Here's a statement of Yoneda's lemma for n-category.
SoientLet C be a n-category and $C^{\wedge}=[C^o,n-1Cat]$ be the n-category of presheaves on C.
$C^o$ is the opposite n-category of C and $n-1Cat$ is the n-category of (n-1)-categories.
We have the Yeneda embedding:
$h_C:C \rightarrow C^{\wedge}$ , X being sent to the hom-presheaf $hom_C(-,X)$.
Now Yoneda's lemma says:
- For $X \in C $ and $A \in C^{\wedge}$,
$hom_{C^{\wedge}}(h_C(X),A) \approx A(X) $ in the n-category $n-1Cat$ up to n-equivalence;
- For $A \in C^{\wedge}$,
$hom_{C^{\wedge}}(h_C(-),A) \approx A $ in the n-category $C^{\wedge}$ up to n-equivalence.
I want to test the Yoneda in the case n=0.
Now a 0-category is, supposedly, an ensemble and a (-1)-category is either 1 or 0 (truth values). So the 0-category of (-1)-categories is
$$-1Cat=\{0,1\}$$.
Now soitlet C be a 0-category ; $C^{\wedge}=[C^o,-1Cat]$ is nothing but the power set of C since a function of C in {0,1} defines a subset of C.
The Yoneda embedding is
$h_C:C \rightarrow C^{\wedge}$ , x being sent to the hom-presheaf $hom_C(-,x)=\{x\}$ (since $hom_C(y,x)=0$ if $y \neq x$), that is, x being sent to the singleton {x}.
So the Yoneda lemma gives:
$hom_{C^{\wedge}}(h_C(x),A) \approx A(x) = 0$ if $x \notin A$
and
$hom_{C^{\wedge}}(h_C(x),A) \approx A(x) = 1$ if $x \in A$
in the 0-category $-1Cat$ up to 0-equivalence which is equality.
Thus
$hom_{C^{\wedge}}(h_C(x),A) = 1$ as long as $x \in A$,
so this indeed makes $C^{\wedge}$ not just a set but a set with equivalence relation, also called setoid in nLab.
I don't actually expect this: Yoneda's lemma forces us to consider setoids instead of plain sets as 0-categories.
So now my question is: Given two 0-categories C and D,
[C, D] (functions of C in D) should be a 0-category, i.e., a setoid.
So what is the equivalence on [C,D]?
In the case of [C,{0,1}], a subset A and a singleton {x} are equivalent ifififf x is an element of A. Two functions A and B (i.e. two sunsets of C) are equivalent ifififf their intersection is not empty; indeed any subset is either equivalent to the empty set or the whole set C itself (if the 0-category C is strictly a set).
This is indeed rather confusing. I hope someone can perhaps clarify.
Or perhaps the Yoneda should not be applied in this rather trivial case at all?
Notes:
I use words like "soit" and "soient"; they're French words for "let be". I think you all know this. I just don't like seperate "let" and "be".
I use "ifif" instead of "iff".
Sorry for these bizzare usage.