Intereting Posts

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Ok, so I was reading the Wikipedia article on Yonedas lemma. And I’ve heard before that when you prove things in category theory you automatically get a lot of results by proving it in abstract categories.

One of my favorite things in math is the whole Lagrange’s theorem $\rightarrow$ Euler’s theorem $\rightarrow$ Fermat’s little theorem chain.

Are there examples of other “semi-mainstream” theorems trivialized by Yoneda?

- Minimal polynomial of $\alpha^2$ given the minimal polynomial of $\alpha$
- Identifying a quotient group (NBHM-$2014$)
- Can we conclude that this group is cyclic?
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- Prove that any two left cosets $aH, bH$ either coincide or are disjoint, and prove Lagrange's theorem
- Step in Proof of Cardinality of Product of two Groups

Regards

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- Irreducible but not prime in $\mathbb{Z} $
- Show group of order $4n + 2$ has a subgroup of index 2.
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- Prime elements of ring $\mathbb{Z}$
- Can you give me an example of topological group which is not a Lie group.
- Show that $\mathbb{K}$ is a field

The most beautiful and yet elementary things I noticed when I was learning the basics were about Yoneda Lemma; these are not “theorems” in the real sense of the term; instead they are *principles* one sees in action when proving things.

- There this thing called Yoneda lemma where you prove that given a contravariant functor $\mathcal C\to \bf Set$, the correspondence $F\mapsto \text{Nat}(\hom(-,A),F)$ “acts as an evaluation” giving you a set with as much elements as $FA$ back. Ok then, let’s play this game where we evaluate at the object $B$ the functor $\hom(-,B)$: we get that

$\text{Nat}(\hom(-,A),\hom(-,B))\cong \hom(A,B)$, i.e. that the correspondence which sends an object to its representable presheaf is

fully faithful.

- Yoneda Lemma allows you to reduce statements about complicated categories to statements about
*sets*, or better to say, functors which take value in $\bf Set$. This is because

the previous point gives you a fully faithful functor $\mathcal C \to [\mathcal C^\text{op},\bf Set]$, the

Yoneda embedding$A\mapsto \hom(-,A)$;

- there’s this thing you maybe learned about a “product of $A,B$” in a category being an object $P$ endowed with two maps $A\leftarrow P\to B$ such that blablabla; ok then, let’s feed the Yoneda representable functor $\hom(X,-)$ with this diagram: we obtain something which tells you that $\hom(X,P)$ is precisely the product (i.e. the
*set theoretic*product, the one you learn in your first day as a freshman in any “calculus 0” course) of $\hom(X,A)$ and $\hom(X,B)$. This gives you that

$P\cong A\times B$ in a category $\cal C$ if and only if for

any$X\in\cal C$ the set $\hom(X,P)$ is in bijection the product of $\hom(X,A)$ and $\hom(X,B)$, and this bijection isnatural, ie. respects the fact that I can have arrows $Y\to X$, generating arrows $\hom(X,P)\to \hom(Y,P)$.

The same argument (even if it’s far more involved to grasp “visually”) works well for any shape of diagram: equalizers/kernels, pullbacks, inverse limits … and properly dualized, it works well with colimits!

Whenever you are able to characterize a universal object (i.e. a limit/colimit) in the category of sets, then you can define that very universal object in any category exploiting the former principle: an object $K$ with maps $A_i\to K$ in $\mathcal C$ is universal (say, a colimit for the diagram $F\colon i\mapsto A_i$) if and only if passing its arrows $A_i\to K$ through the yoneda embedding $\hom(-,X)$, I obtain the set theoretic universal (in fact the limit, since $\hom(-,X)$ is contravariant), *naturally* in $X$.

Everything I said deeply relies on the fact that you are using *sets*. Or maybe not? There this thing called “enriched Yoneda lemma”, which is

the same statement, but for functors between any $\bf Ab$-enriched category [where each $\hom(X,Y)$ is a $\mathbb Z$-module] and the category of abelian groups…

The Yoneda lemma allows you to prove one of the most useful results in basic category theory, namely that left/right adjoint functors preserve colimits/limits of any shape. I can’t even estimate the number of times this simple remark saved my life in practice. All you have to do is follow a suitable string of isomorphisms, and then say:

Since the yoneda embedding is fully faithful, it

reflects isomorphisms, i.e. whenever $\hom(A,X)\cong \hom(B,X)$, or $\hom(X,A)\cong \hom(X,B)$, for any $X\in\cal C$,naturally in $X$, then $A\cong B$.

Cayley’s theorem. Yoneda embedding. Uniqueness of representing objects

The correspondence between characteristic classes on the one hand and cohomology classes on a classifying space on the other is a nice application of Yoneda’s lemma, though it probably doesn’t deserve the name of “theorem”.

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