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For $C$ a locally small category, every object $X$ of $C$ induces a presheaf on $C$: the representable presheaf $h_X$ represented by $X$. This assignment extends to a functor $C \to [C^{op}, Set]$ from $C$ to its category of presheaves. The Yoneda lemma implies that this functor is full and faithful and hence realizes $C$ as a full subcategory inside its category of presheaves.
Recall from the discussion at representable presheaf that the presheaf represented by an object $X$ of $C$ is the functor $h_X :C^{op} \to Set$ whose assignment is illustrated by
which sends each object $U$ to $Hom_C(U,X)$ and each morphism $\alpha:U'\to U$ to the function
Moreover, for $f : X \to Y$ a morphism in $C$, this induces a natural transformation $h_f : h_X \to h_Y$, whose component on $U$ in $X$ is illustrated by
For this to be a natural transformation, we need to have the commuting diagram
but this simply means that it doesn’t matter if we first “comb” the strands back to $U'$ and then comb the strands forward to $Y$, or comb the strands forward to $Y$ first and then comb the strands back to $U'$
which follows from associativity of composition of morphisms in $C$.
The Yoneda embedding for $C$ a locally small category is the functor
from $C$ to the category of presheaves over $C$ which is the image of the hom-functor
under the Hom adjunction
in the closed symmetric monoidal category Cat.
Hence $Y$ sends any object $c \in C$ to the representable presheaf which assigns to any other object $d$ of $C$ the hom-set of morphisms from $d$ into $c$:
The Yoneda embedding is sometimes denoted by よ, the hiragana kana for “Yo”; see the references below.
We can also curry the Hom functor in the other variable, thus obtaining a contravariant functor
which is explicitly given by $c \mapsto C(c,-)$. This is sometimes jokingly called the contravariant Yoneda embedding.
However, since $C^{op}(-,c)=C(c,-)$, it is easy to see that the contravariant Yoneda embedding is just the Yoneda embedding $Y: C^{\op} \to [(C^{op})^{op}, Set]=[C, Set]$ of $C^{op}$, and hence does not require special treatment.
(Yoneda embedding is a fully faithful functor)
For $\mathcal{C}$ any category, the functor
is fully faithful.
We need to show that for $c_1, c_2 \in \mathcal{C}$ any two objects, we have that every morphism of presheaves between their represented presheaves
is of the form
for a unique morphism
in $\mathcal{C}$. This follows by the Yoneda lemma, which says that morphisms $\phi$ as above are identified with the elements in
It is also limit preserving (= continuous functor), but does in general not preserve colimits.
The Yoneda embedding of a small category $S$ into the category of presheaves on $S$ gives a free cocompletion of $S$.
If the Yoneda embedding of a category has a left adjoint, then that category is called a total category .
Early accounts:
For more see at Yoneda lemma the list of references given there.
It seems that the notation “よ” for the Yoneda embedding (the hiragana kana for “Yo”) was first used in
Subsequent references that use this notation include:
Emily Riehl, Dominic Verity, p. 10 of Elements of $\infty$-category theory (web)
David Li-Bland, p. 5 of The stack of higher internal categories and stacks of iterated spans (arXiv:1506.08870)
Fosco Loregian, p. 4 of This is the (co)end, my only (co)friend (arXiv:1501.02503)
Michael Hill, Michael Hopkins, Douglas Ravenel, p. 53 of Equivariant stable homotopy theory and the Kervaire invariant problem, (web)
Last revised on June 18, 2021 at 14:36:20. See the history of this page for a list of all contributions to it.