2
$\begingroup$

Let $\text{Cat}$ be the category of small categories. I am interested in the Yoneda embedding

$$ Y : \text{Cat}^{op} \rightarrow [\text{Cat}, \text{Cat}]$$

$Y$ is a $2$-functor- it can be applied to categories (objects in $X$), functors ($1$-morphisms in $X$), and natural transformations ($2$-morphisms). $Y(C)$ is a functor (an object in $[\text{Cat}, \text{Cat}]$). $Y(C)(D) = [C, D]_{\text{Cat}}$. $Y(F)$ is a natural transformation (a $1$-morphism in $[\text{Cat}, \text{Cat}]$).

My question is about $Y(\eta)$ for a natural transformation $\eta : F \implies G$ ($2$-morphism in $C$). I can't seem to work out what the data of $Y(\eta)$ is - it should be a $2$-morphism in $[\text{Cat}, \text{Cat}]$.

$\endgroup$
2
$\begingroup$

It seems like your question is more about what the 2-morphisms in $\newcommand\Cat{\mathbf{Cat}}[\Cat,\Cat]\newcommand\C{\mathcal{C}}\newcommand\D{\mathcal{D}}$ are, rather than what the data of $Y(\eta)$ is specifically.

Let's do this a little more generally. Let $\C$, $\D$ be (strict) 2-categories. Then $[\C,\D]$ should also be a (strict) 2-category, and we want to understand the 0, 1, and 2-cells.

0-cells:

The objects are strict 2-functors, i.e., functors $F:\C\to \D$ which act on objects, morphisms, and 2-morphisms subject to compatibility criteria. More concretely, once we've decided where $F$ sends objects, then the maps on hom categories $$F_{X,Y} : \C(X,Y)\to \D(X,Y)$$ should all be functors, and moreover, $$ \require{AMScd} \begin{CD} \C(Y,Z)\times \C(X,Y) @>\circ_{\C,X,Y,Z}>>\C(X,Z)\\ @VF_{Y,Z}\times F_{X,Y}VV @VVF_{X,Z}V\\ \D(FY,FZ)\times \D(FX,FY) @>\circ_{\D,FX,FY,FZ}>>\D(FX,FZ)\\ \end{CD} $$ should strictly commute.

1-cells:

The morphisms are (strictly) natural families of 1-cells. I.e., given $F,G:\C\to \D$, a 1-cell from $F$ to $G$ is a family $T_X : FX\to GX$ of 1-cells in $\D$, subject to the requirement that the usual diagram commute strictly for each 1-cell $f:X\to Y$ in $\C$: $$ \begin{CD} FX @>Ff>> FY\\ @VT_X VV @VVT_Y V \\ GX @>Gf>> GY. \\ \end{CD} $$

2-cells:

Let $F,G :\C \to \D$ be 2-functors, $T,S : F\to G$ be 1-cells between them. A 2-cell $\alpha : T \to S$ is a natural family of 2-cells. More concretely, it is the choice for every $X\in C$ of a 2-cell in $\D$, $\alpha_X : T_X\to S_X$ natural in the sense that for every 1-cell of $\C$, $f:X\to Y$, we have that the following 2-cells from $G(f)\circ T_X = T_Y\circ F(f)$ to $G(f)\circ S_X = S_Y\circ F(f)$ are equal. The two cells are the whiskered composites $G(f).\alpha_X$ and $\alpha_Y.F(f)$.

Applying this to $\C=\D=\Cat$

Given a 2-cell $\eta : F\to G$ in $\Cat$, we need to produce for each category $C$ a 2-cell $Y(\eta)_C : Y(F)_C\to Y(G)_C$.

If $X$ and $Y$ are the categories such that $F,G:X\to Y$, then $Y(F)_C: [Y,C]\to [X,C]$ is the functor $-\circ F$, and similarly for $G$. Then $Y(\eta)_C$ should be the whiskered composite $-.\eta$.

In other words, for any functor $K:Y\to C$, for all $x\in X$, by definition, $\eta_X : FX\to GX$, so $K.\eta_X = K(\eta_X) : KFX\to KGX$ is a natural transformation.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.