A student argues: 'Since the Yoneda embedding is fully faithful, every presheaf F: C^op → Set must be naturally isomorphic to Hom(−, X) for some object X in C.' What is wrong with this reasoning?
AFull faithfulness does imply that every presheaf is representable — the student is correct
BFull faithfulness means the embedding preserves and reflects morphisms between representables, but says nothing about presheaves outside the image of Y — many presheaves are non-representable
CThe error is that Hom(−, X) is a covariant functor, but presheaves are required to be contravariant
DFull faithfulness only holds when C is a small category; the claim fails in general
Full faithfulness is a property about morphisms: the embedding is injective on morphisms (faithful) and surjective on morphisms between objects in its image (full). It says nothing about whether every presheaf is in the image of Y at all — that would be essential surjectivity. Many presheaves are non-representable (e.g., in the category of sets, the presheaf sending every set to the empty set unless the set is empty is not representable). The non-representable presheaves are a feature, not a flaw.
Question 2 Multiple Choice
The Yoneda embedding proves that if Hom(−, X) ≅ Hom(−, Y) as functors, then X ≅ Y in C. Which property of the embedding does this follow from?
AFull faithfulness — the embedding is bijective on hom-sets, so isomorphic hom-functors force the representing objects to be isomorphic
BEssential surjectivity — every presheaf is representable by a unique object, so isomorphic functors must have the same representing object
CCocompleteness of the presheaf category — colimits force the objects to coincide
DThe fact that Hom(−, X) is always a sheaf, making representable functors uniquely determined
Full faithfulness means Hom_C(X, Y) ≅ Hom_{[C^op,Set]}(Y(X), Y(Y)) — every morphism in C corresponds to exactly one natural transformation between the representables, and vice versa. If Y(X) ≅ Y(Y) (as presheaves), then there are natural transformations in both directions that compose to identities — and full faithfulness pulls these back to morphisms in C that compose to identities, giving X ≅ Y. Essential surjectivity plays no role here.
Question 3 True / False
The Yoneda embedding Y: C → [C^op, Set] is an equivalence of categories — nearly every presheaf is naturally isomorphic to Hom(−, X) for some X.
TTrue
FFalse
Answer: False
An equivalence of categories requires the functor to be fully faithful AND essentially surjective (every object in the target is isomorphic to something in the image). The Yoneda embedding is fully faithful but not essentially surjective — there are many non-representable presheaves that are not naturally isomorphic to any Hom(−, X). The gap between C and [C^op, Set] is precisely the non-representable presheaves, which are a rich and important class of objects in their own right.
Question 4 True / False
Two objects X and Y in a category C are isomorphic if and only if their representable functors Hom(−, X) and Hom(−, Y) are naturally isomorphic.
TTrue
FFalse
Answer: True
This is the key corollary of full faithfulness of the Yoneda embedding. If X ≅ Y, then post-composition with the isomorphism gives a natural isomorphism Hom(−, X) ≅ Hom(−, Y). Conversely, if Hom(−, X) ≅ Hom(−, Y), full faithfulness pulls the natural isomorphism back to an isomorphism X ≅ Y in C. An object is therefore completely characterized — up to isomorphism — by its hom-functor.
Question 5 Short Answer
What does it mean for the Yoneda embedding to be 'fully faithful,' and what philosophical conclusion does this imply about how objects are determined by their relationships?
Think about your answer, then reveal below.
Model answer: Fully faithful means the Yoneda embedding Y: C → [C^op, Set] is bijective on hom-sets: Hom_C(X, Y) ≅ Hom_{[C^op,Set]}(Y(X), Y(Y)) for all X, Y. Every morphism in C corresponds to exactly one natural transformation between the representable functors, and every natural transformation between representables comes from a unique morphism. The philosophical consequence is that an object X is completely determined — up to isomorphism — by the functor Hom(−, X), which encodes how every other object maps into X. Two objects with the same mapping-in behavior are the same object. An object is known entirely by its relationships.
This is the categorical formulation of a deep principle: the intrinsic properties of an object are less important than its position in the web of morphisms. The Yoneda embedding makes this precise — you can replace any object with its hom-functor without losing any information. This idea pervades modern mathematics: sheaves, spectra, and moduli spaces are all defined by what maps into them, not by explicit internal structure.