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Can you please suggest some (relatively simple) exercises to practice the use of the Yoneda Lemma? Harder exercises are welcome too, but I would like to start with simpler ones.

The answers to this question helped me understand the general context of using it, but I would like to practice it myself.

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Mildly related: in this question: I try to elucidate a proof that right adjoints preserve limits which uses the Yoneda lemma. – lentic catachresis Feb 10 '12 at 0:29
Everything in category theory uses the Yoneda lemma ;). – Martin Brandenburg Mar 1 '12 at 17:44
up vote 7 down vote accepted

Exercise A: Let $U : \mathrm{Grp} \to \mathrm{Set}$ be the forgetful functor. What are the endomorphisms of $U$? If you can, even describe $\mathrm{End}(U)$ as a monoid and $\mathrm{Aut}(U)$ as a group. What about other forgetful functors from algebraic structures? Choose your favorite example.

Exercise B: What does Yoneda's Lemma say for functors which are defined on a category with just one object? As a corollary, deduce Cayley's "Theorem".

Exercise C: Let $M \subseteq N$ be normal subgroups of a group $G$. Deduce $(G/M)/(N/M) \cong G/N$ with the help of the Yoneda lemma. If you are bored, go through some arbitrary algebra text book and prove all these canonical isomorphisms (direct sums, tensor products, localization, Kähler differentials, ...) with the Yoneda lemma, thereby getting rid of irrelvant element chases.

Exercise D (a little bit more advanced, but still easy): Let $D$ be a category with all small colimits and $C$ be a small category. Define $\widehat{C}$ to be the category of "presheaves" on $C$, that is, functors $C^{\mathrm{op}} \to \mathrm{Set}$. Find an equivalence of categories between functors $C \to D$ and cocontinuous functors $\widehat{C} \to D$. This is induced by the Yoneda embedding $Y : C \to \widehat{C}$; which thus is the universal cocompletion of $C$. What happens when $C$ has just one object?

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One exercise (or series of exercises) involving Yoneda's lemma that was particularly helpful for me was to verify the equivalence of two definitions of a group object in a category with a final object and products. One definition is in terms of the existence of morphisms that make certain diagrams derived from the usual axioms of a group commute, and the other is in terms of group functors (the contravariant functor represented by the object admits a factorization through the category of groups via the forgetful functor from groups to sets). For details about the various statements one proves are equivalent, see Tate's article on finite flat group schemes in Cornell-Silverman-Stevens.

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