Finn Lawler 2-extranatural transformation

I don’t think anyone has written down a definition of ‘weak’ extranatural transformations for bicategories, so let’s do that. There are no surprises, but all we’ll be doing is defining one particular kind and showing that they compose with ordinary pseudonatural transformations, so there may well be unexpected wrinkles elsewhere.

Let $T \colon K^{op} \times K \to L$ be a (pseudo)functor and $\ell$ be an object of $L$. An $ob K$-indexed family $\beta_a \colon \ell \to T(a,a)$ of morphisms of $L$ is extranatural in $a$ if for each $f \colon a \to b$ in $K$ there is an invertible 2-cell

$$\array{ \ell & \overset{\beta_a}{\longrightarrow} & T(a,a) \\ \mathllap{\beta_b} \downarrow & \cong\,\beta_f & \downarrow \mathrlap{T(a,f)} \\ T(b,b) & \underset{T(f,b)}{\longrightarrow} & T(a,b) }$$

satisfying some fairly obvious axioms corresponding to the usual ones for pseudonatural transformations:

1. The assignment $f \mapsto \beta_f$ is natural with respect to 2-cells $m \colon f \Rightarrow g$.
2. $\beta_1$ is an identity, modulo the unitors of $T$.
3. For $a \overset{g}{\to} b \overset{h}{\to} c$, $\beta_{h g}$ is equal to a suitable pasting composite of $\beta_g$ and $\beta_h$. (It’s pretty obvious if you try to draw it: paste $\beta_g$ and $\beta_h$ along $\beta_b$ and complete the square (so to speak) using the bifunctoriality isomorphisms of $T$.)

The notion of a modification between such transformations is the obvious one, so we get a category $Exnat(\ell, T)$.

Now let $S$ be another functor and $\alpha \colon T \Rightarrow S$ be an ordinary pseudonatural transformation. We want to show that the family $\alpha_{a,a} \circ \beta_a$ is again extranatural. The naturality (1) and unit (2) axioms are obvious from the diagrams; the only non-trivial part is axiom (3). To draw the diagrams here would be quite painful, so I’ll just point out that $(g,h) \cong (a,h)(g,b) \cong (g,c)(b,h)$ in $K^{op} \times K$, and that this gives you the equations you need between $\alpha_{g,h}, \alpha_{g,b}, \alpha_{a,h},$ etc.

Yoneda

It’s easy to see that, for a bicategory $K$, the morphisms $1_a \colon \mathbf{1} \to \hom_K(a,a)$ are extranatural in $a$. In fact, this is the universal extranatural transformation out of the terminal category $\mathbf{1}$, in the following sense.

Lemma (Yoneda)

Let $H \colon K^{op} \times K \to Cat$ be a pseudofunctor. Then there is an equivalence of categories $Exnat(\mathbf{1}, H) \simeq Nat(\hom_K, H)$, given in one direction by composition with $1 \colon \mathbf{1} \to \hom_K$.

Proof

An extranatural $\beta \colon \mathbf{1} \to H$ yields a pseudonatural $\hat \beta \colon \hom_K \to H$ with components $\hat\beta_{a,b} \colon f \mapsto H(a,f) \circ \beta_a$ (one could also choose the isomorphic $H(f,b) \circ \beta_b$). The mediating $\beta_{g,h}$ are obtained from the compositors of $H$, the unitors of $K$, and $\beta_g$. These are all suitably natural, so $\hat \beta$ is indeed a pseudonatural transformation. For a natural $\alpha \colon \hom_K \to H$, the mediating isomorphisms $(\alpha_{a,f})_{1_a}$ provide the components of an invertible modification $\hat{\alpha 1} \to \alpha$, which is in fact natural in $\alpha$. The (natural) isomorphisms $\beta_a \cong H(a,1_a) \circ \beta_a$ complete the equivalence.

In fact, because $Nat(\hom_K, H) \simeq \{\hom_K, H\}$ (the $\hom_k$-weighted limit of $H$), what we have shown is that the 2-category $Cat$ admits bicategorical ends.