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.

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