Finn Lawler 2-extranatural transformation (Rev #1)

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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 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.)

Now let SS be another functor and α:ST\alpha \colon S \Rightarrow T 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.

Revision on May 25, 2011 at 22:26:04 by Finn Lawler?. See the history of this page for a list of all contributions to it.