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:K op×KLT \colon K^{op} \times K \to L be a (pseudo)functor and \ell be an object of LL. An obKob K-indexed family β a:T(a,a)\beta_a \colon \ell \to T(a,a) of morphisms of LL is extranatural in aa if for each f:abf \colon a \to b in KK there is an invertible 2-cell

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

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

Now let SS be another functor and α:TS\alpha \colon T \Rightarrow S be an ordinary pseudonatural transformation. We want to show that the family α a,aβ a\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)(a,h)(g,b)(g,c)(b,h)(g,h) \cong (a,h)(g,b) \cong (g,c)(b,h) in K op×KK^{op} \times K, and that this gives you the equations you need between α g,h,α g,b,α a,h,\alpha_{g,h}, \alpha_{g,b}, \alpha_{a,h}, etc.

Yoneda

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

Lemma (Yoneda)

Let H:K op×KCatH \colon K^{op} \times K \to Cat be a pseudofunctor. Then there is an equivalence of categories Exnat(1,H)Nat(hom K,H)Exnat(\mathbf{1}, H) \simeq Nat(\hom_K, H), given in one direction by composition with 1:1hom K1 \colon \mathbf{1} \to \hom_K.

Proof

An extranatural β:1H\beta \colon \mathbf{1} \to H yields a pseudonatural β^:hom KH\hat \beta \colon \hom_K \to H with components β^ a,b:fH(a,f)β a\hat\beta_{a,b} \colon f \mapsto H(a,f) \circ \beta_a (one could also choose the isomorphic H(f,b)β bH(f,b) \circ \beta_b). The mediating β g,h\beta_{g,h} are obtained from the compositors of HH, the unitors of KK, and β g\beta_g. These are all suitably natural, so β^\hat \beta is indeed a pseudonatural transformation. For a natural α:hom KH\alpha \colon \hom_K \to H, the mediating isomorphisms (α a,f) 1 a(\alpha_{a,f})_{1_a} provide the components of an invertible modification α1^α\hat{\alpha 1} \to \alpha, which is in fact natural in α\alpha. The (natural) isomorphisms β aH(a,1 a)β a\beta_a \cong H(a,1_a) \circ \beta_a complete the equivalence.

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

Last revised on October 31, 2012 at 03:22:19. See the history of this page for a list of all contributions to it.