usually care more about when two objects, in a given category, are isomorphic than when they are actually equal. In two objects are isomorphic if and only if they have the same cardinality. As the set

of maps from the singleton to has the same cardinality of we can recover (up to isomorphism) by considering the set (the definition of depends of course on the underlying category; being in this case )

Of course the same idea does not work for a general category. For example if we take a topological space then the set

only recovers the *underlying set* of , but completely forgets about its *topology*.

The incredibly brilliant idea is that, in order to recover (as an object of ), one should consider *all* the sets

at once, where varies over all topological spaces.

More precisely: start from a category , then any object determines a functor

(where stands for *opposite category*),

Actually one can check that the assignment determines a functor

(where stands for the category of functors)

**Theorem**(The Yoneda Lemma) The functor above is fully faithful.

Actually we have a sharper result.

**Theorem**Let be any object of and let be any object in Then

We therefore have our desired result: any category can be embedded into

From now on we shall adopt the following notation

(*presheaves* on ), which shall be explained in the next post.

Our next task is to come up with a gluing operation inside which will lead to the definition of a subcategory (*sheaves* on ). But for that we need to know what a *site* is.