So I think we’re ready, at least for the statement of Yoneda’s lemma. It says that for any locally small category ![\mathcal C]( "\mathcal C")
, if ![A]( "A")
is an object in ![C]( "C")
, and ![F:\mathcal C\to\textsc{Set}]( "F:\mathcal C\to\textsc{Set}")
a functor, then
![\mbox{Nat}(h^A,F)\cong F(A)]( "\mbox{Nat}(h^A,F)\cong F(A)")
Moreover, the isomorphism is natural in both ![A]( "A")
and ![F]( "F")
.
Wow that looks complicated. Let’s parse some of the notation that I haven’t even explained yet. So certainly we know what the righthand-side means. It’s ![F]( "F")
applied to ![A]( "A")
. That’s just a set. As for the lefthand-side, ![\mbox{Nat}]( "\mbox{Nat}")
and ![h^A]( "h^A")
I haven’t explained.
So ![h^A]( "h^A")
is a functor we mentioned briefly, but I used a different notation. It’s the representable functor ![\hom(A,-)]( "\hom(A,-)")
. I write it as ![h^A]( "h^A")
here to avoid over using parentheses and therefore complicating this business well beyond it’s current level of complication. As a reminder, ![\hom(A,-)]( "\hom(A,-)")
is a functor from ![\mathcal C]( "\mathcal C")
to ![\textsc{Set}]( "\textsc{Set}")
(the same as ![F]( "F")
). It takes objects ![B]( "B")
in ![\mathcal C]( "\mathcal C")
to the set of homomorphisms ![\hom(A,B)]( "\hom(A,B)")
. Remember we’re in a locally small category, so ![\hom(A,B)]( "\hom(A,B)")
really is a set. It takes morphisms ![f:B\to C]( "f:B\to C")
to a map ![h^A(f):\hom(A,B)\to\hom(A,C)]( "h^A(f):\hom(A,B)\to\hom(A,C)")
by sending
![h^A(f):\phi\mapsto f\circ\phi]( "h^A(f):\phi\mapsto f\circ\phi")
We checked all the necessary details in an earlier post to make sure this really was a functor.
So the last thing is that ![\mbox{Nat}]( "\mbox{Nat}")
. It’s the collection of all natural transformations between the functors ![h^A]( "h^A")
and ![F]( "F")
. So Yoneda’s lemma claims that there is a one to one correspondence between the the natural transformations from ![h^A]( "h^A")
to ![F]( "F")
and the set ![F(A)]( "F(A)")
.
In particular, it claims that ![\mbox{Nat}(h^A,F)]( "\mbox{Nat}(h^A,F)")
is a set. This is because ![\textsc{Set}]( "\textsc{Set}")
is a locally small category, and so each natural transformation (defined as a collection of morphisms, each of which was a set) is a set. But there are only so many collections of morphisms, not even all of which are natural transformations. The collection is small enough to be a set. If you don’t care about this set theory business. Then disregard the paragraph you probably just read angrily.
It’s worth mentioning now, that Yoneda’s lemma is a generalization of some nice theorems. We can (and will) use it to derive Cayley’s theorem (every group embeds into a symmetric group). We can (and will) use it to derive the important fact that ![\hom_R(R,M)\cong M]( "\hom_R(R,M)\cong M")
in the category of ![R]( "R")
-modules. I bet in that one you can already start to see the resemblance.
We’ll prove Yoneda’s lemma over the next few posts.
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