EXERCISES IN CATEGORY THEORY 4
1. The Yoneda Lemma
In Set elements of a set X correspond to maps 1 —>• X where 1 is the set with 1 element. From one perspective, the Yoneda lemma says that representable functors have a similar behaviour: maps C(X, —) —>• F in [C, Set] correspond to elements of the set FX.
(1) Use the Yoneda lemma to show that a morphism 6 : F —>• G G [C, Set] is mono if and only if each of its components 6x ■ FX —>• GX is mono is Set: an injective function. Note: one direction is straightforward and does not use the Yoneda lemma.
(2) Given F,G G [C, Set] we want to work out what the product functor F xG looks like. Use the Yoneda lemma and the universal property of products to show that we must have (F x G)(X) = F(X) x G(X).
(3) Set F x G(X) — F(X) x G(X). Use the universal property of the product projections in Set
F(X) F
G(X) G
to define FxGon morphisms and to construct a product diagram in [C, Set] as above right.
(4) To each object X of C we have assigned a functor C(X,—) : C —>• Set. For each / : X —>• Y describe a natural transformation C(f, —) : C(Y, —) —>• C(X, —).
(5) Prove that these assignments define a functor Y : Cop —>• [C, Set]. T/iis is called the Yoneda embedding.
(6) A functor F : G ->• D is said to be fully faithful if given f : FX ^ FY e D there exists a unique g : X —>• Y such that = /. Use the Yoneda Lemma to prove that the Yoneda embedding Cop —>• [C, Set] is fully faithful.
Date: October 8, 2014.
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