## Nobuo

In the previous post we started with ${\mathsf{C}}$ (the category of open subsets of ${\mathbb{R}^n}$‘s), embedded it inside ${\mathsf{Top}}$ and formed topological manifolds via gluing. (not much of a surprise there…)

We mentioned that there is another way to enlarge ${\mathsf{C}}$ using the Yoneda lemma. Let’s (briefly) see how.

Let’s start from the easiest case: take ${X}$ to be a set. Pick any set with one element ${\bullet}$, in other words a final object in ${\mathsf{Set}}$.

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

$\displaystyle h_X(\bullet) := Hom_{\mathsf{Set}}(\bullet,X),$

of maps from the singleton ${\bullet}$ to ${X,}$ has the same cardinality of ${X,}$ we can recover (up to isomorphism) ${X}$ by considering the set ${h_X(\bullet).}$ (the definition of ${h_X}$ depends of course on the underlying category; being in this case ${\mathsf{Set}}$)

Of course the same idea does not work for a general category. For example if we take a topological space ${X \in \mathsf{Top},}$ then the set

$\displaystyle h_X(\bullet) = Hom_{\mathsf{Top}}(\bullet,X)$

only recovers the underlying set of ${X}$, but completely forgets about its topology.

The incredibly brilliant idea is that, in order to recover ${X}$ (as an object of ${\mathsf{Top}}$), one should consider all the sets

$\displaystyle h_X(Y) = Hom_{\mathsf{Top}}(Y,X)$

at once, where ${Y}$ varies over all topological spaces.

More precisely: start from a category ${\mathsf{C}}$, then any object ${X \in \mathsf{C}}$ determines a functor

$\displaystyle h_X : \mathsf{C}^\circ \rightarrow \mathsf{Set}$

(where ${\circ}$ stands for opposite category), ${h_X(Y) := Hom_{\mathsf{C}}(Y,X).}$

Actually one can check that the assignment ${X \mapsto h_X}$ determines a functor

$\displaystyle h : \mathsf{C} \rightarrow Fun(\mathsf{C}^\circ,\mathsf{Set}).$

(where ${Fun}$ stands for the category of functors)

• Theorem (The Yoneda Lemma) The functor ${h}$ above is fully faithful.

Actually we have a sharper result.

• Theorem Let ${P}$ be any object of ${Fun(\mathsf{C}^\circ,\mathsf{Set}),}$ and let ${X}$ be any object in ${\mathsf{C}.}$ Then
$\displaystyle Hom(h_X,P) \simeq P(X).$

We therefore have our desired result: any category ${\mathsf{C}}$ can be embedded into ${Fun(\mathsf{C}^\circ,\mathsf{Set}).}$

From now on we shall adopt the following notation

$\displaystyle PSh\mathsf{C}:= Fun(\mathsf{C}^\circ,\mathsf{Set})$

(presheaves on ${\mathsf{C}}$), which shall be explained in the next post.

Our next task is to come up with a gluing operation inside ${PSh\mathsf{C},}$ which will lead to the definition of a subcategory ${Sh\mathsf{C}}$ (sheaves on ${\mathsf{C}}$). But for that we need to know what a site is.