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Appeared in some form in [Yoneda-homology]. Used by Grothendieck in a generalized form in [Gr-II].

Lemma 4.3.5 (Yoneda lemma). Let $U, V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Given any morphism of functors $s : h_ U \to h_ V$ there is a unique morphism $\phi : U \to V$ such that $h(\phi ) = s$. In other words the functor $h$ is fully faithful. More generally, given any contravariant functor $F$ and any object $U$ of $\mathcal{C}$ we have a natural bijection

\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ U, F) \longrightarrow F(U), \quad s \longmapsto s_ U(\text{id}_ U). \]

Proof. For the first statement, just take $\phi = s_ U(\text{id}_ U) \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U, V)$. For the second statement, given $\xi \in F(U)$ define $s$ by $s_ V : h_ U(V) \to F(V)$ by sending the element $f : V \to U$ of $h_ U(V) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U)$ to $F(f)(\xi )$. $\square$

Comments (2)

Comment #1841 by Peter Johnson on

This is really a comment on the Reference Gr-II here. In the bibliography Gr-I through Gr-VI are listed as [Gro95a--f]! These were of course first published c1959 and THOSE years (someone should know them) should be given.

Comment #1878 by on

Not sure what you want me to do. The data entry "YEAR" in the bib file refers to the publication date of the publication and not of the date the text was written. So if there is a new edition of a book and we refer to that one, then the date of that publication will be entered in that field. Anyway, I think it is completely clear which document this reference is referring to and that is what is most important I think.

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  • 13 comment(s) on Section 4.3: Opposite Categories and the Yoneda Lemma

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